New Approaches to Cooperative Game Theory:
Core and Value

by
Yusuke Kamishiro

B.Sc., Tokyo Institute of Technology, 2002
M.Sc., Tokyo Institute of Technology, 2004
A.M., Brown University, 2006

A dissertation submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in the Department of Economics at Brown University

PROVIDENCE, RHODE ISLAND
May 2010

c Copyright 2010 by Yusuke Kamishiro


This dissertation by Yusuke Kamishiro is accepted in its present form
by the Department of Economics as satisfying the
dissertation requirement for the degree of Doctor of Philosophy.

Date
Roberto Serrano, Advisor

Recommended to the Graduate Council

Date
Pedro Dal Bó, Reader

Date
Geoffroy de Clippel, Reader

Approved by the Graduate Council

Date
Sheila Bonde, Dean of the Graduate School

iii

Vitae
Date and Place of Birth:
March 7th, 1980 in Tokyo, Japan.

Citizenship:
Japan

Education:
Ph.D., Department of Economics, Brown University, May 2010.
A.M., Department of Economics, Brown University, May 2006.
M.Sc., Department of Value and Decision Science (VALDES), Tokyo Institute of
Technology, March 2004.
B.Sc., Department of Information Science, Tokyo Institute of Technology, March
2002.

Awards:
Merit Dissertation Fellowship, Graduate School, Brown University, Spring 2009.
Abramson Award, given for the third-year paper, Brown University, June 2008.
Graduate Fellowship, Brown University, 2005–2006.

iv

Acknowledgements

I would like to express my deepest gratitude to my principal advisor, Professor
Roberto Serrano, for his guidance throughout the process of writing my dissertation
at Brown. Chapter 2 was jointly written with him and I thank him for providing
me the opportunity to work with him. Many thanks to Professor Pedro Dal Bó and
Professor Geoffroy de Clippel who served as readers on my dissertation committee.
Their perceptive comments and suggestions have proved invaluable. Professor Shigeo Muto at Tokyo Institute of Technology has convinced me to pursue my doctoral
research in the United States. I thank him for drawing my attention to this possibility. Finally, I thank my family members to support me during my study abroad
experience.

v

Abstract of “ New Approaches to Cooperative Game Theory: Core and Value ” by
Yusuke Kamishiro, Ph.D., Brown University, May 2010

In this dissertation, I investigate the core with asymmetric information (Chapters 2
and 3) and the Shapely value with externalities (Chapter 4). In Chapter 2 (jointly
with Professor Roberto Serrano), we investigate to what extent the core convergence results hold for core notions with asymmetric information. We concentrate on
the core with respect to equilibrium blocking, a core notion in which information
is transmitted endogenously within coalitions, as blocking can be understood as an
equilibrium of a communication mechanism used by players in coalitions. We identify conditions under which asymmetric information remains as an externality and
non-market outcomes stay in the core, as well as those for the core to converge to the
set of incentive compatible ex-post Walrasian allocations. In Chapter 3, I investigate
the non-emptiness of the incentive compatible coarse core. I show that the incentive
compatible coarse core is non-empty in quasilinear economies, if agents are informationally small and the strict core in each state is non-empty. This result means that
in quasilinear economies, the non-emptiness result in Vohra (1999) is robust to the
relaxation of non-exclusive information. In Chapter 4, I analyze a situation where
several players entail cooperation in the presence of externalities by using games in
partition function form. I concentrate on the axioms of anonymity, monotonicity,
and weak dummy on a restriction operator, which is defined in Dutta, Ehlers and
Kar (2008) for the potential approach. I connect the Shapley value of the associated
characteristic function constructed from a restriction operator with values of games
in partition function form proposed in previous literature.

Contents

Vitae

iv

Acknowledgments

v

1 Introduction

1

2 Equilibrium Blocking in Large Quasilinear Economies
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 An Interim Exchange Economy . . . . . . . . . . . .
2.2.2 Allocation Rules as Mechanisms . . . . . . . . . . . .
2.2.3 Core w.r.t. Equilibrium Blocking . . . . . . . . . . .
2.2.4 Market Equilibrium . . . . . . . . . . . . . . . . . . .
2.3 Equilibrium Inclusion in the Core . . . . . . . . . . . . . . .
2.4 Independent Replica Economies . . . . . . . . . . . . . . . .
2.4.1 Non-Convergence Results . . . . . . . . . . . . . . .
2.4.2 Convergence of the Core of Unrestricted Mechanisms
2.5 Ex-Post Replica Economies . . . . . . . . . . . . . . . . . .
2.6 Signal-Based Replica Economies . . . . . . . . . . . . . . . .
2.6.1 Notation and definitions . . . . . . . . . . . . . . . .
2.6.2 Convergence Result . . . . . . . . . . . . . . . . . . .
2.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . .

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3 Informational Size and the Incentive Compatible Coarse Core in
Quasilinear Economies
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Informational smallness . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Non-emptiness of the incentive compatible coarse core . . . . . . . .
3.4.1 The incentive compatible coarse core . . . . . . . . . . . . .
3.4.2 Non-emptiness of the incentive compatible coarse core . . . .

vi

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4 Axioms on Restriction Operators and Values of Games in
tion Function Form
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Framework and notation . . . . . . . . . . . . . . . . . . . .
4.3 Restriction operators . . . . . . . . . . . . . . . . . . . . . .
4.4 Anonymity of restriction operators . . . . . . . . . . . . . .
4.5 Monotonicity of restriction operators . . . . . . . . . . . . .
4.6 Weak dummy axiom and values in Macho-Stadler et al . . .

vii

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Chapter

One

Introduction

2
Game theory is the study of decision problems with multiple decision makers, whose
decisions impact one another. It is divided into two branches: non-cooperative game
theory and cooperative game theory. While non-cooperative game theory analyzes
the situation where each action is taken by a single agent autonomously, cooperative
game theory mainly studies interactions among coalitions of agents.

The main question of cooperative game theory is the following: given the sets
of feasible outcomes for each coalition, what outcome will occur? One can take a
positive approach or a normative one to answer this question, and different solution
concepts for predictions or recommendations have been given in the literature. From
each of these points of view, I discuss the most widely-used solution concepts: the
core and the Shapley value.

The core and the Shapley value concepts have been successful each in its own
way. The literature on the core proposes a foundation for a Walrasian equilibrium
of large economies. First, the core of replicated economies converges to the set of
Walrasian allocations (Debreu and Scarf (1963)). Second, there are meaningful sufficient conditions that guarantee the non-emptiness of the core (Bondareva (1963),
Shapley (1967), Scarf (1967)). The Shapley value, which was first proposed in Shapley (1953), has been used not only to recommend reasonable rules of allocation, but
also used as a practical tool to measure voter power and to allocate cost.

However, a drawback of most of the theory and of these concepts in particular is
that they cannot differentiate between various situations in which agents are asymmetrically informed, or in which the feasible outcomes of a coalition depend on an
external coalitional arrangement of agents. For instance, consider a situation where
there are many sellers and many buyers of used cars. In this situation, it would be
natural to assume that only each seller knows the value of his/her own car and that

3
the agents have differential information. Examples of economies with externalities
include international environmental agreements and the use of public resources.

In this dissertation, I investigate how these properties would affect cooperative
behavior among agents. In particular, I focus on the core with asymmetric information (Chapter 2 and Chapter 3) and the Shapley value with externalities (Chapter
4).

In the second chapter of my thesis (jointly with Professor Roberto Serrano),
“Equilibrium Blocking in Large Quasilinear Economies”, we investigate to what extent the core convergence results hold for core notions with asymmetric information.
We concentrate on the core with respect to equilibrium blocking, a core notion in
which information is transmitted endogenously within coalitions, as blocking can
be understood as an equilibrium of a communication mechanism used by players in
coalitions. We consider independent, ex-post and signal-based replicas of the basic
economy. For each, we offer an array of negative and positive convergence results
as a function of the complexity of the mechanisms used by coalitions. We identify
conditions under which asymmetric information remains as an externality and nonmarket outcomes stay in the core, as well as those for the core to converge to the
set of incentive compatible ex-post Walrasian allocations. Further, all the results
are robust to the relaxation of the incentive constraints, and hence suggest a process
through which information may get incorporated into a fully revealing equilibrium
price function.

The third chapter, which is entitled “Informational Size and the Incentive Compatible Coarse Core in Quasilinear Economies”, investigates the non-emptiness of
the incentive compatible coarse core defined in Vohra (1999). In general, the incentive compatible coarse core might be empty, as shown in Vohra (1999) and Forges,

4
Mertens and Vohra (2002). Hence identifying sufficient conditions of non-emptiness
is an important question. Vohra (1999) shows that non-exclusive information is a
sufficient condition for non-emptiness, where non-exclusive information, introduced
by Postlewaite and Schmeidler (1986), means that the true state can be identified
even if the type of any one individual is not known. In this chapter, I investigate the
non-emptiness of the incentive compatible coarse core when agents are informationally small in the sense of McLean and Postlewaite (2002). Informational smallness
is a concept encompassing non-exclusive information. I show that the incentive
compatible coarse core is non-empty in quasilinear economies, if agents are informationally small and the strict core in each state is non-empty. This result means that
in quasilinear economies, the non-emptiness result in Vohra (1999) is robust to the
relaxation of non-exclusive information.

In the last chapter, entitled “Axioms on Restriction Operators and Values of
Games in Partition Function Form”, I analyze a situation where several players entail cooperation in the presence of externalities. In order to take externalities into
consideration, Thrall and Lucas (1963) defined the games in partition function form
as a generalization of those in characteristic function form. Several solution concepts
have been given for games in partition function form, generalizing the Shapley value.
Recently, Dutta, Ehlers and Kar (2008) followed the methodology of the potential
approach, originally proposed in Hart and Mas-Colell (1989) for games in characteristic function form, to offer a value of games in partition function form. In order to
do so, Dutta et al. defined a restriction operator. Loosely speaking, a restriction operator (artificially) specifies a game in partition function when one player is removed.
They limit the discussion to the case where the order to remove a group of players is
irrelevant (Path Independence Axiom). They show that the value derived from the
potential is the Shapley value of the associated characteristic function constructed

5
from a restriction operator. They also consider a case where several axioms are imposed on a restriction operator. In this chapter, I draw connections between values
of games in partition function form proposed in previous literature and the Shapley
value constructed from a restriction operator satisfying the axioms of anonymity,
monotonicity, and weak dummy on a restriction operator. Particularly, I focus on
the values obtained in de Clippel and Serrano (2008) and the ones in Macho-Stadler,
Pérez-Castrillo and Wettstein (2007).

Chapter

Two

Equilibrium Blocking in Large
Quasilinear Economies

7

2.1

Introduction

This chapter studies information transmission in large economies and sheds light on
how information may get incorporated into a fully revealing rational expectations
equilibrium price function. To do so, we use an approach based on the core. We
concentrate on the core based on endogenous information transmission. We shall
call such a core the core with respect to (w.r.t.) equilibrium blocking, in that each
blocking move is identified with an equilibrium of a communication mechanism used
by coalitions. This was the unifying approach proposed in Serrano and Vohra (2007),
and we pursue it further here. Indeed, as shown in that paper, this general notion
of core encompasses other notions previously proposed in the literature: if coalitions
are restricted to use deterministic mechanisms that involve only one coalition, it
boils down to the credible core of Dutta and Vohra (2005); if they can use random
mechanisms that are subjected to a measurability restriction explained below, the
approach yields Myerson’s (2007) virtual utility core; and if such restrictions in random mechanisms are removed, one obtains the randomized mediated core of Serrano
and Vohra (2007).1

The core convergence theorem is an important cornerstone of the relationship
between the predictions of game theory in large economies and competitive equilibrium allocations. Many results have followed the seminal works of Debreu and Scarf
(1963) and Aumann (1964); see Anderson (2008) for a recent survey. This chapter
1

The literature has been asking the question of which is the “right” notion of asymmetric
information core for some time. In this respect, we think that understanding our results as “this
core converges” and “that core does not converge” is not the best reading of our contribution.
As already stated, several core concepts in the literature are all illustrations of the same core,
one in which the tools of mechanism design are used to set up each blocking move as a Bayesian
equilibrium of a communication mechanism among the participants. Also, to avoid the long name
of the concept, we shall shorten it: for example, instead of saying “core w.r.t. equilibrium blocking
of deterministic mechanisms,” we shall say from now on “core of deterministic mechanisms,” and
so on.

8
focuses on interim economies with hidden types, a leading class of problems within
asymmetric information (in it, each agent has private information that is not public
knowledge). As already stated, we shall concentrate on a core concept that models
the information transmission that goes on within each coalition endogenously. We
shall present an array of negative and positive results for the main replica processes
proposed in earlier literature (independent, ex-post and signal-based), leading to a
fairly comprehensive picture of the problem at the interim stage.

The core convergence question in the ex-ante stage (i.e., that in which no agent
has received any private information) has also been studied. For this stage, if no
restrictions are imposed on the set of allocations, the question is simply a reformulation of the standard problem in an Arrow-Debreu economy framework. Indeed, the
most interesting cases happen when either measurability or incentive compatibility
restrictions are imposed.2

There are also results obtained for the ex-post stage (i.e., that in which all private
information has been made common knowledge).3 The results in the current paper
will connect with some of these, due to the information transmission embodied in
the interim core we study, which, if one allows unrestricted communication blocking
mechanisms, places it closer to an ex-post solution concept.

For the interim stage, our concern here, we shall consider independent replicas,
2

Forges, Heifetz and Minelli (2001) study the convergence of the ex-ante incentive compatible
core using the independent replication process. McLean and Postlewaite (2005) establish a positive
convergence result for the ex-ante incentive compatible core in a signal-based replication process.
The reader is referred to Allen and Yannelis (2001) and Forges, Minelli and Vohra (2002) for general
surveys of the area.
3
See, for example, Einy, Moreno and Shitovitz (2000a), who work with atomless economies,
assuming the number of states is finite. Although we shall argue that in ex-post replicas the
continuum case is covered, it is not clear to us how to extend our other replicas analysis to atomless
economies: the state space would also become “atomless,” and then, it would be difficult to define
the join information of a coalition, for example, hindering the use of the arguments in that paper.

9
ex-post replicas and signal-based replicas. Independent replicas were introduced in
Gul and Postlewaite (1992) and also studied in Forges, Heifetz and Minelli (2001).
Independent replicas also replicate the states of the world, and in them, each agent’s
preferences depend only on the information pertinent to his replica. Thus, agents’
information is relevant to a negligible set of agents in large economies. Serrano, Vohra
and Volij (2001) studies ex-post replicas, i.e., replicas in which the set of states of the
world is not replicated with the economy. This implies that, already in the second
replica, the economy is one of non-exclusive information in the sense of Postlewaite
and Schmeidler (1986) and incentive constraints become redundant. The signalbased replicas of McLean and Postlewaite (2005) lie somewhere in between: the set
of underlying unobservable states is not replicated, and agents receive signals that,
if pooled, come close to identifying the true state, although this never happens. The
unobservable state is a common value in agents’ preferences, while types –signals–
are purely informational, affecting only probability assessments. Agents in the three
replica processes become informationally small in different ways and it is important
to understand the different properties of information flows in each case.

Our starting point is to consider economies in which the state is not verifiable
ex-post, and thus, incentive constraints become important. For these economies,
one can use the tools of mechanism design to propose a core notion in which communication within coalitions is endogenous; this notion of “core w.r.t. equilibrium
blocking” takes center stage in our study and we describe it in the sequel. However,
importantly, we shall also study how our main results extend to economies without
incentive constraints.4
4

A word about the proof method followed here is called for. The core we employ relies heavily
on the machinery of mechanism design and the revelation principle. Thus, incentive constraints are
at the center of the analysis. These cause problems to non-emptiness of the core. We are careful
in our proofs to construct arguments that rely on information transmission, with independence of
whether or not it incorporates incentive constraints. The scope of our conclusions is enhanced as a
result.

10
We shall assume that agents in our economies have quasilinear preferences. For
this class, different core existence results have been provided (see Dutta and Vohra
(2005), Myerson (2007)). It is important to stress that non-emptiness is in general
hard to achieve for the interim core once incentive constraints are in place, as pointed
out by Vohra (1999) and Forges, Mertens and Vohra (2002). Aside non-emptiness,
some of our proofs also rely on the quasilinearity assumption (especially the generalized version of the first welfare theorem, i.e., the inclusion of equilibrium in the core).
Myerson (2007) notes that, even in quasilinear settings, random blockings make a
difference to the definition of the core in the presence of asymmetric information
(unlike what is known for complete information games). The current work shows
that, even in quasilinear economies, depending on the replica process and on how
much information transmission one allows in random blocking plans, the answer that
one obtains to the core convergence question is also very different.

If blocking communication mechanisms are deterministic and use only one coalition, our core yields the credible core. The credible core, proposed in Dutta and
Vohra (2005), is the set of incentive compatible allocations immune to credible objections. A coalition has a credible objection if it can identify an informational event
such that the types of agents involved in the event are the only ones that prefer
the alternative proposed to the status quo, given that the other types behave as
prescribed in the objection. This self-selection ensures that a type wishes to participate in the objection if and only if it is consistent with the objection’s event.
Therefore, the information transmitted in the objection via the event is “credible”
in that the types that are not part of it have no incentive to join the objection;
hence the name of the concept. But there is no good reason to restrict attention
to such communication mechanisms, the main point being that blocking is to be
understood as an equilibrium phenomenon from which information transmission is

11
derived endogenously. For instance, if a certain class of measurable random communication mechanisms is permitted, our core yields the concept proposed in Myerson
(2007). Based on the virtual utility construct, Myerson (2007) proposes a core notion that imposes the credibility requirements over random coalition formation and
random allocations for each coalition. Serrano and Vohra (2007), which sets up the
Dutta-Vohra and Myerson objections as communication mechanisms played by the
agents in each coalition and derives their objections’ inequalities from the equilibria
of such communication games, also relax the measurability restriction used in Myerson’s mechanisms. Indeed, the use of unrestricted random blockings is one difference
between Myerson’s (2007) concept and that arrived at in Serrano and Vohra (2007),
which had not been previously emphasized, and such a difference plays an important
role in some of our results. The issue concerns the possibility of information transmission within a random blocking plan across the different coalitions that comprise
it. Correspondingly, we shall talk about the core of an increasingly large class of
communication mechanisms: deterministic mechanisms (the credible core of DuttaVohra), measurable random mechanisms (the virtual utility core of Myerson), and
unrestricted random mechanisms (the randomized mediated core of Serrano-Vohra).
Details will be provided in Section 2.

Our results show both negative and positive convergence conclusions, and both
are important in the understanding of information flows in large economies. First, in
independent replicas, we shall propose a simple robust example to demonstrate that
the core of measurable random mechanisms –and thus, of deterministic mechanisms
a fortiori– does not shrink to any price equilibrium notion. However, if information transmission is allowed to flow within the coalitions that participate in a given
random blocking plan, the strictly incentive compatible allocations in the core of unrestricted mechanisms converge to the set of incentive compatible ex-post Walrasian

12
allocations. We remark again that incentive constraints are not of the essence.

In ex-post replicas, Serrano, Vohra and Volij (2001) have exhibited a robust
failure of the core convergence theorem for cores in which there is only exogenous
information transmission. Essentially, for cores that use a limited amount of exogenously specified information transmission (e.g., the coarse core of Wilson (1978)),
the core is “too large” and does not shrink down to the set of price allocations. On
the other hand, the core that exogenously allows any kind of communication (Wilson’s fine core) is often “too small,” even empty, still when price allocations can be
supported. In contrast, in quasilinear economies, the fine core is non-empty as it
contains the ex-post core. Furthermore, we shall show that the core of deterministic mechanisms converges to the set of ex-post Walrasian allocations (recall that in
these replicas incentive constraints can be dropped in the second replica).5 Thus, in
this case, there is no need to resort to random communication mechanisms to obtain
a full revelation of information result; modelling communication as an equilibrium
phenomenon within a coalition suffices.

In signal-based replicas, we show a probabilistic convergence result for the core
of unrestricted mechanisms. Specifically, for large enough replicas the probability
of a specific informational event is arbitrarily close to 1. The informational event
in question consists of those signal profiles for which at any core allocation the expost utility of almost all agents is arbitrarily close to that obtained at an ex-post
Walrasian allocation in some state. For these replicas, we do not know the answer
to the convergence question for the core of restricted mechanisms (deterministic, for
instance).
5

Indeed, it follows that for large enough ex-post replicas of quasilinear economies, the fine
core, the ex-post core, the core of deterministic mechanisms and the core of random mechanisms
(measurable or not) all coincide.

13
Regarding the quasilinearity assumption, we note that our core shrinking argument in independent and ex-post replicas does not rely on it. That is, even without
it, one can show that if such a replica of an allocation is in the core in all replicas, it
will have the ex-post Walrasian property. In contrast, the probabilistic core shrinking in signal-based replicas does rely on quasilinearity. And more importantly, what
definitely fails in the absence of quasilinearity is the other part of the argument, that
is, the inclusion of equilibrium in the core, since insurance opportunities exist that
render the equilibrium allocation interim inefficient (Laffont (1985)).

To conclude with the introduction, our findings shed light on the rather more
mysterious information transmission that goes on within a price function in a fully
revealing rational expectations equilibrium (Radner (1979), Allen (1981)). Indeed,
such an equilibrium yields ex-post Walrasian allocations. Our convergence result
for the core suggests that the full revelation of information that accompanies the
price function is a consequence of the communication mechanisms –unrestricted random blocking plans– used by coalitions in large economies: an allocation without
the market equilibrium property does not survive these blocking moves, after such
communication devices are employed. That is, a Bayesian equilibrium blocking of
some such communication mechanism, in which information is credibly transmitted,
knocks that allocation out of the core. The allocations that remain in the core of
every replica are therefore immune to any conceivable credible communication, and
those happen to be the ex-post Walrasian allocations. On the other hand, the core
does not shrink enough if either information transmission is prevented to circulate
across different coalitions that participate in a given random blocking plan (e.g.,
when replicas are independent), or if it is not an equilibrium phenomenon (e.g.,
when one works with ex-post replicas).

By taking an approach based on the core, the current paper reaches conclusions

14
about informational flows in large economies that are independent of the specific
trading procedure. Our findings therefore complement other results in the literature,
directly grounded on non-cooperative game-theoretic models, in terms of providing
foundations/criticisms of rational expectations equilibria in studies of specific procedures. These include Holden and Subrahmanyan (1992) and Foster and Viswanathan
(1996) in models á la Kyle (1985); Dubey, Geanakoplos and Shubik (1987) and Forges
and Minelli (1997) in market games á la Shapley and Shubik (1977); Milgrom (1981),
Pesendorfer and Swinkels (1997), Reny and Perry (2006) and Peters and Severinov
(2008) in auctions; and Wolinsky (1990), Blouin and Serrano (2001) and Gottardi
and Serrano (2005) in matching& bargaining and related models. The latter paper
also includes a detailed description of the relevant issues in these and other papers.

2.2

2.2.1

Preliminaries

An Interim Exchange Economy

The basic model of an exchange economy with asymmetric information that we shall
use in this chapter can be formulated as follows. Let Ti denote the (finite) set of agent
i’s types. The interpretation is that ti ∈ Ti denotes the private information possessed

by agent i. With N = {1, . . . , n} as the finite set of agents, let TN = i∈N Ti denote
the set of all information states. We will use the notation t−i to denote (tj )j=i .



Similarly T−i = j=i Tj , TS = j∈S Tj and T−S = j ∈S
/ Tj . We assume that agents
have a common prior probability distribution q defined on TN , and that no type is
redundant, i.e., q(ti) > 0 for all ti ∈ Ti for all i. At the interim stage, nature chooses
tN ∈ TN , and each agent i knows her type, ti. Hence, conditional probabilities will
be important: for each i ∈ N and ti ∈ Ti, the conditional probability of t−i ∈ T−i ,

15
given ti is denoted q(t−i | ti).
We assume that there are |L| = l, a finite number of commodities, and that
commodity L is a numeraire. The consumption set of agent i is Xi = Rl−1
+ × R.
Agent i’s utility function in state tN is denoted ui (·, tN ) : Xi × TN → R, and it is
l
quasilinear in the numeraire: ui (xi , tN ) = vi (x−l
i , tN ) + xi . The endowment of agent i

of type ti is ωi ∈ Xi (assumed to be independent of the state – with this assumption,
all private information concerns agents’ preferences and beliefs.)
We can now define an admissible exchange economy as E = (ui , Xi , ωi , Ti )i∈N , q.
In our analysis, the contracts will be signed at the interim stage. However, when
a coalition gets together and considers to upset a given allocation, some information
may be transmitted within the members of the coalition. We shall allow coalitions
to use random incentive compatible mechanisms, as we explain next.

2.2.2

Allocation Rules as Mechanisms

Deterministic Mechanisms

For coalition S ⊆ N, a feasible deterministic (state contingent) S-allocation, x :
TN → Rls (where s denotes the cardinality of S), consists of a commodity bundle for


each consumer in S in each state such that i∈S xi (tN ) ≤ i∈S ωi for all tN ∈ TN ,
and satisfying that x(tS , t−S ) = x(tS , t−S ) for all tS ∈ TS and for all t−S , t−S ∈ T−S .
(The latter assumption is made to exclude basic externalities across coalitions, i.e.,
the set of feasible allocations to a coalition is independent of the information held by
the complement, although this may affect the utilities of agents in the coalition). We

16
will denote by AS the set of feasible deterministic state contingent allocations of S.
With confusion being avoided by the context, we shall also use AS to denote the set of


feasible deterministic allocations in a given state: AS = {(xi) ∈ i∈S Xi | i∈S xi ≤

i∈S ωi }. Similarly, deterministic state contingent N-allocations are simply referred
to as deterministic allocations, and the set of such deterministic allocations is denoted
by AN .
For much of this chapter, we shall work in environments in which the information
state will not be verifiable, not even at the ex-post stage. Thus, it becomes necessary
to impose the incentive compatibility constraints into the analysis.

We begin again by considering deterministic allocations. Then, if agent i of type
ti pretends to be of type ti (while all other agents are truthful), she gets interim
utility:
Ui (x, ti | ti) =



q(t−i | ti )ui (xi (t−i , ti), (t−i , ti )).

t−i ∈T−i

An allocation x is incentive compatible (IC) if for every i ∈ N, and for every ti ∈ Ti,
Ui (x | ti ) ≥ Ui (x, ti | ti )
for every ti ∈ Ti \ {ti}. We denote the set of IC allocations by A∗N . An allocation x
is strictly IC if all these inequalities are strict.

Information transmission concerns ruling out some states as impossible, through
the identification of smaller events. For an event E ⊆ TN and ti ∈ Ti , let
E−i (ti ) = {t−i ∈ T−i | (ti , t−i ) ∈ E}

17
and
Ei = {ti ∈ Ti | E−i (ti ) = ∅}.
Consider an allocation rule x ∈ AN , agent i of type ti and an event E. Suppose
q(E−i (ti )) > 0. Then the interim utility conditional on E can be expressed as:


Ui (x|ti, E) =

t−i ∈E−i (ti)

q(t−i | ti)
ui (xi (t−i , ti ), (t−i , ti)).
q(E−i (ti )|ti)

The corresponding interim utility (conditional on E) if type ti pretends to be of
type ti, while the others are truth-telling, is:
Ui (x, ti | ti, E) =


t−i ∈E−i(ti )

q(t−i | ti )
ui (xi (t−i , ti), (t−i , ti )).
q(E−i (ti )|ti)

Given E ⊆ TN , an S-allocation x ∈ AS is IC over E if for every i ∈ S and for every
ti , ti ∈ Ei :
Ui (x | ti, E) ≥ Ui (x, ti | ti, E).

Random Mechanisms

At this point it will be convenient to introduce the notion of a status quo. By a status
quo we refer to an IC, state contingent allocation x ∈ A∗N with the interpretation
that in every state tN , the outcome is x(tN ), unless there is an agreement to change
it. This means, in particular, that if there is an attempt to change the status quo
but the attempt fails, the outcome in state tN is x(tN ), i.e., any discussion about
a possible change does not by itself allow any agent to strategically manipulate the
status quo x.

18
To potentially destabilize a status quo, now we expand coalitional interactions
and information transmission to also consider random plans. A random coalitional
plan μ consists of a collection of probability distributions over feasible allocation
rules for various coalitions. In particular, μ(S, y S , tN ), where y S ∈ AS , denotes the
probability with which coalition S ⊆ N is receiving y S ∈ AS when the (reported)
state is tN ∈ TN .
We shall say that such a random plan is measurable with respect to coalitional
information, or simply measurable, if for every S, for every y S and for every tS ,
μ(S, y S , (tS , t−S )) = μ(S, y S , (tS , t−S )) for every t−S , t−S ∈ T−S . Otherwise, we shall
say that the random plan is non-measurable. Non-measurabilities in this sense are
in principle possible, although we shall always require that, if P is the union of all
coalitions in the support of μ, μ(S, y S , (tP , t−P )) = μ(S, y S , (tP , t−P )) for all (S, y S )
in the support of μ, all tP ∈ TP and all t−P , t−P ∈ T−P . The idea is to think of
the random plan as a mediated communication mechanism used by the members
of the coalition; thus, information can potentially flow in any possible way within
the mechanism, but only information available to someone that participates in the
random plan.
Note that, if μ(S, y S , tN ) > 0 implies μ(S, z S , tN ) = 0 for all z S = y S , we
would associate with each coalition S one deterministic allocation y S ∈ AS . Also, if
μ(S  , ·, ·) = 0 for every S  = S, the plan relies only on a single coalition. But in general a random plan may include random allocations within each coalition, as well as
random coalition formation. The plan specifies for each tN ∈ TN , 0 ≤ μ(S, y S , tN ) ≤
 
1 for all coalitions S and S-allocations y S , and S yS μ(S, y S , tN ) ≤ 1.
For type ti of agent i, the interim utility of such a random plan μ, completed
with the implementation of the status quo x as needed when the probabilities of the

19
plan do not add up to 1, is this:

Ui ([μ/x]|ti) =



q(t−i | ti )[

t−i

 

μ(S, y S , tN )ui (yiS (tN ), tN )

S⊇{i} yS

+νi (tN )ui (xi (tN ), tN )],
where νi (tN ) := 1 −





S⊇{i}

yS

μ(S, y S , tN ).6

If everyone else is truthful and type ti pretends to be type ti , his interim expected
utility from the random plan μ completed with the status quo x is:
Ui ([μ/x], ti|ti ) =



q(t−i | ti )[

t−i

 

μ(S, y S , (t−i , ti ))ui(yiS (t−i , ti ), tN )

S⊇{i} yS

+νi (t−i , ti )ui (xi(tN ), tN )],
where νi (ti , t−i ) := 1 −




S⊇{i}

yS

μ(S, y S , (ti , t−i )).

Note how the type misreport affects the implementation probabilities μ(·) of each
coalitional allocation, and therefore also νi (·) of the status quo. In addition, it garbles
the outcomes in each y S , but not those in x. In the next subsection, we shall go over
this, which is related to the timing of events in the blocking plan versus the status
quo; see also Myerson (2007) and Serrano and Vohra (2007).
A random plan μ is IC if for every i ∈ N and for every ti, ti in the support of μ:
Ui ([μ/x] | ti) ≥ Ui ([μ/x], ti | ti ).
6

More precisely, if a random plan μ includes random allocations within each coalition, then μ
might have uncountable support, and hence, the interim utility should be defined incorporating
this. For simplicity, we assume that μ allows only countable support. None of our arguments rely
on this assumption.

20

2.2.3

Core w.r.t. Equilibrium Blocking

The crux of this chapter is the endogenous information transmission that goes on
within a coalition of asymetrically informed agents (unlike, say, the concepts of coarse
and fine core in Wilson’s (1978) seminal work). To study such endogenous information transmission, the concepts presented so far can be used to define appropriate versions of the core. It is important to remark that, when coupled with an IC allocation
x ∈ A∗N , interpreted as the status quo, a random plan trying to upset it may allow for
the status quo to prevail with some positive probability. That is, since a random plan
μ against the presence of status quo x allows for each tN ∈ TN , 0 ≤ μ(S, y S , tN ) ≤ 1
 
for all coalitions S and S-allocations y S , and S yS μ(S, y S , tN ) ≤ 1, one must
assign the rest of probability ν(tN ) to the implementation of the status quo.7
We next define a random plan as a communication mechanism. Following Myerson (2007) and Serrano and Vohra (2007), the discussions of a blocking plan against
a status quo in a context of incomplete information take place after the truthful
reports for the status quo have occurred, but before the implementation of such a
status quo. Therefore, one can specify the following timing of actions within the
blocking plan μ against an IC x:

• Stage 0: types are reported to sustain x as an IC allocation. These type reports
t0N will be used every time the status quo needs to be implemented, and their
reports are made with independence of any potential blocking plan.8 This
stage is therefore not part of the random plan as a communication mechanism.
7

When the appropriate payoff inequalities are imposed versus a given status quo, one describes
a “blocking plan” used in the definition and characterization of the inner core; see Myerson (1991),
Qin (1993) and de Clippel and Minelli (2005).
8
If this is not the case and there is complete forward-looking behavior, i.e., agents envisioning
each possible blocking plan before they report their types to sustain the status quo, one would
probably arrive at a concept far afield from the core.

21
• Stage 1: this and the next stages describe the timing of actions within the
blocking plan μ itself. Types are reported again after players have been informed about μ. Each player is privately informed about the instances in
which μ calls him to act. The type reports of this stage are used only if and
when the blocking plan is implemented.
• Stage 2: players are invited to participate in the blocking plan, i.e., phone
calls are made by the blocking mediator, taking into account the probabilities
μ(S, y S , tN ), which use already the types reported within μ. Note in particular
how a non-measurable plan is perfectly possible, allowing information transmission from coalition to coalition within μ. On the other hand, the plan must
be measurable with respect to the information of the union of coalitions within
μ.
• Stage 3: each agent in the support of μ is asked to either accept or reject the
blocking plan, and he does so in private communications with the blocking
mediator. An allocation proposed to coalition S is implemented if and only if
every agent in S accepts the random plan; otherwise, the status quo x(t0N ) is
implemented.

Two observations are in order that justify this communication mechanism, in
terms of how it would work under complete information. First, in a complete information economy, the arrangements behind the status quo and those behind a
blocking move are kept “separate,” in the sense that agents believe that if the blocking move fails, they can get back to the status quo, which is still available as it was
before the blocking conversations began. This justifies the use of the reports t0N every
time the blocking plan is not approved. Second, a player, being informed only of the
probabilities μ(·) of coalitions that include him, assigns the rest of probability to the

22
status quo prevaling. Again, under complete information, if a player is not called to
participate in a blocking move, he believes the status quo will happen (even though
it may not be even feasible, once the blocking plan succeeds).

The approach just described is quite general. As explained in Serrano and Vohra
(2007), one could conceivably consider arbitrary communication mechanisms, but
there is a version of the revelation principle at work. Any equilibrium of such general
communication mechanism can be made outcome equivalent to a Bayesian equilibrium of the mechanism just described, in which types are reported truthfully.

We shall say that there is an equilibrium rejection of x if there exists a Bayesian
equilibrium of the above communication mechanism in which the plan μ is accepted
with positive probability.9 Without loss of generality, an equilibrium rejection of x
will be termed a random blocking plan against x.
Given an IC allocation x ∈ A∗N , a random plan μ is a random blocking plan
against x whenever for every type ti ∈ Ti :
Ui ([μ/x] | ti) ≥ Ui (x | ti )

with at least one strict inequality, and
Ui ([μ/x] | ti ) ≥ Ui ([μ/x], ti | ti )
for all ti ∈ Ti .
9

We shall think of such an equilibrium to be in pure strategies. A mixed equilibrium adds
another layer of randomization (on top of the uncertainty about the states, the random coalition
formation and the random allocations of physical goods). Mixed equilibria can be handled in the
approach, but then the expressions written below need an adjustment in the probabilities.

23
Equivalently, one can normalize probabilities and simplify these inequalities because of identical terms on both sides, and write:


q(tN )

t−i

 
S⊇{i}

μ(S, y S , tN )[ui(yiS (tN ), tN ) − ui (xi (tN ), tN )] ≥ 0

(∗)

yS

for all ti ∈ Ti , with at least one strict inequality, and

t−i

≥

q(tN )

t−i

 

μ(S, y S , tN )[ui(yiS (tN ), tN ) − ui (xi (tN ), tN )]

S⊇{i} yS

q(tN )

 

μ(S, y S , (t−i , ti))[ui(yiS (t−i , ti ), tN ) − ui (xi (tN ), tN )]

S⊇{i} yS

for all i ∈ N and ti , ti ∈ Ti.

(∗∗)

Thus, information transmission is endogenous in a random blocking plan. The
equilibrium rejection of x implies that those types that are part of the objection
best-respond to the others by accepting it, whereas those types that are not do not
have an incentive to turn their zero probability of participation into a positive one
by pretending to be one of the invited types. This is the content of equations (*)
and (**), which respectively express the equilibrium acceptance condition and the
IC constraints over the event described by the equilibrium probabilities.
An IC allocation x ∈ A∗N is in the core w.r.t. equilibrium blocking of some class
of random communication mechanisms whenever there does not exist an equilibrium
rejection of x, i.e., a random blocking plan against x, for a communication mechanism
in that class.

Particular cases of this definition are the following. We shall progressively enlarge
the class of blocking communication mechanisms considered:

24
An IC allocation x ∈ A∗N is in the credible core (Dutta and Vohra (2005)) if and
only if it is in the core of single-coalition deterministic mechanisms. That is, whenever
there does not exist a random blocking plan μ against x such that μ(S, y S , tN ) > 0

only for one coalition S and one deterministic rule y S ∈ AS , and for tN ∈ i∈S Ei ×
T−S , where Ei is the set of types of agent i in the support of μ. Note how, in
particular, the definition of IC over an event E for an S-allocation for a fixed coalition
S, as defined above, applies to these blocking plans.

Thus, in a credible objection to an allocation, a fixed coalition S identifies an
information event over which the objection takes place. All the types consistent with
that event prefer the objection to the status quo allocation after Bayesian updating
of their beliefs given the event. Furthermore, no type within the relevant event wants
to misrepresent its information. Finally, the types not consistent with the event do
not wish to participate in the objection by pretending they are one of the types in it:
to see this, in the second condition of a random blocking plan, suppose ti is one of
these excluded types and ti one of the types in the support of μ. The latter is what is
called the credibility restriction in this objection. Hence the name credible objection.
As can be easily seen, all these conditions are particular cases of equations (*) and
(**) above.
An IC allocation x ∈ A∗N is in the virtual utility core (Myerson (2007)) if and
only if it is in the core of measurable random mechanisms. That is, whenever there
does not exist a random blocking plan μ against x that is measurable.

Thus, in these objections, an agent is invited to the blocking plan, but there
is a process of random coalition formation. Each agent in the support of μ gets a
phone call inviting him to participate in the blocking plan, and those “phone calls”
are made according to μ. On the basis of μ, contingent on the “phone call,” each

25
agent updates his interim beliefs using Bayes’ rule. Again, given this, each type is
part of the equilibrium rejection of x if and only if he wishes to go along with it
instead of remaining at the status quo, and the appropriate IC constraints are also
imposed given μ – equations (*) and (**). The measurability of the plan implies
that a given coalition S cannot be called with different probabilities in two states
(tS , t−S ) and (tS , t−S ) that the coalition cannot discern (observe in particular that
the Dutta-Vohra credible objections are measurable).
An IC allocation x ∈ A∗N is in the randomized mediated core (Serrano and Vohra
(2007)) if and only if it is in the core of unrestricted random mechanisms. That is,
whenever there does not exist a random blocking plan μ against x.

For these objections, the same story applies, except that the non-measurability
of μ allows for information transmission across coalitions within the blocking plan.

Under the assumptions made so far, and if one allows average feasibility in the
numeraire (while exact feasibility is required for the other commodities), Myerson
(2007) establishes the non-emptiness of the virtual utility core. Furthermore, his existence argument does not use at all the measurability of blocking plans. Therefore,
it follows that, under exactly the same assumptions, the core of unrestricted mechanisms is also non-empty. An alternative approach to non-emptiness is provided
in Dutta and Vohra (2005). In quasilinear economies, assuming that there exists
an IC ex-post core allocation, they show that the credible core is non-empty. It is
easy to complete their argument to show that the core of unrestricted mechanisms
is also non-empty (see the proof of our Proposition 1). That is, adding randomness
to blocking plans does not jeopardize the non-emptiness of the core in this case.

The following is an obvious, but useful observation:

26
Observation 1: The core of unrestricted mechanisms is a subset of the same concept
based on measurable mechanisms, itself a subset of the concept if one only uses
single-coalition deterministic mechanisms.

10

When incentive constraints can be dropped, one can still define the corresponding
core w.r.t. equilibrium blocking. Then, a core allocation x ∈ AN , and the equilibrium rejection of x amounts only to condition (*). As above, as a function of the
class of communication mechanisms considered, one can create similar nested sets
of allocations.

11

In this case, it can be shown (Dutta and Vohra (2005), Serrano

and Vohra (2007)) that all fine objections can be made credible, and hence, the
core of single-coalition deterministic mechanisms coincides with the fine core. Without incentive constraints, non-emptiness results are easier to obtain: in particular,
it follows from Dutta and Vohra (2005) that in quasilinear economies this core is
non-empty.

We shall use in some of our proofs the following additional observation:

Observation 2: An IC allocation that is in the fine core (which is in the IC fine
core) is in the core of single-coalition deterministic mechanisms.
10

It is easy to relate these notions to Wilson’s (see Dutta and Vohra (2005)). If in a credible
objection one drops the credibility restriction (the self-selection of types into the objection), one
constructs an (IC) fine objection. The IC fine core is the set of IC allocations x ∈ A∗N immune
to IC fine objections. On the other hand, an IC objection based on a common knowledge event
for a coalition is an IC coarse objection, which is always credible. The IC coarse core is the set of
incentive compatible x ∈ A∗N against which there is no IC coarse objection. It follows that the IC
fine core is a subset of the credible core, itself a subset of the IC coarse core.
11
If one drops the IC constraints from both the status quo allocations and the objections in the
IC coarse core and IC fine core, one arrives at the coarse and fine core definitions (Wilson (1978)).

27

2.2.4

Market Equilibrium

The competitive market equilibrium concept that we shall employ in this chapter
is the ex-post Walrasian equilibrium: an allocation rule x is an ex-post Walrasian
equilibrium allocation rule if x(tN ) constitutes a Walrasian equilibrium allocation in
each tN ∈ TN . We shall assume that there exists an ex-post Walrasian equilibrium
allocation rule that is IC. Under this assumption, our negative convergence result
extends also to any price-taking equilibrium concept that satisfies Property P, as
first suggested in Serrano, Vohra and Volij (2001):

A price-taking equilibrium concept is said to obey Property P if, whenever it is
non-empty, in an economy that includes some fully informed agents, each of them
receives in equilibrium a bundle that maximizes his ex-post utility over his ex-post
budget constraint.

2.3

Equilibrium Inclusion in the Core

In this section we begin by showing that, in our domain of quasilinear economies, the
set of ex-post Walrasian allocation rules that satisfy IC is in the core of unrestricted
mechanisms. The section ends by extending the result when incentive constraints
are dropped.

Proposition 1. Consider a quasilinear exchange economy E in our admissible class.
Let x∗ ∈ A∗N be an ex-post Walrasian equilibrium allocation rule satisfying IC. Then,
x∗ is in the core of unrestricted mechanisms.

28
Proof: By hypothesis, x∗ is IC. So we need to show that there does not exist any
random blocking plan that improves upon x∗.

Since an ex-post Walrasian allocation is an element of the ex-post core, we can
say that for all S ⊆ N, all tN ∈ TN and y S ∈ AS ,


ui (yiS (tN ), tN ) ≤

i∈S



ui (x∗i (tN ), tN ).

i∈S

This corresponds to an ex-post objection. Now, consider a random plan μ (without
measurability). This inequality implies that for all S ⊆ N, all y S ∈ AS and all
tN ∈ TN ,


μ(S, y S , tN )ui (yiS (tN ), tN ) ≤

i∈S



μ(S, y S , tN )ui (x∗i (tN ), tN ).

i∈S

Thus,
 



q(tN )

S⊆N yS ∈AS tN ∈TN

 

≤



μ(S, y S , tN )ui (yiS (tN ), tN )

i∈S



q(tN )

S⊆N yS ∈AS tN ∈TN



μ(S, y S , tN )ui (x∗i (tN ), tN ).

i∈S

If a blocking plan μ existed, we would obtain (recall equation (*)):

i∈N tN

=

q(tN )

 

μ(S, y S , tN )[ui(yiS (tN ), tN ) − ui (x∗i (tN ), tN )]

S⊇{i} yS

 
S⊆N yS

tN

q(tN )



μ(S, y S , tN )[ui(yiS (tN ), tN ) − ui (x∗i (tN ), tN )]

i∈S

> 0,

which contradicts the inequality in the previous paragraph.

29
Thus, in quasilinear economies, ex-post Walrasian allocation rules that satisfy IC
are elements of the smallest core w.r.t. equilibrium blocking. It is immediate to drop
the incentive constraints from this result, and with the same proof (note it does not
rely on the IC condition (**)), demonstrate the following:

Proposition 2. Consider a quasilinear exchange economy E in our admissible class,
but where incentive constraints are dropped. Let x∗ ∈ AN be an ex-post Walrasian
equilibrium allocation rule. Then, x∗ is in the core of unrestricted mechanisms.

Remark: Ignoring incentive constraints, Dutta and Vohra (2005) show that in
quasilinear economies the ex-post core is included in the fine core, which is therefore
non-empty. The proof of Proposition 2 strengthens this result, as without incentive
constraints, the fine core coincides with the core of single-coalition deterministic
mechanisms.

Remark: Propositions 1 and 2 apply to all quasilinear economies, including their
different kinds of replicas, as we are about to define in the next sections.

Remark: The conclusions of Propositions 1 and 2 do not extend to non-quasilinear
economies. In fact, ex-post Walrasian rules may be even interim inefficient.

2.4

Independent Replica Economies

We turn to define the replicas of the basic economies and allocations. We shall
first use the independent replicas introduced in Gul and Postlewaite (1992), also
used in Forges, Heifetz and Minelli (2001). Each agent’s utility depends only on the

30
information contained in his replica. Because replicas are independent, the set of
states in the replicated economy is the product of the sets of states for each replica.
Formally, given an economy E = (ui , Xi , ωi , Ti)i∈N , q, and an allocation x ∈
AN , independent replicas of E and x are defined as follows. For every positive
integer m, let Im = {1, 2, . . . , m}. The m-th replica of E is the economy E m =
(u(i,j) , X(i,j), ω(i,j) , T(i,j))(i,j)∈N ×Im , qm , where for all (i, j) ∈ N × Im , X(i,j) = Xi ,


T(i,j) = Ti over k=i T(k,j), ω(i,j) = ωi , u(i,j) = ui : Xi × i∈N T(i,j) → R, and
qm (t(1,1), . . . , t(n,1), . . . , t(1,m), . . . , t(n,m)) =



q(t(1,j), . . . , t(n,j)).

j∈Im

The m-th replica of x is denoted xm where xm
(i,j) = xi for all (i, j) ∈ N × Im .
We shall say that a set of allocation rules satisfies the equal treatment property
if, in any replicated economy, it contains only replicas of allocations x of the basic
economy.

Note that the set of information states changes with replication. Since types
are independent, each agent potentially retains a piece of private information even
after replication, and incentive constraints do not disappear. On the other hand,
since preferences depend only on information held by agents in the same copy of the
economy, an agent’s private information affects a vanishingly small fraction of agents
if one replicates the economy sufficiently many times.

31

2.4.1

Non-Convergence Results

In the current subsection we show that, even if one restricts attention to equal
treatment allocations, the core of measurable mechanisms does not converge to any
price-taking equilibrium allocations satisfying Property P, no matter how many times
the economy is replicated. Later in this subsection we shall argue that this core,
in any replicated economy, may in addition contain allocations that violate equal
treatment. Thus, if the communication mechanisms used by coalitions are restricted
to be measurable random blocking plans, the core remains “too large.”

Equal Treatment Allocations

Our first non-convergence result follows:
Proposition 3. Consider independent replicas. There exists an allocation x̃ ∈ A∗N in
an admissible economy E satisfying that for every m, x̃m is in the core of measurable
mechanisms in the replicated economy E m and that cannot be supported by any pricetaking equilibrium notion that obeys Property P.

Proof: Consider the following economy E. There are two consumers and two commodities. Suppose T1 = {s, t} while agent 2 is uninformed (and therefore has only
one type). The information state can then be described by s or t. Suppose s and t
are equally probable. Let ω1 = ω2 = (1.5, 1). The utility functions are as follows:
u1 (x1, x2, s) = ln x1 + x2 ,

u2 (x1, x2, s) = 2 ln x1 + x2,

u1 (x1, x2, t) = 2 ln x1 + x2, u2 (x1, x2, t) = ln x1 + x2.
(Recall that throughout we use superscripts to index commodities and subscripts to

32
index consumers.) Thus, the two individuals are ex-ante identical, but the realized
type of individual 1 determines ex-post which of the two has a higher utility from
consumption of the first good.12

We consider first the allocation x̂ defined by:
xˆ1 (s) = (1, 1.5),

xˆ2(s) = (2, 0.5);

xˆ1 (t) = (2, 0.5),

xˆ2(t) = (1, 1.5).

Then each agent’s ex-post utility is:
u1(xˆ1 (s), s) = 1.5,

u2 (xˆ2(s), s) = 2 ln 2 + 0.5,

u1(xˆ1 (t), t) = 2 ln 2 + 0.5,

u2 (xˆ2(t), t) = 1.5.

Note how x̂ must be the only allocation prescribed by an equilibrium concept that
obeys Property P in this economy. Note also how many equilibrium concepts will
obey this property here, including the ex-post Walrasian equilibrium, the constrained
market equilibrium found in Wilson (1978), or the rational expectations equilibrium
(which here yields the empty set). Finally, observe that x̂ is IC.

To construct the allocation we are interested in, we modify x̂ by requiring an
additional transfer of 0.05 units of good 2 from individual 2 to individual 1 in each
state. We denote the resulting allocation as x̃:
x˜1 (s) = (1, 1.55),

x˜2(s) = (2, 0.45);

x˜1 (t) = (2, 0.55),

x˜2(t) = (1, 1.45).

12
This example was first proposed in Kreps (1977) to illustrate how the set of rational expectations
equilibria might be empty. It was also used in Forges, Heifetz and Minelli (2001) to show that the
equal treatment property does not hold for the ex-ante incentive compatible core in the two-fold
replicated economy.

33
This allocation yields utility levels
u1 (x˜1(s), s) = 1.55,

u2 (x˜2(s), s) = 2 ln 2 + 0.45,

u1 (x˜1(t), t) = 2 ln 2 + 0.55,

u2 (x˜2(t), t) = 1.45.

We consider now the independent replication process described above. Recall
that we denote individual i in the j-th replica as (i, j). We present our argument in
two claims.

Claim 3.1: For all m, both the m-th replicated allocations x̂m and x̃m are in the
core of single-coalition deterministic mechanisms in E m .

Proof of Claim 3.1: We will show that the m-th replication of x̃ is in the core
of single-coalition deterministic mechanisms in the replicated economy E m . (The
argument for x̂m is similar and we omit it.)
Since we can check that x̃m is IC (each informed agent in each replica does not
wish to pretend that he is of the other informed type), it suffices to show that x̃m is
in the fine core of E m ; see Observation 2 above.
It is easy to show that x̃ is maximizing the sum of ex-post utilities for each
state and it is ex-post individually rational. Hence, x̃ is in the fine core of the
original economy E. Thus, for j = 1, 2, . . . , m, if the allocation is blocked by a
coalition S that includes both (1, j) and (2, j), then the allocation is also blocked
by S \ {(1, j), (2, j)}. This implies that it is sufficient to check whether coalition
S = {(1, 1), . . . , (1, k), (2, k + 1), . . . , (2, m)} blocks the allocation x̃m for each m
and k(0 ≤ k ≤ m). For this possible blocking coalition, we can restrict attention

34
to information events of the form E =

k

j=1 {r(1,j) }

×

m

j=k+1

T(1,j), where r(1,j) =

s or t(1 ≤ j ≤ k).
Without loss of generality, we may assume that
⎧
⎪
⎨ s (1 ≤ j ≤ j  )
r(1,j) =
⎪
⎩ t (j  + 1 ≤ j ≤ k)
for some j (0 ≤ j  ≤ k). Here, j  = 0(j  = k) means that r(1,j) = t(s) for every
(1, j)(1 ≤ j ≤ k), respectively.
Note that any feasible allocation for coalition S has to be constant on E, i.e., each
uninformed agent must receive the same bundle in both relevant states. It follows
from the quasilinearity that the optimal allocation of good 1 can be obtained as the
solution to


max

j


ln x1(1,j)

+

k


2 ln x1(1,j)

j=j  +1

j=1

s.t.

k


x1(1,j)

j=1

m

1
(2 ln x1(2,j) + ln x1(2,j))
+
2
j=k+1

+

m


x1(2,j) ≤ 1.5m.

j=k+1

By the first-order condition, the solution of this problem is

x1(1,i)

=

⎧
⎪
⎨λ

(1 ≤ j ≤ j )

⎪
⎩ 2λ (j  + 1 ≤ j ≤ k)

, x1(2,j) = 1.5λ(k + 1 ≤ j ≤ m),

where
λ=

j

1.5m
1.5m
.
=

+ 2(k − j ) + 1.5(m − k)
1.5m + 0.5k − j 

35
Then the sum of interim utilities is
j  · ln λ + (k − j  ) · 2 ln(2λ) + (m − k) · 1.5 ln(1.5λ) + m.

(1)

On the other hand, the sum of interim utilities of the original allocation x̃m for
the coalition is
1
1.55j  + (2 ln 2 + 0.55)(k − j ) + {(2 ln 2 + 0.45) + 1.45} · (m − k).
2

(2)

Now it suffices to show that (2) − (1) is non-negative for all m, k(0 ≤ k ≤ m)
and j (0 ≤ j  ≤ k). We denote the difference [(2) − (1)] as g(j , k, m).
For fixed k and m(0 ≤ k ≤ m), the first-order condition with respect to the
variable j  is
ln(1.5m) − ln(1.5m + 0.5k − j  ).
Thus the function g is minimized at j  = 0.5k. Then
g(0.5k, k, m) = 1.05k + (ln 2 + 0.95)(m − k) − 1.5(m − k) ln 1.5 − m
= (ln 2 − 1.5 ln 1.5 − 0.05)(m − k) + 0.05k ≥ 0
(since ln 2 − 1.5 ln 1.5 − 0.05 > 0 and m ≥ k ≥ 0). Therefore, g(j  , k, m) ≥ 0 for
every j  , k and m(0 ≤ k ≤ m, 0 ≤ j  ≤ k). Thus, the proof of Claim 3.1 is complete.

Claim 3.2: For all m, both the m-th replicated allocations x̂m and x̃m are in the
core of measurable mechanisms in E m .

36
Proof of Claim 3.2: We will show that the m-th replication of x̃ is in the core
of measurable mechanisms in the replicated economy E m . (The argument for x̂m is
similar and we omit it.)

In the proof of Claim 3.1, we have argued that the m-th replication of x̃ is in the
fine core. Therefore, we have shown that for all S ⊆ N × Im, all tS ∈ TS and all
y S ∈ AS ,
 

qm (t−S )ui(yiS (tN ×Im ), tN ) ≤

i∈S t−S ∈T−S

 

qm (t−S )ui (x̃i (tN ×Im ), tN ).

i∈S t−S ∈T−S

(Recall that qm is the probability distribution over the states in the replicated economy E m ). The earlier inequality corresponds to a fine objection in which all information within the coalition is transmitted to its members. Now, consider a random plan
μ that satisfies T S -measurability. This inequality implies that for all S ⊆ N × Im,
all y S ∈ AS and all tS ∈ TS ,




μ(S, y S , tN ×Im )

i∈S



≤

qm (t−S )ui (yiS (tN ×Im ), tN )

t−S ∈T−S
S

μ(S, y , tN ×Im )

i∈S



qm (t−S )ui (x̃i (tN ×Im ), tN ).

t−S ∈T−S

Thus,


qm (tS )

tS ∈TS

≤





μ(S, y S , tN ×Im )

i∈S

tS ∈TS

qm (tS )


i∈S



qm (t−S )ui (yiS (tN ×Im ), tN )

t−S ∈T−S
S

μ(S, y , tN ×Im )


t−S ∈T−S

qm (t−S )ui (x̃i (tN ), tN ),

37
which implies that






qm (tN ×Im )

S⊆N ×Im yS ∈AS tN ×Im ∈TN ×Im



≤







μ(S, y S , tN ×Im )ui(yiS (tN ×Im ), tN )

i∈S

qm(tN ×Im )

S⊆N ×Im yS ∈AS tN ×Im ∈TN ×Im



μ(S, y S , tN ×Im )ui (x̃i(tN ), tN ).

i∈S

If a blocking plan μ existed, we would obtain (recall equation (*)):




qm (tN ×Im )

i∈N ×Im tN ×Im



=

 

μ(S, y S , tN ×Im )[ui(yiS (tN ×Im ), tN )

S⊇{i} yS

 

S⊆N ×Im yS tN ×Im

qm (tN ×Im )



−ui (x̃i(tN ), tN )]
μ(S, y S , tN ×Im )[ui(yiS (tN ×Im ), tN )

i∈S

−ui (x̃i(tN ), tN )]
> 0,

which contradicts the inequality in the previous paragraph.

Claims 3.1 and 3.2 complete the proof of Proposition 3.

Remark: Note the parallel steps followed by the proofs of Claim 3.2 and of Proposition 1. There is an important difference, though. While the allocation x̃m of the proof
of Claim 3.2 is in the fine core of every replica, if x∗ is IC ex-post Walrasian, x∗m is
in the ex-post core of every replica. This difference matters: in the former case, the
non-existence of a fine objection leads, through the use of measurable blocking plans,
to the non-existence of a measurable equilibrium blocking. In the case of the latter,
not having ex-post objections in any state leads, through the use of (measurable or
not) blocking plans, to the non-existence of any kind of equilibrium blocking.

38
If incentive constraints are not relevant (as in many general equilibrium analyses,
and also in Wilson’s (1978) original paper), one can state a similar result to Proposition 3, using an identical proof. The observation is simple enough: the proofs of
Claims 3.1 and 3.2 do not rely on IC conditions (recall Proposition 2 for the inclusion
of the ex-post Walrasian allocations in the core of unrestricted mechanisms):

Proposition 4. Consider independent replicas and suppose incentive constraints are
dropped. There exists an allocation x̃ ∈ AN in an admissible economy E satisfying
that for every m, x̃m is in the core of measurable mechanisms in the replicated economy E m and that cannot be supported by any price-taking equilibrium notion that
obeys Property P.

Proof: Consider the same economy as in the proof of Proposition 3. Clearly, even if
non-IC random blocking plans are possible, no such blocking exists for x̃. Note how
the proof of Proposition 3 makes no use of conditions (**).

Non-Equal Treatment Allocations

In this subsection we demonstrate that the core of measurable mechanisms in replicated economies also contains allocations that violate equal treatment across replicas,
even if the economy is replicated an arbitrary number of times.

Indeed, in the same economy used in the proof of Proposition 3, consider the
allocation (x̂, . . . , x̂, x̃). We shall show that this allocation belongs to the core of
measurable mechanisms in the (m + 1)-fold replicated economy E m+1 for all m ≥ 1.
That is, in the first m replicas, the bundles in x̂ are allocated, whereas those in x̃ are

39
assigned in the last replica, thereby violating the equal treatment property of this
core.

We follow analogous steps to those in Claim 3.1, and first show that this allocation
is in the core of single-coalition deterministic mechanisms in the (m + 1)-replicated
economy. Since we can easily check that the proposed allocation is IC, it suffices to
show that it is in the fine core.13

Now we make the following observations:

Observation 3: For each state, both x̂ and x̃ are maximizing the sum of ex-post
utilities. Both allocations are in the fine core of the original economy.

Observation 4: For k = 1, 2, . . . , m, m + 1, if the allocation is blocked by a coalition
S that includes both (1, k) and (2, k), then the allocation is also blocked by S \
{(1, k), (2, k)}.
Observation 5: For k = 1, 2, . . . , m, if the allocation is blocked by a coalition that
includes (2, k) but does not include (1, k), then the allocation is also blocked by the
coalition that includes (1, k) but does not include (2, k) (by symmetry of allocation
x̂).

From these observations, we can say that it is sufficient to check whether coalition
S is one of two possibilities: indeed, either S = {(1, 1), . . . , (1, m),
(2, m + 1)} or S = {(1, 1), . . . , (1, m), (1, m + 1)} block the proposed allocation for
each m.
13
Again, with the same example, we are showing that this core violates equal treatment in
economies in which incentive constraints are dropped.

40
For the first possible blocking coalition, we can confine our attention to events

of the form E = i∈S {ri } × T−S , where ri = s or t(i ∈ S \ {(2, m + 1)}).
Without loss of generality, we may assume that

r(1,i)

⎧
⎪
⎨ s (1 ≤ i ≤ j)
=
⎪
⎩ t (j + 1 ≤ i ≤ m)

for some j(0 ≤ j ≤ m). Here, j = 0(j = m) means that the types of all individuals
in S \ {(2, m + 1)} are s (t), respectively.
Note that any feasible allocation has to be constant on E and utility of every
individual other than (2, m + 1) is determined on E. It follows from quasilinearity
that the optimal allocation of good 1 can be obtained as the solution to

max

j


ln x1(1,i)

+

i=1

m


2 ln x1(1,i) +

i=j+1
m


s.t.

1
1
ln x1(2,m+1) + · 2 ln x1(2,m+1)
2
2

x1(1,i) + x1(2,m+1) ≤ 1.5(m + 1).

i=1

By the first-order condition, the solution of this problem is

x1(1,i) =

⎧
⎪
⎨λ

(1 ≤ i ≤ j)

⎪
⎩ 2λ (j + 1 ≤ i ≤ m)

and x1(2,m+1) = 1.5λ, where
λ=

1.5(m + 1)
.
j + 2(m − j) + 1.5

41
Then the sum of interim utilities is:
j · ln λ + (m − j) · 2 ln(2λ) + 1.5 ln(1.5λ) + (m + 1).

(3)

On the other hand, the sum of interim utilities of the original allocation is:
1
1.5j + (2 ln 2 + 0.5)(m − j) + {(2 ln 2 + 0.45) + 1.45}.
2

(4)

Now it suffices to show that (4) − (3) is non-negative for all m and j(0 ≤ j ≤ m).
We write the difference:
1
1.5j + (2 ln 2 + 0.5)(m − j) + {(2 ln 2 + 0.45) + 1.45}
2
− [j · ln λ + (m − j) · 2 ln(2λ) + 1.5 ln(1.5λ) + (m + 1)]

Take partial derivative with respect to j:
1 − 2 ln 2 − ln λ + 2 ln(2λ) − (2m − j + 1.5)[∂ ln λ/∂j]
= 1 + ln λ − (2m − j + 1.5)[∂ ln λ/∂j]
= ln λ = ln[

1.5m + 1.5
],
2m − j + 1.5

which vanishes at j = 0.5m, is negative for j < 0.5m and positive thereafter. There-

42
fore, it suffices to check the value of the function at j = 0.5m for a fixed m:
1
0.75m + (2 ln 2 + 0.5)(0.5m) + {(2 ln 2 + 0.45) + 1.45}
2
−[(0.5m) · 2 ln 2 + 1.5 ln(1.5) + (m + 1)]
1
= ln 2m + {(2 ln 2 + 0.45) + 1.45} − [(0.5m) · 2 ln 2 + 1.5 ln(1.5) + 1]
2
= ln 2m + ln 2 + 0.95 − [ln 2m + 1.5 ln(1.5) + 1]
= ln 2 − 0.05 − [1.5 ln(1.5)] = 0.0340... > 0.

Since this value is independent of m, we have shown that for all j satisfying that
0 ≤ j ≤ m for all m ≥ 1, the difference (4) − (3) is positive, which contradicts that
the coalition blocks the original allocation.

If the blocking coalition contains agent (1, m + 1) instead of (2, m + 1), one can
have two cases:

First, it is agent (1, m + 1) in state s. We write the utility difference, similar to
(4) − (3), for this case:
1.5j + (2 ln 2 + 0.5)(m − j) + 1.55
− [(j + 1) · ln λ + (m − j) · 2 ln(2λ) + (m + 1)]
and now λ = (1.5m + 1.5)/(2m − j + 1).
Again, we partially differentiate the aggregate utility difference with respect to

43
j:

1 − 2 ln 2 + ln λ + 2 ln 2 − (2m − j + 1)[∂ ln λ/∂j]
= ln λ = ln[

1.5m + 1.5
],
2m − j + 1

which vanishes at j = 0.5m−0.5, is negative if j < 0.5m−0.5 and positive thereafter.
We thus check the value of the aggregate utility difference at j = 0.5m − 0.5:
1.5(0.5m − 0.5) + (2 ln 2 + 0.5)(0.5m + 0.5) + 1.55
−[(0.5m + 0.5)2 ln 2 + (m + 1)]
= 0.75m − 0.75 + ln 2(m + 1) + 0.25m + 0.25 + 1.55 − [(m + 1)(1 + ln 2)]
= 0.05 > 0,

and also independent of m, so we are also done with this case. Finally, we have case
2: agent (1, m + 1) acts in state t. The utility difference similar to (2)−(1) is now:
1.5j + (2 ln 2 + 0.5)(m − j) + 2 ln 2 + 0.55
− [j · ln λ + (m + 1 − j) · 2 ln(2λ) + (m + 1)],
where now λ = (1.5m + 1.5)/(2m − j + 2).
Differentiating partially with respect to j, one gets:
1 − 2 ln 2 − [ln λ − 2 ln(2λ) + (2m − j + 2)(∂ ln λ/∂j)]
= −2 ln 2 − [ln λ − 2 ln(2λ)]
= ln λ = ln[

1.5m + 1.5
],
2m − j + 2

44
which vanishes at j = 0.5m + 0.5, is negative at j < 0.5m + 0.5, and positive
thereafter. Therefore, it suffices to check the value of the aggregate utility difference
at j = 0.5m + 0.5:
1.5(0.5m + 0.5) + (2 ln 2 + 0.5)(0.5m − 0.5)
+2 ln 2 + 0.55 − [(m + 1)(ln 2 + 1)]
= 0.05 > 0,

also independent of m.

Thus, now the proof is complete. No such blocking coalition exists and the
proposed allocation is in the IC fine core, and hence, also in the core of singlecoalition deterministic mechanisms (Observation 2).

The arguments to show that the allocation is in the core of measurable mechanisms are similar to the inequalities derived in Claim 3.2, and we therefore omit
them.

2.4.2

Convergence of the Core of Unrestricted Mechanisms

In this subsection we show that the allocations of the core of unrestricted mechanisms converge to the set of ex-post Walrasian allocations as the economy is replicated enough times. Thus, in a large enough replica, unrestricted random blocking
plans are capable of knocking out any allocation without the ex-post market equilibrium property. Recall that the inclusion of equilibrium in this core was shown in
Proposition 1.

45
Specifically, we begin by showing that the strictly IC allocations in the core of
unrestricted mechanisms satisfy equal treatment, and that they converge, as the
economy is replicated enough times, to the set of IC ex-post Walrasian allocations.
As we have been doing in previous ones, incentive constraints are dropped later in
this subsection.

Proposition 5. Consider independent replicas, and suppose that, in every ex-post
state, the core convergence theorem holds.14

• (i) The strictly IC allocations in the core of unrestricted mechanisms satisfy
the equal treatment property.
• (ii) If the economy is sufficiently replicated, x is a strictly IC allocation rule in
the economy E and its replica xm is in the core of unrestricted mechanisms in
E m for every m, then x must be an ex-post Walrasian allocation rule.

Proof: (i) First, we show that for every m ≥ 2, the equal treatment property holds
for the core of unrestricted mechanisms under strict IC. Suppose not. That is, for
m
some i, some j, j , and some tN , xm
(i,j) (t̄m ) = x(i,j  ) (t̄m), where t̄m := (tN , tN , . . . , tN ).

We shall show that there exists a random blocking plan that improves upon
(xm
(i,j) )(i,j)∈N ×Im .
By the ex-post equal treatment property, there exists a coalition S ⊂ N × Im and
a feasible plan y S , such that ui (yiS (t̄m ), tN ) > ui (xi(t̄m ), tN ) for all i ∈ S. Consider
another allocation rule x arbitrarily close to x, also strictly IC, satisfying that xi = xi
14

Strictly speaking, we cannot use the theorem in Debreu and Scarf (1963): their consumption
set is the non-negative orthant of the L-dimensional Euclidean space, while we allow negative
consumption in the numeraire. This difference does not affect the validity of the ex-post core
convergence result, though.

46
for all i ∈
/ S.
Let us consider the following blocking plan μ for N × Im : μ(S, y S (t̄m ), t̄m ) = ε >
0, μ(N × Im, x (t̄m ), t̄m) = 1 − ε > 0 and μ assigns zero for any other state.
Since x satisfies IC, we can see that both conditions (*) and (**) of Section 2.2
are satisfied for any i ∈
/ S. For i ∈ S, conditions (*) are clearly satisfied. For the
conditions (**) –IC–, one has two cases:
Case 1: Agent i’s true type is ti and he reports ti . Then, the LHS of (**) is,
because of the state probability, approximately proportional to ε[ui(yiS (t̄m ), tN ) −
ui (xm
i (t̄m ), tN )] > 0, while the RHS is 0.
Case 2: Agent i’s true type is ti and he reports ti . Let t := (ti , tN \{i}). Then,
the LHS of (**) is 0, whereas the RHS is proportional to ε[ui(yiS (t̄m ), t)


−ui (xi(t ), t)] + (1 − ε)[ui(xi(t̄m ), t) − ui (xm
i (t ), t )].



Since ui (xi(t̄m ), t) − ui (xm
i (t ), t ) < 0 by strict IC, by taking ε small enough, we

can obtain that the RHS is negative. Hence the equal treatment property holds.
(ii) Next we show that if x is a strictly IC allocation rule in the economy E and
its replica xm is in the core of unrestricted mechanisms in E m for every m, x must
be an ex-post Walrasian allocation rule. Suppose not. That is, let x be a strictly IC
allocation rule whose replica is in this core of every replicated economy, but suppose
that x(tN ) is not a Walrasian allocation for some tN .
By the ex-post core convergence theorem, there exists m such that in the m-th
replication of the ex-post economy in tN , there exists a coalition S and a feasible
plan y S ∈ AS , S ⊆ N × Im , such that ui (yiS (t̄m), tN ) > ui (xi (tN ), tN ) for all i ∈ S,

47
where t̄m = (tN , tN , . . . , tN ). Consider another allocation rule x arbitrarily close to
/ S. By considering a mechanism
x, also strictly IC, satisfying that xi = xi for all i ∈
similar to the one used above, we can say, using the same arguments, that there
exists a randomized mediated blocking for xm .

Remark: The proof of Proposition 5 does not use the assumption that the set of IC
ex-post Walrasian allocations is non-empty. In its absence, the proposition implies
the non-existence of allocations in the core of unrestricted mechanisms in sufficiently
large replicas.

To illustrate the shrinking of the core of unrestricted mechanisms, we shall consider again the economy in the proof of Claim 3.1. Indeed, we show now that x̃,
which was shown to be in the core when only measurable blocking mechanisms were
considered, is blocked by a non-measurable random blocking plan.

Example 1. Consider the economy in the proof of Proposition 3. First, note that
the second replication of x̃ is not in the ex-post core. For instance, it is blocked by
coalition S = {(2, 1), (1, 2), (2, 2)} in state (s, s) (this notation means that for each
of the two replicas the type of individual 1 is s). We denote an ex-post blocking
allocation bundle for S as y S (s, s).


For coalition S  = {(1, 1), (2, 1)}, we define an allocation bundle y S (s, s) as fol



S
S
lows: y(1,1)
(s, s) = (1, 1.56) and y(2,1)
(s, s) = (2, 0.44), which can be obtained from

x̃(s) by an additional transfer of 0.01 units of commodity 2 from individual (2,1) to
individual (1,1).
Then, we consider the following blocking plan μ: μ(S, y S , (s, s)) = 1 − ε (ε being

48


a very small positive number), μ(S  , y S , (s, s)) = ε, and μ(T, y T , r) = 0 for all
coalitions T and states r = (s, s).

Then, the blocking plan μ makes all individuals involved better off. The ex-post
objection “almost works” for coalition S, except that they do not have the information
relevant to agent (2,1). This is the reason to bring agent (1,1) into the blocking
plan. Indeed, choosing ε sufficiently small, individual (2,1) is better off. Because
agent (1,1) only participates in the blocking plan with probability ε, in the event he
receives the phone call from the blocking mediator, he accepts the plan because he
also improves. Effectively, in this event he “sells” his information to agent (2,1) for
0.01 additional units of the numeraire. Recall that each agent believes that the status
quo is still available in the event he is not part of the blocking plan, and under these
beliefs, types are reported truthfully to the blocking plan.
Finally, it is easy to choose y S (s, s) appropriately to ensure that the blocking plan
satisfies IC. Note in particular how this can be done so that the other types of agents
(1,1) and (1,2) do not have an incentive to misrepresent their types or accept the
blocking plan. They optimally choose to truthfully stay out of it (equations (*) and
(**) applied to them).

If one drops the incentive constraints, one obtains the following parallel result:
Proposition 6. Consider independent replicas, and suppose that incentive constraints are dropped, and that, in every ex-post state, the core convergence theorem
holds.

• (i) The allocations in the core of unrestricted mechanisms satisfy the equal
treatment property.

49
• (ii) If the economy is sufficiently replicated, x is an allocation rule in the economy E and its replica xm is in the core of unrestricted mechanisms in E m for
every m, then x must be an ex-post Walrasian allocation rule.

Proof: The arguments in the proof of Proposition 5 do not rely on x being IC and
x being strictly IC. All that matters is the equal treatment property of the ex-post
core and the ex-post core convergence theorem. After one realizes this, the proof is
simpler since one does not need to check conditions (**).

2.5

Ex-Post Replica Economies

Different from the independent replica process studied so far, Serrano, Vohra and
Volij (2001) propose ex-post replicas. Ex-post replicas result in information becoming
non-exclusive already in the second replica, which in turn makes incentive constraints
redundant. This is a different sense of defining a negligible size of agents from the
informational point of view.

Formally, the only difference of this replica process with respect to the one defined
n
in the previous section is the following. Let T = i=1 Ti be the set of ex-post states
in the basic economy. This will remain the set of ex-post states even after replication.
For each (i, j) ∈ N × Im, t(i,j) = t(i,j ) = ti , and qm (t−i |ti ) = q(t−i |ti).
For this replica process, Serrano, Vohra and Volij (2001) exhibited a robust failure
of the core convergence theorem for core notions that did not model the information
transmission by means of equilibrium blocking. However, the main result of this
section is the following:

50
Proposition 7. Consider ex-post replicas, and suppose that incentive constraints
are either present or dropped, and that, in every ex-post state, the core convergence
theorem holds.

• (i) The allocations in the core of single-coalition deterministic mechanisms
satisfy the equal treatment property.
• (ii) If the economy is sufficiently replicated, x is an allocation rule in the economy E and its replica xm is in the core of single-coalition deterministic mechanisms in E m for every m, then x must be an ex-post Walrasian allocation
rule.

Proof: First, as observed above, already in the second replica, the economy is one of
non-exclusive information, and therefore, even if incentive constraints were relevant
in the basic economy, they can be dropped from the second replica onwards.
Now consider an allocation x whose replica xm is in the core of single-coalition
deterministic mechanisms in every ex-post replicated economy. When incentive constraints are dropped, this core is identical to the fine core (Dutta and Vohra (2005),
Serrano and Vohra (2007)). Moreover, in quasilinear economies, the ex-post core
is a subset of the fine core (Dutta and Vohra (2005)), and adapting an argument
for the continuum (Einy, Moreno and Shitowitz (2000b)) to large enough replica
economies , the fine core is a subset of the ex-post core. Therefore, by combining
these statements, we obtain that allocations in the core of single-coalition deterministic mechanisms satisfy the equal treatment property and converge to the set of
ex-post Walrasian allocations.

51
Remark: Putting together Propositions 7 and 2, we find out that, when incentive
constraints are dropped, in large enough quasilinear economies, the core of singlecoalition deterministic mechanisms is a subset of the ex-post core, itself a subset of
the core of unrestricted mechanisms. Therefore, all three converge to the same set
in large economies. This is not true if incentive constraints remain relevant.

Remark: Making use of the same steps as in the proof of Proposition 7 and of
Aumann’s (1964) core equivalence theorem for the economy with an atomless continuum of agents, one can easily establish the same equivalence theorem for the core
of single-coalition deterministic mechanisms. Recall that in this section it is very
important that the set of states of the world is finite.

Therefore, a convergence/equivalence theorem has been established for this case.
Interestingly, when the set of states of nature is not replicated with the economy,
there is no reason to resort to the full force of random blocking plans to obtain it.
It suffices to apply the equilibrium logic to blocking plans, even for those involving
only a deterministic allocation rule coming from a single coalition.

2.6

Signal-Based Replica Economies

We consider now a different replica process, which we call signal-based replicas. This
was first considered in McLean and Postlewaite (2002). Agents’ utility functions will
depend on an underlying but unobserved state of nature θ, and each agent will receive
a private signal that is correlated with the state of nature. A replication of this initial
economy consists of a set of agents whose utility functions and endowments are the
same as those in the underlying initial economy, but whose private signals across

52
cohorts are independent conditional on θ. As the number of replicas increases, each
agent becomes small in the economy in terms of endowment, and each agent is also
“informationally small”: the conditional distribution on the state of nature does
not vary much in that agent’s signal if other agents’ signals are known. Note that
no agent’s information is redundant in this replication process: regardless of the
number of replications, each agent still has information that cannot be inferred from
the aggregate information of other agents.

2.6.1

Notation and definitions

Since the model employed in these replicas is slightly different from the rest of the
paper, we begin with a few preliminaries. Let Θ = {θ1, . . . , θm } denote the (finite)
state space and let T1 , . . . , Tn be finite sets, where Ti represents the set of possible
signals that agent i ∈ N might receive. In this model, nature chooses an element
θ ∈ Θ. The state of nature is unobservable but each agent i receives a “signal” that
is correlated with nature’s choice of θ. We denote the probability distribution on
Θ × T as P .
The consumption set of each agent Xi is Rl−1
+ × R and for each θ ∈ Θ, ωi ∈ Xi
denotes the (state independent) initial endowment of agent i in state θ ∈ Θ. The
preferences of agent i are given by a utility function ui : Rl−1
+ × R × Θ → R where
l
ui (xi , θ) = vi−l (x−l
i , θ) + xi is the utility function of agent i in state θ. We note

that in this model agents’ utility functions do not depend on TN . The collection
e = ({ui , Xi , ωi , Ti}i∈N , Θ, P ) will be called a private information economy. It will be
assumed that the data defining the private information economy is common knowledge among the agents. A private information economy allocation x = (x1 , . . . , xn )
for the private information economy is a collection of functions xi : TN → AN .

53
For each state θk ∈ Θ, the collection {ui (·, θk ), ωi }i∈N defines an associated (complete information) economy. Let (zi∗(θk ))i∈N denote a Walrasian equilibrium allocation in state θk .
Let us define a sequence of signal-based replica economies. Recall that Im =
{1, 2, . . . , m}. Given the collection {ωi , ui}i∈N and m, let {ω(i,j) , u(i,j)}(i,j)∈N ×Im denote the m replication of {ωi , ui }i∈N satisfying:
(1) ω(i,j) = ωi for all i ∈ N and all j ∈ Im,
(2) u(i,j)(x, θ) = ui (x, θ) for all x ∈ Rl−1
+ × R, i ∈ N and j ∈ Im .
For any positive integer m, let T m = T × · · · × T denote the m-fold Cartesian
product and let tm denote a generic element of T m . Let Pm be a probability distribution on Θ × T m . Then em = ({u(i,j), X(i,j) , ω(i,j), T(i,j)}(i,j)∈N ×Im , Θ, Pm ) is a private
information economy with n × m agents.
A sequence of replica economies {({u(i,j), X(i,j), ω(i,j) , T(i,j)}(i,j)∈N ×Im , Θ, Pm )}∞
m=1
is a conditionally independent sequence if
(a) For each m, each j ∈ Im , and each (θ, t1, . . . , tn ) ∈ Θ × T ,
m
m
P (θ, tm
(1,j) , t(2,j) , . . . , t(n,j) ) = P (θ, t1 , t2 , . . . , tn );

(b) for each m and each θ, the probability distributions over
m
m
m
m
m
m
, T(2,1)
, . . . , T(n,1)
), . . . , (T(1,m)
, T(2,m)
, . . . , T(n,m)
)
(T(1,1)

are independent given θ;

54
(c) for every θ, θ with θ = θ, there exists a tN ∈ TN such that P (tN | θ) =
P (tN | θ ).
Thus a conditionally independent sequence is a sequence of private information
economies with n × m agents containing m “copies” of each agent i ∈ N. It follows
that, for large enough m and for almost every signal profile in the replica economy,
one could assign a probability to a state that is arbitrarily close to 1.

2.6.2

Convergence Result

In this subsection we provide a convergence result for this model. In each state θk ,
we make the following (strongly) ex-post core convergence assumption:15

Assumption 1: For every ε > 0, there exists an integer m̂ such that for all
m > m̂, if a feasible allocation in the m-fold economy, xm ∈ AN ×Im satisfies
∗
S
ui (xm
∈ AS such
i , θk ) − ui (zi (θk ), θk ) > ε, then there exist a coalition S and y


ε
that j∈S uj (yjS , θk ) > j∈S uj (xm
j , θk ) + 2 .

This assumption implies the following:

Assumption 1’: In each state θk , for all ε > 0 there exists an integer m̂ such that
for all m > m̂, if a feasible allocation in the m-fold economy, xm ∈ AN ×Im satisfies
∗
#{j ∈ Im | ui(xm
(i,j) , θk ) − ui (zi (θk ), θk ) > ε} > εm
15

Strongly ex-post refers to the timing after the state of the world would be observed (something
that never happens in this model), while ex-post would refer to the signal profile being commonly
known by all agents, something that could happen if they chose to pool their information.

55
for some i ∈ N, then there exist S ⊂ N ×Im and y S ∈ AS such that

ε
m
j∈S uj (xj , θk ) + 2 × (εm).


j∈S

uj (yjS , θk ) >

We show that when m is large, and except for very unlikely signal profiles, allocations in the strict IC core of unrestricted mechanisms for the m-fold private
information economy will give most agents utility that is close to that of some Walrasian equilibrium allocation in some state θk . The following proposition formalizes
this statement. It says that for large enough replicas the joint probability of a specific informational event is arbitrarily close to 1. The informational event in question
consists of those signal profiles for which at any core allocation the ex-post utility of
almost all agents is arbitrarily close to that obtained at a strongly ex-post Walrasian
allocation in some state:

Proposition 8. Consider conditionally independent signal-based replicas of a quasilinear economy in our admissible class. For every ε > 0, there exists an m̂ > 0 such
that for all m > m̂, every strictly IC allocation in the core of unrestricted mechanisms
xm of m-replicated economies satisfies
Pr tm ∈ T m | ∃k s.t.
m
∗
#{j ∈ Im | |ui(xm
(i,j) (t ), θk ) − ui (zi (θk ), θk )| < ε} ≥ (1 − ε)m

for all i ∈ N

≥ 1 − ε,

where (zi∗(θk ))i∈N is a Walrasian equilibrium allocation in θk .

Proof: Let m1 be the number satisfying Assumption 1’. That is, in each state θk ,

56
for all m > m1, if a feasible allocation in m-fold economy, xm ∈ AN ×Im satisfies
∗
#{j ∈ Im | ui(xm
(i,j) , θk ) − ui (zi (θk ), θk ) > ε} > εm

for some i ∈ N, then there exist S ⊂ N ×Im and y S ∈ AS such that

ε
m
j∈S uj (xj , θk ) + 2 × (εm).
2

ε
Let η := min{ε, ε2 +2M
}, where M := maxθ maxi vi−l (
n

note that η satisfies (1 − η) ·

ε2
2

n

j=1


j∈S

ωj−l , θ)+

uj (yjS , θk ) >

n

j=1

ωjl . We

> η · M · n.

Applying the argument in Gul and Postlewaite (1992, p.1290), it follows that for
every η > 0 there exists an integer m2 satisfying the following: for all m > m2,
Pr{tm ∈ T m | for some k, Pr(θ = θk | tm ) ≥ 1 − η} ≥ 1 − η.

Let m̂ := max{m1, m2 }.
Suppose that m > m̂. Let Bkm := {tm ∈ T m | Pr(θ = θk | tm ) ≥ 1 − η}. We
show that for every tm ∈ Bkm , every strictly IC allocation in the core of unrestricted
m
∗
mechanisms xm satisfies #{j ∈ Im | |ui (xm
(i,j) (t ), θk )−ui (zi (θk ), θk )| < ε} ≥ (1−ε)m

for all i ∈ N.
m
∗
Suppose not. Then #{j ∈ Im | ui (xm
(i,j) (t ), θk ) − ui (zi (θk ), θk ) > ε} > εm for

some i ∈ N. By considering a random mechanism similar to the one in the proof of
Proposition 5, it would be enough to show that there exists an ex-post objection in
tm .
By using assumption 1’, there exists a coalition S ⊂ N × Im and y S ∈ AS such

57
that


j∈S



uj (yjS , θk ) >


j∈S

m
uj (xm
j (t ), θk ) +

uj (yjS , θ)P (θ | tm ) > (1 − η)

j∈S θ∈Θ



ε
2

× (εm). Then

uj (yjS , θk )

j∈S

> (1 − η)



m
uj (xm
j (t ), θk ) + (1 − η) ·

j∈S

≥ (1 − η)
>





ε2
m
2

m
uj (xm
j (t ), θk ) + η · Mn · m

j∈S
m
m
uj (xm
j (t ), θ)P (θ | t ) (because |S| ≤ nm).

j∈S θ∈Θ

Since agents’ utilities are quasilinear, this inequality implies that y S is an ex-post
objection to xm in tm .

Remark: Unlike Proposition 5, this proof relies on quasilinearity.

Remark: McLean and Postlewaite (2005) establish a positive convergence result
for the ex-ante IC core under strongly conditionally independent sequences, which
means that the distributions over each Tj are also independent given θ in an original
economy (compare with part (b) in the definition above).

Once again, noting that the core shrinking argument in the proof of Proposition
8 does not rely on the use of incentive constraints, one realizes that these can be
dropped. The relevant framework now would be one in which signals are publicly
verifiable, but not the state of nature, which remains unobservable. Then, one can
state our final result, whose proof can be omitted:

58
Proposition 9. Consider conditionally independent signal-based replicas of a quasilinear economy in our admissible class in which incentive constraints are dropped.
For every ε > 0, there exists an m̂ > 0 such that for all m > m̂, every allocation in
the core of unrestricted mechanisms xm of m-replicated economies satisfies
Pr tm ∈ T m | ∃k s.t.
m
∗
#{j ∈ Im | |ui(xm
(i,j) (t ), θk ) − ui (zi (θk ), θk )| < ε} ≥ (1 − ε)m

for all i ∈ N

≥ 1 − ε,

where (zi∗(θk ))i∈N is a Walrasian equilibrium allocation in θk .

2.7

Concluding Remarks

This chapter has studied equilibrium blocking in large quasilinear economies. Results
vary as a function of the class of communication mechanisms employed by coalitions
and also of the replica process used. The main factor that accounts for the difference
in results is the amount of information transmission that one permits each coalition
to use. If random coalition formation is possible and information can be used,
within a blocking plan, from coalition to coalition in the plan, a positive convergence
result was obtained to the set of incentive compatible ex-post Walrasian allocations,
whenever this is non-empty. But such a positive convergence result is prevented in
independent replicas if blocking communication mechanisms are restricted to being
either measurable or deterministic. On the other hand, in ex-post replicas there is
no need to go beyond deterministic mechanisms to obtain a convergence result. In
signal-based replicas, where our results are weaker, a probabilistic convergence result
was obtained for the core of unrestricted mechanisms.

59
We close by reiterating a very important observation. We note that, while we
have been using the incentive constraints to motivate our analysis –in part, because
the approach is rooted in mechanism design–, all our results extend to the case
in which such constraints are not imposed. It then follows that the relevance of
the results in the current paper is enhanced substantially, as existence of ex-post
Walrasian allocations is readily obtained.

16

16

See also de Clippel (2007) for a different core convergence result when incentive constraints are
ignored.

Chapter

Three

Informational Size and the
Incentive Compatible Coarse Core
in Quasilinear Economies

61

3.1

Introduction

We study the core of an interim exchange economy with quasilinear utilities in which
agents are asymmetrically informed (see Forges, Minelli and Vohra (2002) for a
survey of the area). If coalitions are formed at the interim stage, then we need to
specify the information that agents in a coalition are allowed to use in constructing
an objection. For example, we may assume that a coalition can focus its potential
objection on an event which is commonly known to all members of the coalition.
This is the basis of the coarse core developed in Wilson (1978). He proved that,
under standard assumptions, the coarse core is non-empty.

We assume that private information does not become publicly verifiable. Then, it
is appropriate to introduce feasible mechanisms which satisfy incentive compatibility
constraints in addition to the resource constraints. Vohra (1999) defines the incentive
compatible coarse core, which takes incentive constraints into account. The incentive
compatible coarse core is based on the assumption that no information is transmitted
within a coalition, and it is larger than other incentive compatible interim core concepts in which possible blockings are facilitated by information transmission among
agents (e.g., the credible core in Dutta and Vohra (2005)). Although the incentive
compatible coarse core would be the best hope for obtaining non-emptiness, it still
might be empty, as shown in Vohra (1999). Furthermore, Forges, Mertens and Vohra
(2002) established that specifically the incentive compatible coarse core of a quasilinear economy might also still be empty. Hence, identifying sufficient conditions
of non-emptiness is an important question. Vohra (1999) shows that non-exclusive
information is a sufficient condition for non-emptiness, where non-exclusive information, introduced by Postlewaite and Schmeidler (1986), means that the true state
can be identified even if the type of any one individual is not known.

62
In this chapter, we investigate the non-emptiness of the incentive compatible
coarse core when agents are informationally small in the sense of McLean and Postlewaite (2002). Informational smallness is a concept encompassing non-exclusive information in the sense that non-exclusive information corresponds to the case where
the informational size of each agent is zero.

We consider a situation in which the agents’ utility functions depend on an underlying but unobserved state of nature and in which each agent will receive a private
signal that is correlated with the state of nature. This corresponds to a “common
value” model in which signals do not directly affect the underlying payoff functions
but affect expected utilities. Roughly speaking, agents are informationally small if
the conditional distribution on the state of nature does not vary much in that agent’s
signal if other agents’ signals are known. McLean and Postlewaite (2002) show that
it is possible to approximate almost any allocation of the underlying complete information economy by an incentive compatible mechanism whenever all agents are
informationally small. They also use this characterization in McLean and Postlewaite
(2003) to prove that the ex ante incentive compatible core is non-empty if agents are
informationally small and the strict core of an underlying Arrow-Debreu economy
is non-empty. We show that the incentive compatible coarse core is non-empty in
quasilinear economies, similarly if agents are informationally small and the strict
core in each state is non-empty. This result means that in quasilinear economies the
non-emptiness result by Vohra (1999) is robust to the relaxation of non-exclusive
information.

63

3.2

Basic notation

Let N = {1, . . . , n} be the set of agents. Let Θ = {θ1, . . . , θm } denote the (finite)
state space and let T1 , . . . , Tn be finite sets, where Ti represents the set of possible

signals that agent i might receive. For each S ⊆ N, let TS := i∈S Ti and T−S :=

i∈N \S Ti . Elements of TS and T−S will be written tS and t−S , respectively. For
notational simplicity, we will simply write T for TN and T−i for TN \{i}.
In our model, nature chooses an element θ ∈ Θ. The state of nature is unobservable but each agent i receives a “signal” that is correlated with nature’s choice of θ.
We denote the probability distribution on Θ × T as P . In spite of a slight abuse of
notation, we also denote as P the marginal probability distribution of P over Θ or
T.
The consumption set of each agent is Rl−1
+ × R (the l-th commodity is numeraire)
and for each θ ∈ Θ, wi ∈ Rl++ denotes the (state independent) initial endowment
of agent i in state θ. The preferences of agent i are given by a utility function
−l −l
l
ui : Rl−1
+ × R × Θ → R where ui (xi , θ) = vi (xi , θ) + xi is the utility function of

agent i in state θ. The following assumptions with regard to vi−l are imposed in this
chapter:
(i) vi−l (·, θ) is continuous and strictly concave,
(ii) vi−l (0, θ) = 0,
−l
(iii) vi−l (·, θ) is (strongly) monotonic: if x−l , y −l ∈ Rl−1
≥ y −l and x−l = y −l ,
+ , x

then vi−l (x−l , θ) > vi−l(y −l , θ).

64
The collection ({ui , wi }i∈N , θ̃, t̃, P ) will be called a private information economy
(PIE for short). It will be assumed that the data defining the PIE is common
knowledge.

Let
ΦS = {(ξi )i∈S | ξi ∈ Rl−1
+ × R,


i∈S

ξi ≤



wi }.

i∈S

A private information economy allocation z = (z1, . . . , zn ) for the PIE is a collection of functions zi : T → ΦN .
For each state θk ∈ Θ, the collection {ui (·, θk ), wi}i∈N defines an associated (complete information) economy. A commodity vector for agent i in the economy is denoted as xi (θk ). We refer to this economy as the auxiliary economy in state θk .
Under the assumptions we made, in each θk ∈ Θ, a Walrasian equilibrium allocation
(x∗i (θk ))i∈N ∈ ΦN exists. We assume that in each state there exists a Walrasian
equilibrium such that the amount of numeraire for every agent is positive. (The last
assumption is just to simplify the argument. Our non-emptiness result goes through
without it.)

3.3

Informational smallness

In formulating the conditions under which the incentive compatible coarse core is
non-empty, we need the notions of informational size, aggregate uncertainty and
distributional variability introduced in McLean and Postlewaite (2002). Formally,
these concepts are described as follows. (The reader can be referred to McLean and
Postlewaite (2002) for elaboration of the concepts.)

65
1. Informational size
Let PΘ be the conditional probability distribution on Θ given t ∈ T . Any vector
of agents’ types t = (t−i , ti) ∈ T induces a conditional distribution on Θ and, if
agent i unilaterally changes his announced type from ti to ti , this conditional
distribution will change in general. We consider agent i to be informationally
small if, for each ti, there is a small probability that he can induce a large
change in the conditional distribution on Θ by changing his announced type
from ti to some other ti . McLean and Postlewaite (2002) formalized this in the
following definition. Let
Iεi(ti , ti ) := {t−i ∈ T−i | PΘ (· | t−i , ti ) − PΘ (· | t−i , ti ) > ε},
where  ·  is the 1-norm.
The informational size of agent i is defined as
˜ ∈ Iεi (ti , ti) | t˜i = ti } ≤ ε}.
min{ε ≥ 0 | Pr{t−i
νiP = max max

ti ∈Ti ti∈Ti

Loosely speaking, we will say that agent i is informationally small with respect
to P if his informational size νiP is small.
2. Negligible aggregate uncertainty
Secondly, aggregate uncertainty is quantified. Let
μPi := max min{ε | Pr{PΘ (· | t̃) − Iθ  > ε for all θ ∈ Θ | t˜i = ti} ≤ ε},
ti ∈Ti

where Iθ denotes the measure that puts probability 1 on state θ.
If μPi is small for each i, then we will say that P exhibits negligible aggregate
uncertainty. In this case, each agent knows that, conditional on his own signal,

66
the aggregate information of all agents will, with high probability, provide a
good prediction of the true state.
3. Distributional variability
To define the measure of variability, we first define a metric d on simplex ΔΘ
as follows: for each α, β ∈ ΔΘ , let

d(α, β) :=

α
β
−
α2 β2

,
2

where ·2 denotes the 2-norm. Hence, d(α, β) measures the Euclidean distance
between the Euclidean normalizations of α and β. We define
ΛPi = min  min d(PΘ (· | ti), PΘ (· | ti ))2.
ti ∈Ti ti∈Ti \{ti}

This is the measure of the variability of the conditional distribution PΘ (· | ti )
as a function of ti .

3.4

Non-emptiness of the incentive compatible coarse
core

3.4.1

The incentive compatible coarse core

Let ({ui , wi}i∈N , θ̃, t̃, P ) be a PIE. For each S ⊆ N, S-feasible allocations for the
PIE are of the form z : TS → ΦS , where we recall that ΦS = {(ξi )i∈S | ξi ∈


Rl−1
+ × R,
i∈S ξi ≤
i∈S wi }.

67
An S-feasible allocation (zi )i∈S is incentive compatible if




ui (zi (tS\{i}, ti), θ)P (θ, tS\{i} | ti)

θ∈Θ tS\{i} ∈TS\{i}

≥





ui (zi (tS\{i}, ti), θ)P (θ, tS\{i} | ti)

θ∈Θ tS\{i} ∈TS\{i}

for each ti , ti ∈ Ti and i ∈ S.
We focus on the incentive compatible coarse core defined in Vohra (1999).
Definition 1. The incentive compatible coarse core consists of all incentive compatible allocations z to which there exists no incentive compatible coarse objection.
Coalition S has an incentive compatible coarse objection to z if there exist an incentive compatible allocation y S and a common knowledge event E for S such that




P (θ, tS\{i} | ti )ui(yiS (tS\{i}, ti ), θ)

θ∈Θ tS\{i} ∈TS\{i}

>

 

P (θ, t−i | ti)ui (zi (t−i , ti), θ)

θ∈Θ t−i ∈T−i

for all ti ∈ Ei and all i ∈ S, where Ei = {ti ∈ Ti | (t−i , ti ) ∈ E for some t−i ∈ T−i }.

We note that in this definition, an objection from the grand coalition corresponds
to domination in the sense of the interim incentive efficiency defined in Holmström
and Myerson (1983). That is, an incentive compatible allocation z is interim incentive
efficient if the grand coalition N has no incentive compatible coarse objection to z.
Coalition S is said to have an ex post objection to z if there exist yS ∈ ΦS and
t ∈ T such that

θ∈Θ

P (θ | t)ui (yiS , θ) >


θ∈Θ

P (θ | t)ui (zi(t), θ)

68
for all i ∈ S. (Our use of the term “ex post” refers to events that occur after the
realization of the signal profile t, but before the realization of the state θ (allocations
can depend on agents’ types but not on θ, which is assumed to be unobservable.))

Without incentive constraints, the result in Dutta and Vohra (2005, Proposition
5.1) implies that in quasilinear economies, if coalition S has a coarse objection, then
S has an ex post objection in some t ∈ T . (They show that in quasilinear economies,
if coalition S has a fine objection, then it has an ex post objection, where a fine
objection allows all members of the coalition to decide how much of their private
information they wish to share with each other. Hence, a coarse objection must
be a fine objection.) This means that in quasilinear economies, if coalition S has
no ex post objection, then it has no incentive compatible coarse objection. (Note
that an incentive compatible coarse objection is one of the coarse objections without
incentive constraints.) We use this fact to prove the non-emptiness.

3.4.2

Non-emptiness of the incentive compatible coarse core

In order to show the non-emptiness of the incentive compatible coarse core, we
impose an assumption with respect to auxiliary economies: existence of a strict core
allocation in each state.
Definition 2. For each state θk ∈ Θ, a core allocation in state θk is strict if the
inequality of blocking for every coalition S = N is defined as the weak one.

The strict core in state θk is defined in such a way that if all agents of coalition
S = N are indifferent we go on with the objection. We can give a characterization of
each allocation in the strict core by the following lemma. (The proof of this lemma

69
is a reproduction of McLean and Postlewaite’s (2003) Step 1 in the proof of Theorem
2.)
Lemma 1. If (xi (θk ))i∈N is a strict core allocation of the auxiliary economy in state
θk , then there exists a σk > 0 such that the following condition holds: for all coalition
S = N, there is no allocation some y S ∈ ΦS such that
ui (xi (θk ), θk ) − σk < ui (yiS , θk )
for all i ∈ S.

We now present a result concerning the non-emptiness of the incentive compatible
coarse core.
Theorem 1. Suppose that agents’ utilities are quasilinear and the strict core is
non-empty in each state. Then there exists a δ > 0 such that, whenever agents’
information structure satisfies
max μPi ≤ δ min ΛPi
i

i

and

max νiP ≤ δ min ΛPi ,
i

i

the incentive compatible coarse core of the PIE ({ui , wi }i∈N , θ̃, t̃, P ) is nonempty.

Proof: In order to prove this theorem, we use the following lemma (its proof is given
at the end of this section. The proof of this lemma is similar to the one in McLean
and Postlewaite (2002, Theorem 1), but several changes are needed):
Lemma 2. Let A = (ζi (θ1), . . . , ζi (θm ))i∈N be a feasible allocation in each state
θk ∈ Θ. For each η > 0, there exists a δ > 0 such that, whenever agents’ information
structure satisfies
max μPi ≤ δ min ΛPi
i

i

and

max νiP ≤ δ min ΛPi ,
i

i

70
there exist an incentive compatible PIE allocation z(·) for the PIE ({ui , wi}i∈N , θ̃, t̃, P )
and a collection B1 , . . . , Bm of disjoint subsets of T such that
(i) Pr{t̃ ∈ ∪m
k=1 Bk } ≥ 1 − η,
(ii) Pr{θ̃ = θk | t̃ = t} ≥ 1 − η for all k = 1, . . . , m and all t ∈ Bk ,
(iii) for all i ∈ N, all k = 1, . . . , m and all t ∈ Bk ,
ui (zi (t); θk ) ≥ ui(ζi (θk ), θk ) − η.

(iv) for all i ∈ N and all t ∈ T \ [∪m
k=1 Bk ],
zi(t) = x̂i(t),

where (x̂i(t))i∈N is an ex post core allocation in type profile t.

By using this lemma, we prove the non-emptiness of the incentive compatible
coarse core.

Let
σ := min σk
k

and η :=

σ
,
1+M +σ

where σk is a number obtained in Lemma 1 in each state θk and

M :=

max max vi−l (
i
θ

n

j=1

wj−l , θ)

+

n


wjl .

j=1

We pick a strict core allocation (ζi (θ1 ), . . . , ζi (θm ))i∈N of the auxiliary economy in
each state θk such that every agent has a positive amount of numeraire. (This

71
procedure is possible because we are assuming that in each state there exists a
Walrasian equilibrium such that the amount of numeraire for every agent is positive.)

Given this allocation and η, we are able to choose an incentive compatible allocation rule z in accordance with Lemma 2. First, we show that this allocation z is
not ex post blocked by a proper subset S = N. If t ∈ T \ [∪m
k=1 Bk ], then there is no
such blocking from condition (iv) in Lemma 2. So suppose that for some k, there
exist (yiS )i∈S ∈ ΦS and t ∈ Bk satisfying


ui (yiS , θ)P (θ | t) >

θ∈Θ



ui (zi (t), θ)P (θ | t)

θ∈Θ

for all i ∈ S. We will show that this leads to a contradiction.
For each i ∈ S and the given t ∈ Bk ,


ui(zi (t), θ)P (θ | t) ≥ (1 − η)ui (zi(t), θk )

θ∈Θ

≥ (1 − η)[ui(ζi (θk ), θk ) − η]
> (1 − η)ui (ζi (θk ), θk ) − η.

On the other hand,

θ

ui (yiS , θ)P (θ | t) ≤ (1 − η)ui (yiS , θk ) + η · M.

72
These imply that for each i ∈ S,
(1 − η)ui (yiS , θk ) + η · M ≥



ui (yiS , θ)P (θ | t)

θ∈Θ

>



ui (zi(t), θ)P (θ | t)

θ∈Θ

> (1 − η)ui (ζi (θk ), θk ) − η.
Thus we conclude that for each i ∈ S,
ui (yiS , θk ) > ui (ζi (θk ), θk ) − σ
contradicting the assumption that (ζi (θk ))i∈N is a strict core allocation of the auxiliary economy in state θk . Since utility functions are quasilinear, this result implies
that coalition S = N has no incentive compatible coarse objection to z from the
argument in section 3.4.1.

Finally, if z is not interim incentive efficient then we can find an interim incentive
efficient mechanism z  that dominates z. Since the utility functions are continuous
and the set of feasible allocations satisfying both individual rationality and incentive
compatibility is compact, we can find an interim incentive efficient mechanism such
that no coalition S ⊆ N has an incentive compatible coarse objection.

Remarks.

(1) In order to obtain the non-emptiness result, we assumed that the strict core in
each state is non-empty. If we define a strict Walrasian equilibrium allocation in
each state θ ∈ Θ in an analogous way to McLean and Postlewaite (2003, Definition
4), then we can say that every Walrasian equilibrium in each state θ is strict under
the assumptions we made. Hence, similar to McLean and Postlewaite (2003, Section

73
5.2), we are able to show that the non-emptiness of the strict core in each state is
generically satisfied with regard to the space of endowments.

(2) Bahçeci (2003) introduces a version of almost complete information and investigates the non-emptiness of the incentive compatible coarse core (without the assumption of quasilinearity). We note that it is difficult to connect our non-emptiness
result with his result, since each agent’s informational size does not approach zero
even though the noise given in Bahçeci’s model gets close to zero as long as the noise
exists.

(3) Much previous literature works with utilities ûi(·, t) that depend on the type
profile t, but, as explained in McLean and Postlewaite (2003), the two approaches
are formally interchangeable. Roughly speaking, the conditions on informational
smallness can be represented by using utility profile ûi (·, t) as follows: informational
smallness implies that for each agent i’s possible types ti , he assigns small probability
to the event in which he has a large influence on the ex post utility profile (ûj )j∈N ,
given his observed type. Negligible aggregate uncertainty implies that with high
probability every agent’s ex post utility ûi (·, t) is close to ui (·, θ) for some θ ∈ Θ.
Finally, distributional variability means that the distributions on the ex post utility
profile, conditional on different types the agent might receive, are not close.

(4) The allocation z given in the proof satisfies incentive compatibility and has no
ex post objection by any subcoalition S = N. Serrano and Vohra (2007) analyze
the core concepts based on endogenous information transmission among members
of a coalition. Since, in quasilinear economies, an objection defined for these core
concepts implies an ex post objection for some t ∈ T (Proposition 1 in Kamishiro
and Serrano (2009)), we can say the following: under the same conditions, there
exists an incentive compatible allocation which has no blocking based on endogenous

74
information transmission by any subcoalition S = N.

However, this observation does not immediately imply the non-emptiness of these
core notions, because z given in the proof is not necessarily immune to possible
objections (based on endogenous information transmission) by the grand coalition
N. (In these core notions, stronger conditions are imposed on the grand coalition
than interim incentive efficiency.) Analyzing the non-emptiness of the core based
on endogenous information transmission under informational smallness is a topic of
further research.

We hereafter prove Lemma 2.
Proof of Lemma 2: Let A = (ζi (θ1 ), . . . , ζi (θm ))i∈N be a feasible allocation in each
state θk ∈ Θ, and η > 0. Choose δ so that

0 < δ < min

c(η, A) η 1
√
, ,
20 mM 4 6


,

where c(η, A) is the number given in Lemma A.1 in McLean and Postlewaite (2002)
introduced below (we use this lemma later, but we omit to write down the definition
of the function c since we will only directly use the lemma.).
Define μ̂P := maxi μPi , ν̂ P := maxi νiP , and ΛP := mini ΛPi , and suppose that
ν̂ P ≤ δΛP , μ̂P ≤ δΛP (conditions of the Theorem). If ΛP = 0, then we are able to
prove in a similar way to McLean and Postlewaite (2002, page 2444).
So suppose that ΛP > 0. For each k = 1, . . . , m, let
Ak = {t ∈ T | PΘ (· | t) − Iθk  ≤ μ̂P },

Bk = {t ∈ T | PΘ (· | t) − Iθk  ≤ μ̂P + ν̂ P },

75
where we recall that  ·  is the 1-norm. Let
A0 = T \ [∪k Ak ] and

B0 = T \ [∪k Bk ].

Then Ak ⊆ Bk for all k = 1, . . . , m and A0 ⊇ B0 .
Since ΛP ≤ 2 (by the definition of ΛP ), it follows that
1
1
μ̂P ≤ δΛP < ΛP ≤ ,
6
3

1
1
ν̂ P ≤ δΛP < ΛP ≤ ,
6
3

and the collections Π = {A0, A1, . . . , Am} and Π = {B0 , B1 , . . . , Bm } are partitions
of T , respectively.

Here, we use the following lemma:
Lemma 3. (McLean and Postlewaite (2002, Lemma A.1))
Let A = (ζi (θ1 ), . . . , ζi (θm ))i∈N be a feasible allocation in each state θk ∈ Θ. For
each η ≥ 0, there exists a collection {{yi (θk , ti)}(ti ,θk )∈Ti×Θ }i∈N satisfying:
(i) yi (θk , ti ) ∈ Rl+ and



i∈N (yi (θk , ti)

− wi ) ≤ 0 for all ti ∈ Ti and all θk ∈ Θ,

(ii) ui(ζi (θk ); θk ) ≥ ui (yi (θk , ti)) ≥ ui(ζi (θk ); θk ) − η for all ti ∈ Ti and all θk ∈ Θ,
(iii) for each ti, ti ∈ Ti with ti = ti ,

θk

[ui (yi (θk ; ti), θk ) − ui (yi (θk ; ti), θk )]P (θk | ti ) ≥

c(η, A)
√
min ΛPi .
2 m i

Applying this lemma, there exists a collection ({yi (θk , ti)}(θk ,ti )∈Θ×Ti )i∈N satisfy-

76
ing the above three conditions. Let z(·) be the PIE allocation defined as

zi (t) :=

⎧
⎪
⎨ yi (θk , ti) if t ∈ Bk (k = 1, . . . , m),
⎪
⎩ x̂i (t)

if t ∈ B0 ,

where (x̂i (t))i∈N is an ex post core allocation in state t. Before proving that the PIE
allocation z(·) is incentive compatible, we first show two claims.

Claim 1: For each i ∈ N and ti ∈ Ti,


|PΘ (θk |ti) − Pr{t ∈ Ak | t˜i = ti }| ≤ 2μ̂P .

k

Proof of claim 1 is the same as that of Claim 1 in McLean and Postlewaite (2002,
page 2444).

Claim 2: For each i ∈ N and ti , ti ∈ Ti ,
m




k=1

t−i
(t−i ,ti )∈Ak
(t−i ,ti )∈B
/ k

P (t−i | ti) ≤ ν̂ P .

Proof of Claim 2: Choose ti , ti ∈ Ti and define
Ψ=

m


{t−i ∈ T−i | (t−i , ti) ∈ Ak and (t−i , ti ) ∈
/ Bk }

k=1

and
Φ = {t−i ∈ T−i | PΘ (· | t−i , ti ) − PΘ (· | t−i , ti) > ν̂ P }.
By a similar argument to McLean and Postlewaite (2002, Claim 2), it suffices to prove

77
/ Φ. Then for some k(k = 1, . . . , m), we
that Ψ ⊆ Φ. Suppose that t−i ∈ Ψ but t−i ∈
/ Bk , and PΘ (· | t−i , ti) − PΘ (· | t−i , ti ) ≤ ν̂ P . Then
have (t−i , ti) ∈ Ak , (t−i , ti) ∈
PΘ (· | t−i , ti ) − Iθk  ≤ PΘ (· | t−i , ti) − Iθk  + PΘ (· | t−i , ti ) − PΘ (· | t−i , ti )
≤ μ̂P + ν̂ P ,

an impossibility (by the definition of Bk ). This completes the proof of claim 2.

From the definition of z, we can check that ui (zi (t); θ) ≤ M for all t ∈ T and all
θ ∈ Θ.
In order to prove incentive compatibility, we can use exactly the same sequence
of inequalities on pp. 2446–2447 of McLean and Postlewaite (2002) with their K1
replaced with M, their zi (θk ; ti) replaced with yi (θk ; ti ) and their set Ak ∪A0 replaced
with Bk to conclude that


[ui(zi (t−i , ti ), θ) − ui (zi (t−i , ti), θ)]P (θ, t−i | ti)

θ∈Θ t−i

c(η, A)
≥ √ ΛP − 2M
2 m



c(η, A) P
= 0.
5 √
Λ
20 mM

In order to complete the proof, it is sufficient to show that z(·) satisfies conditions
(i), (ii) and (iii) (condition (iv) is trivially satisfied). To prove (i), note that Pr{t̃ ∈
A0 | t̃i = ti } ≤ μ̂P for each i ∈ N and ti ∈ Ti (by the definition of μ̂P ). Thus
Pr{t̃ ∈ A0} =


ti∈Ti

Pr{t̃ ∈ A0 | t̃i = ti}P (ti ) ≤ μ̂P ≤ δΛP ≤

η P
Λ < η,
4

and hence,
m
Pr{t̃ ∈ ∪m
k=1 Bk } ≥ Pr{t̃ ∈ ∪k=1 Ak } = 1 − Pr{t̃ ∈ A0 } ≥ 1 − η

78
(because Ak ⊆ Bk for every k = 1, . . . , m by the definitions of Ak and Bk ). We can
check conditions (ii) and (iii) in a similar way to McLean and Postlewaite (2002, pp.
2447–2448).

Chapter

Four

Axioms on Restriction Operators
and Values of Games in Partition
Function Form

80

4.1

Introduction

The purpose of this chapter is to analyze a situation where several players entail
cooperation in the presence of externalities. In the absence of externatlies, Shapley
(1953) provides a value of characteristic function form games. He shows that there is
a unique value satisfying the axioms of efficiency, symmetry, additivity and dummy
player. Various axiomatizations of the Shapley value have been given so far (see
Winter (2002) and Serrano (2009) for a survey).

In order to take externalities into consideration, Thrall and Lucas (1963) defined
the games in partition function form as a generalization of those in characteristic
function form. Several solution concepts have been given for the games in partition
function form, generalizing the Shapley value. For instance, Macho-Stadler, PérezCastrillo and Wettstein (2007) focus on axioms designed to capture the intuitive
content of Shapley’s original axioms, and de Clippel and Serrano (2008) offer several
values relying on the marginal approach.

Recently, Dutta, Ehlers and Kar (2008) followed the methodology of the potential
approach originally proposed in Hart and Mas-Colell (1989) for games in characteristic function form to offer a value of games in partition function form. In order to
consider the potential of games in partition function form, Dutta et al. defined a
restriction operator. Loosely speaking, a restriction operator (artificially) specifies a
game in partition function when one player is removed. They limit the discussion to
the case where the order to remove a group of players is irrelevant (Path Independence axiom). They show that the value derived from the potential is the Shapley
value of the associated characteristic function constructed from a restriction operator. They also consider a case where several axioms are imposed on a restriction

81
operator.

In this chapter, we draw connections between values of games in partition function form proposed in previous literature and the Shapley value constructed from
a restriction operator satisfying axioms. Particularly, we focus on the values obtained in de Clippel and Serrano (2008) and the ones in Macho-Stadler et al (2007).
de Clippel and Serrano (2008) prove that the axioms of efficiency, anonymity, and
monotonicity on a value bound each player’s payoff from below and from above.
They also show that for the class of three-player partition functions, the bounds
obtained are tight. In this chapter, we show that in three-player games the axioms
of anonymity and monotonicity on a restriction operator (not on a value) can play
an important role to obtain the same bounds.

We also compare the sets of values axiomatized in Dutta et al and Macho-Stadler
et al. The values obtained from both approaches are represented as the Shapley value
of a characteristic function constructed from a weighted average of the partition
function. We show that the set of values obtained from Dutta et al’s approach is a
subset of those from Macho-Stadler et al. We also show that in order to obtain this
relationship, the weak dummy axiom (on a restriction operator) defined in Dutta et
al is necessary.

4.2

Framework and notation

The economic environment we study can be described as follows. Let N be some
finite set of players. Choose any subset N of N . We denote by N = {1, . . . , n}.
A coalition S is a group of players, that is, a non-empty subset of N, S ⊆ N. An

82
embedded coalition is a pair (S, P ), where S is a coalition and P  S is a partition
of N. An embedded coalition hence specifies the coalition as well as the structure
of coalitions formed by the other players. Let P denote the set of all partitions of
N. It represents all the possible ways in which society can be organized. The set of
embedded coalitions is denoted by ECL and defined by
ECL = {(S, P ) | S ∈ P, P ∈ P}.

We denote by v a partition function form game, that is, v : ECL → R is a
function that associates a real number with each embedded coalition. Hence, v(S, P )
with S ∈ P , P ∈ P, is the worth of coalition S when the players are organized
according to the partition P . In our environment, players can make transfers among
themselves. Let V denote the class of all partition games which can be constructed
on player sets that are subsets of N . We assume that the grand coalition worth is
allocated to the agents.

4.3

Restriction operators

Dutta et al (2008) introduce a restriction operator in order to specify subgames
(N \ {i}, v). A restriction operator is a mapping r from V to V which specifies for
each game (N, v) a subgame (N \ {i}, v) for each i ∈ N.
In order to define a restriction operator, we prepare some notations. Let P S =
(S1 , S2, . . . , SK ) be a partition of some set S. Then, for any i ∈
/ S, P +i (P S ) is the
set of partitions of S ∪ {i} where player i either joins one of the coalitions Sk of P S ,
the other coalitions remaining unchanged, or it is the partition (S1, S2 , . . . , SK , {i}).

83
We denote P −i = P N \{i}.
Given any (N, v) ∈ V, a restriction operator is specified below:
N

v −i,r (S, P −i ) = ri,S,P
−i (v(S, S ∪ P )P  ∈P +i (P −i \S) ),

where P  S and i ∈
/ S.
The worth of coalition S in the subgame when player i is absent is some function
of the worths (S, P −i ) where S ∈ P −i . For example, if N = {1, 2, 3, 4} then
v −4,r ({1}, ({1}, {2, 3}))
{1,2,3,4}

= r4,{1},({1},{2,3})(v({1}, ({1}, {2, 3, 4})), v({1}, ({1}, {2, 3}, {4})).
In the subgame the worth of any coalition S for a specific partition of the other
players depends only on the worths of S where player i joins one of the existing
members of the given partition (except S) or remains alone. For simplification, we
will drop the superscript N and the subscripts S and P −i whenever no confusion can
result from this simplification.
We impose the following condition on a restriction operator r: for every i ∈ N,
every S ⊆ N \ {i} and P  S,

min

P  ∈P +i (P −i \S)

v(S, S ∪ P  ) ≤ v −i,r (S, P −i ) ≤

max

P  ∈P +i (P −i \S)

v(S, S ∪ P  ).

This condition means that the estimation should be placed between the minimum
and the maximum of the case where player i exists. This assumption implies that
v −i (N \ {i}) = v(N \ {i}, ({i}, N \ {i})).

84
Dutta et al imposed several axioms on a restriction operator. Their most fundamental axiom is path independence. The path independence axiom means that the
order in which players are removed from a game is irrelevant for the subgame. In
this chapter, we also limit the discussion to the case where this axiom is imposed
on restriction operators as well as Dutta et al. Formally, this axiom is defined as
follows. For any i, j ∈ N, let v −ij = (v −i )−j .

Path Independence. A restriction operator r satisfies Path Independence if for all
(N, v) ∈ V, for all i, j ∈ N, v −ij = v −ji .

If this axiom is imposed, then the subgame v −S,r , where some coalition S ⊂ N
leaves the game v, is well-defined. Given any restriction operator r and game (N, v),
we can define the auxiliary characteristic function wvr : 2N → R as follows:
c

wvr (N) = v(N, {N}), and for all S ⊂ N, wvr (S) = v −S ,r (S).

Dutta et al show that the potential approach gives the Shapley value of the
auxiliary characteristic function (they call r-Shapley value):
Shi (wvr ) =



βi (S)wvr (S),

S⊆N

⎧
⎪
⎨

where
βi (S) =

(|S|−1)!(n−|S|)!
n!

for all S ⊆ N if i ∈ S,

⎪
⎩ − |S|!(n−|S|−1)! for all S ⊆ N if i ∈ N \ S.
n!

We investigate some properties of Shi (wvr ) in the next sections.

85

4.4

Anonymity of restriction operators

Firstly, we focus on the anonymity axiom of a restriction operator. Let π be a
permutation of N. We define π(S) = {j | j = π(i) for some i ∈ S} for any coalition
S and π(P ) = {T | T = π(S) for some S ∈ P } for any partition P .

Anonymity. A restriction operator r satisfies anonymity if
v −i,r (S, P −i ) = v −π(i),r (π(S), π(P )−π(i))
for all i ∈
/ S whenever there exist some S and π such that v(S, P ) = v(π(S), π(P ))
for all P .

Anonymity says that if two coalitions face the same situation, excluding the name
of players, then they construct the same estimation about the subgame. Needless
to say, the large class of linear and non-linear restriction operators are compatible
with the anonymity axiom. However, if a partition function u is symmetric, then the
Shapley value of the auxiliary characteristic function constructed from any restriction
operator r satisfying anonymity leads to the same value. We shall say that a partition
function u is symmetric if π(u)(S, P ) = u(S, P ), for each (S, P ) ∈ ECL and each
permutation π of the players. That is to say, the worth of an embedded coalition
is a function only of its cardinality and of the cardinality of the other atoms of the
partition. If u is symmetric, then Shi (wur ) = u(N)/n for all i ∈ N (see the proof of
Theorem 4.1 below).

Next, we consider a case where one more axiom is added. (This axiom is considered in Dutta et al (2008).)

86
Translation Invariance. A restriction operator r satisfies Translation Invariance if
for any i ∈ N, any S ⊆ N \ {i}, and any P −i  S, if v and ṽ satisfy ṽ(S, S ∪P  ) − c =
v(S, S ∪ P ) for all P  ∈ P +i (P −i \ S), then ṽ −i,r (S, P −i ) − c = v −i,r (S, P −i ).

Translation invariance says that if the original worths are translated by some
constant, then in the subgame the worth should also be translated by the same
constant. There are still many restriction operators satisfying both anonymity and
translation invariance. However, we can obtain a result on a subclass of partition
functions which can be decomposed into the sum of symmetric partition and characteristic function. (This subclass is the same as the one analyzed in de Clippel and
Serrano (2008): they consider the case where the weak marginality axiom is imposed
on a value.)
Theorem 2. Let r be a restriction operator which satisfies the axioms, Path Independence, Anonymity and Translation Invariance. Let u be a symmetric partition
function, and let v be a characteristic function. Then
r
)=
Shi (wu+v

u(N)
+ Shi (v),
n

for each i ∈ N.

Proof. Since u is symmetric, the anonymity axiom implies that the auxiliary characteristic function constructed from any restriction operator r, wur (S), depends only
r
upon the size of S. Let wur (S) = αrs (where |S| = s). Then wu+v
(S) = αrs + v(S) from

the Translation Invariance axiom. Since the Shapley value (of characteristic function
r
form games) satisfies additivity and symmetry, we have Shi (wu+v
)=

u(N )
n

+ Shi (v).

87
As an illustration of this subclass, we consider an example.
Example 2. Consider the following three-player partition function:

v(N) = 24;
v({1, 2}) = 12;

v({1, 3}) = 13;

v({2, 3}) = 14;

v({1}, ({1}, {2, 3})) = v({2}, ({2}, {1, 3})) = 9;
v({1}, ({1}, {2}, {3})) = v({2}, ({2}, {1}, {3})) = 0;
v({3}, ({3}, {1, 2})) = 12;

v({3}, ({3}, {1}, {2})) = 3.

Then the auxiliary characteristic function of singleton coalitions is of the form
v −{2,3}({1}) = α, v −{1,3}({2}) = α, v −{1,2}({3}) = α + 3(0 ≤ α ≤ 9). The Shapley
value of the auxiliary characteristic function is (7, 7.5, 9.5) (note that this value is
independent of α).

This partition function can be decomposed as the sum of a characteristic function
v  (v ({1, 3}) = 1, v ({2, 3}) = 2, v ({3}) = 3 and v (S) = 0 for all other S) and a
symmetric partition function u such that u(N) = 24. Since the Shapley value of v 
is (−1, −0.5, 1.5) and u(N)/3 = 8, we can check the formula of the Proposition.

4.5

Monotonicity of restriction operators

Next we consider the case where a restriction operator satisfies monotonicity.

Monotonicity. Let (N, v), (N, v ) ∈ V such that v and v  are the same except
v(S, P ) > v (S, P ) for some S ⊆ N and some P  S. A restriction operator r satisfies
monotonicity if for all i ∈ N \ S and for all P −i ∈ S, v −i (S, P −i ) ≥ v −i (S, P −i ).

88
There are still non-linear values satisfying all of the above axioms including monotonicity, but we can bound each player’s payoff from below and from above. In a
three-player game, the bounds of the value satisfying the axioms leads to the same
ones in de Clippel and Serrano (2008).
Theorem 3. In three-player games, the bounds obtained from the Shapley value
constructed from a restriction operator r satisfying Path Independence, Anonymity,
Monotonicity, and Translation Invariance are the same as the ones in de Clippel and
Serrano (2008). More specifically, if we denote Shi (wvr ) ∈ [μi (v), νi(v)], then
νi (v) = σi∗(v) +

max{0, i (v) − j (v)} + max{0, i (v) − k (v)}
,
6

μi (v) = σi∗(v) +

min{0, i (v) − j (v)} + min{0, i (v) − k (v)}
,
6

where σi∗(v) is player i’s externality-free value payoff and i(v)
= v({i}, ({i}, {j, k})) − v({i}, ({i}, {j}, {k})) is the externality index associated to
player i.

Proof: Let N = {i, j, k} and let us consider the bounds for agent i. We note that
the auxiliary characteristic function for coalitions consisting of more than one player
is determined independent of r in every three-player game. We also note that the
Shapley value of player i in a three-player game can be rewritten by using a function
f as
Shi (wvr ) = f(wvr (S) : |S| ≥ 2) +

wvr ({i}) − wvr ({j}) wvr ({i}) − wvr ({k})
+
.
6
6

(4.1)

Firstly, we consider the upper bound. We consider separately two cases, i ≥ 0
and i < 0.

89
Case 1: i ≥ 0.
Let us consider the restriction operator r which leads to the following characteristic function wvr :
wvr ({i}) := v({i}, ({i}, {j}, {k})) + i(v)(= v({i}, ({i}, {j, k}))),

wvr ({j}) :=

⎧
⎪
⎨ v({j}, ({i}, {j}, {k})) + j (v)(= v({j}, ({j}, {i, k}))) if j ≤ i ,
⎪
⎩ v({j}, ({i}, {j}, {k})) + i (v)

if j > i .

for j = i. We can check that this restriction operator r satisfies anonymity, monotonicity and translation invariance, and that the Shapley value obtained from this
restriction operator leads to the upper bound νi (v).
Next we show that it is impossible to obtain a value greater than this upper
bound. If there exists a restriction operator r which leads to the value greater than
this bound, then (4.1) implies that for some j,




wvr ({i}) − wvr ({j}) > wvr ({i}) − wvr ({j}).

(4.2)

Case 1-1. j < 0.
Then it is impossible to find r satisfying (4.2) because r assigns the maximum
to player i and the minimum to player j.
Case 1-2. 0 ≤ j ≤ i .
Let δ := (v({i}, ({i}, {j}, {k})) + i (v)) − (v({j}, ({i}, {j}, {k})) + j (v)). Then,
wvr ({i}) − wvr ({j}) ≥ δ by the definition of the restriction operator r. Let us con-

90
sider the following game ṽ: ṽ({j}, P ) = v({j}, P ) + δ for all P  {j}. Then by






translation invariance, wṽr ({j}) = wvr ({j}) + δ. On the other hand, wṽr ({j}) ≥


wvr ({i}) from monotonicity and anonymity (we note that ṽ({j}, ({i}, {j}, {k})) >
v({i}, ({i}, {j}, {k})) and ṽ({j}, ({j}, {i, k})) = v({i}, ({i}, {j, k})) hold). They im



ply that wvr ({i}) − wvr ({j}) ≤ δ, which is a contradiction.
Case 1-3. 0 ≤ i < j .
Let η := v({i}, ({i}, {j}, {k})) − (v({j}, ({i}, {j}, {k})). Then,
wvr ({i}) − wvr ({j}) = η by the definition of r. Let us consider the following game ṽ:
ṽ({j}, P ) = v({j}, P ) + η for all P  {j}. Then by translation invariance,








wṽr ({j}) = wvr ({j}) + η. On the other hand, wṽr ({j}) ≥ wvr ({i}) from monotonic



ity and anonymity. They imply that wvr ({i}) − wvr ({j}) ≤ η, which is again a
contradiction.

Case 2: i < 0.
Let

vj :=

vi := v({i}, ({i}, {j}, {k})),
⎧
⎪
⎨ v({j}, ({i}, {j}, {k})) + (j (v) − i(v)) if j ≤ i ,
⎪
⎩ v({j}, ({i}, {j}, {k}))

if j > i .

for j = i. Then by a similar argument to the first case, we can check that this is the
upper bound.

The lower bound for player i’s value can be obtained by the following restriction
operator, depending upon the sign of i . (Since the proof is quite similar to the case
of upper bound, we omit it.)

91
Case 1: i ≥ 0.

wvr ({i}) := v({i}, ({i}, {j}, {k})),

⎧
⎪
⎨ v({j}, ({i}, {j}, {k}))
if j ≤ i ,
r
wv ({j}) :=
⎪
⎩ v({j}, ({i}, {j}, {k})) + (j (v) − i(v)) if j > i.
for j = i.

Case 2: i < 0.

wvr ({i}) := v({i}, ({i}, {j}, {k})) + i(v)(= v({i}, ({i}, {j, k}))),
⎧
⎪
⎨ v({j}, ({i}, {j}, {k})) + i (v)
if j ≤ i ,
r
wv ({j}) :=
⎪
⎩ v({j}, ({i}, {j}, {k})) + j (v)(= v({j}, ({j}, {i, k}))) if j > i .
for j = i.

4.6

Weak dummy axiom and values in MachoStadler et al

Finally, we compare the values axiomatized by Dutta et al (2008) with the ones
obtained through the average approach in Macho-Stadler et al (2007). To simplify
the argument in this section, we limit our discussion to the case where v −i,r (S, P −i )
is represented as a weighted average of v(S, P ). More specifically, we consider the

92
situation where the linearity axiom is imposed on a restriction operator, which is
defined as follows:

Linearity. A restriction operator r satisfies Linearity if:
(a) For any two games v and v , r(v + v ) = r(v) + r(v  ).
(b) For any game v and any scalar λ ∈ R, r(λv) = λr(v).

Dutta et al (2008) additionally impose an axiom, called the weak dummy axiom,
to a restriction operator. The weak dummy axiom is defined as follows:

Weak dummy. A restriction operator r satisfies the weak dummy axiom if for all
(N, v), if player i is a dummy player in (N, v), then player i is a dummy player in
(N, \{j}, v −j ) for all j = i. Here, player i is a dummy player, if for all S ⊆ N
containing i, and for all partitions P , v(S, P ) = v(S \ {i}, (S \ {i}, P )) for all
P  ∈ P +i (P \ S).

This axiom captures the idea that a player who is useless in the original game
should also be useless in all subgames. The result in Dutta et al implies the following
— if a restriction operator r satisfies the path independence, linearity and weak
dummy axioms, then r is of the form
v −i,r (S, P −i ) = θ



v(S, (S ∪ P  )) + (1 − tθ)v(S, (P −i , {i})),

P  ∈P +i (P −i \S)

where t = |P  : P  ∈ P +i (P −i \ S)|1 and 0 ≤ θ ≤
1

1
.
t−1

(The result follows from the

Since P −i ∪ {i} is an element of P +i (P −i \ S), the total weight on P −i ∪ {i} is 1 − (t − 1)θ.

93
fact that linearity is stronger than the axioms Dutta et al imposed. They imposed
the axioms of Scale Invariance, Sign Independence and Non-negativity.)

Now we compare the Shapley value of an auxiliary characteristic function constructed from a restriction operator satisfying the linearity axiom and the weak
dummy axiom with the value of the average approach proposed in Macho-Stadler et
al (2007). They axiomatize the following value:

ϕi (v) =




βi(S)

S⊆N

⎧
⎪
⎨

where
βi (S) =

(|S|−1)!(n−|S|)!
n!




α(S, P )v(S, P ) ,

P S,P ∈P

for all S ⊆ N if i ∈ S,

⎪
⎩ − |S|!(n−|S|−1)! for all S ⊆ N if i ∈ N \ S.
n!

The weights α(S, P ) used in the average approach are symmetric and satisfy the
following:

α(S, P ) =



α(S \ {i}, (P \ (R, S)) ∪ (R ∪ {i}, S \ {i}))

(∗)

R∈{P \S,∅}

for all i ∈ S and all (S, P ) ∈ ECL with |S| > 1.2
The constraint (∗) is a link between the weight of partition P for the coalition
S and the weights of the partitions that result from moving any player in S to
the coalitions in P other than S. We show that the Shapley value of the auxiliary
characteristic function constructed from the above restriction operators leads to the
values obtained by the average approach in Macho-Stadler et al.
Theorem 4. The Shapley value of the auxiliary characteristic function constructed
2

set.

When R = ∅, we assume that the partition (P \ (∅, S)) ∪ (∅ ∪ {i}, S \ {i}) includes the empty

94
from a restriction operator given by Dutta et al. leads to the value derived from the
average approach by Macho-Stadler et al.

Proof. It is sufficient to show that the weights of the auxiliary characteristic function
constructed from Dutta et al.’s approach satisfy the condition (∗). If S = N, then
it is clear that the condition (∗) holds. Let S  N. Then the construction of the
subgame implies that for all i ∈ N \ S and all P −i  S, the coefficient of v −i (S, P −i )
in order to make wvr (S) is the same as the one of v(S ∪ {i}, (P −i \ S, S ∪ {i})) to
make wvr (S ∪ {i}). Therefore, we can check that the condition (∗) holds.

We can also check that in every three-player game, the values derived from both
approaches lead to the same sets of allocations. However, if there are more than
three players, then the value derived from Dutta et al would be a strict subset of the
one in Macho-Stadler et al. Let us consider a four-player game. The restriction on
weights can be represented in the following table. This table represents the weight
of each embedded coalition structure (S, P ), where b, c and θ are parameters.

(S, P ) for n = 4

Macho-Stadler et al.

Dutta et al.

({i}, ({i}, {j}, {k}, {l}))

1 − b − 2c

(1 − θ)2

({i}, ({i}, {j}, {k, l}))

c

θ(1 − θ)

({i}, ({i}, {j, k, l}))

b−c

θ2

({i, j}, ({i, j}, {k}, {l}))

1−b

1−θ

({i, j}, ({i, j}, {k, l}))

b

θ

({i, j, k}, ({i, j, k}, {l}))

1

1

(N, {N})

1

1

If we take b = θ and c = θ(1 − θ), then the value of Macho-Stadler et al would
be the same as that of Dutta et al. On the other hand, if, for instance, we take

95
(b, c) = (1/2, 1/6), then the value cannot be represented by Dutta et al’s notion.
(This parameter leads to the value which is also axiomatized by Macho-Stadler et
al, adding similar influence axiom to a value. Thus this value is not in the one in
Dutta et al.)

Finally, we show that the weak dummy axiom is crucial in order to obtain the
above inclusive relationship. The following example shows that the Shapley value
of the auxiliary characteristic function from a system of axioms with regard to a
restriction operator, path independence, anonymity, monotonicity and linearity, does
not necessarily belong to the set of values obtained by the average approach in
Macho-Stadler et al.
Example 3. Consider a four-player game. We show that it is possible to construct
a restriction operator, which leads to the following auxiliary characteristic function:
wvr (S) =


P S,P ∈P

1
v(S, P ),
|m(S, P )|

where |m(S, P )| denotes |{(T, Q) ∈ ECL : T = S}|, that is,
⎧
⎪
⎪
5 if |S| = 1,
⎪
⎪
⎨
|m(S, P )| =
2 if |S| = 2,
⎪
⎪
⎪
⎪
⎩ 1 if |S| = 3 or 4.
In this auxiliary characteristic function, the worth of a coalition S is the expected
worth in v when all these coalitions are equally likely. The Shapley value of this
auxiliary characteristic function leads to the value axiomatized in Albizuri, Arin and
Rubio (2005). This auxiliary characteristic function can be constructed from the

96
following restriction operator:
1
v −i({j}, ({j}, {k}, {l})) = v({j}, ({j}, {i}, {k}, {l}))
3
1
1
+ v({j}, ({j}, {i, k}, {l}) + v({j}, ({j}, {k}, {i, l})),
3
3
2
3
v −i ({j}, ({j}, {k})) = v({j}, ({j}, {i}, {k})) + v({j}, ({j}, {i, k})),
5
5
1
1
v −i ({j}, ({j}, {k, l})) = v({j}, ({j}, {i}, {k, l}) + v({j}, ({j}, {i, k, l})),
2
2
1
1
v −i ({j, k}, ({j, k}, {l})) = v({j, k}, ({j, k}, {i}, {l})) + v({j, k}, ({j, k}, {i, l})).
2
2
We can check that this restriction operator satisfies path independence, anonymity,
monotonicity and linearity. We also note that the coefficients of this auxiliary characteristic function violate the condition (∗).

Bibliography

[1] Albizuri, M. J., J. Arin and J. Rubio (2005). “An Axiom System for a Value
for Games in Partition Function Form,” International Game Theory Review 7,
63–73.
[2] Allen, B., (1981). “Generic Existence of Completely Revealing Equilibria for
Economies with Uncertainty when Prices Convey Information,” Econometrica
49, 1173–1199.
[3] Allen, B., Yannelis, N. C., (2001). “Differential Information Economies: Introduction,” Economic Theory 18, 263–273.
[4] Anderson, R. M. (2008). “Core Convergence,” In S. Durlauff and L. Blume
(eds.): The New Palgrave Dictionary of Economics, 2nd edition, McMillan,
London.
[5] Aumann, R. J., (1964). “Markets with a Continuum of Traders,” Econometrica
32, 39–50.
[6] Bahçeci, S. (2003). “The Incentive Compatible Coarse Core when Information
is Almost Complete,” Journal of Mathematical Economics, 39, 127–134.
[7] Blouin, M., Serrano, R. (2001). “A Decentralized Market with Common Values
Uncertainty: Non-Steady States,” Review of Economic Studies, 68, 323–346.
[8] Bondareva, O. N. (1963). “Some Applications of Linear Programming Methods
to the Theory of Cooperative Games (in Russian),” Problemy Kibernetiki, 10,
119–139.
[9] de Clippel, G., (2007). “The Type-Agent Core of Exchange Economies with
Asymmetric Information,” Journal of Economic Theory 135, 144–158.
[10] de Clippel, G., Minelli, E., (2005). “Two Remarks on the Inner Core,” Games
and Economic Behavior 50, 143–154.
[11] de Clippel, G. and R. Serrano (2008). “Marginal Contributions and Externalities in the Value,” Econometrica 76, 1413–1436.

97

98
[12] Debreu, G., Scarf, H., (1963). “A Limit Theorem on the Core of an Economy,”
International Economic Review 4, 235–246.
[13] Dubey, P., Geanakoplos, J., Shubik, M. (1987). “The Revelation of Information
in Strategic Market Games: A Critique of Rational Expectations Equilibrium,”
Journal of Mathematical Economics, 6, 105–137.
[14] Dutta, B., L. Ehlers and A. Kar (2008), “Externalities, Potential, Value and
Consistency,” mimeo.
[15] Dutta, B., Vohra, R., (2005). “Incomplete Information, Credibility and the
Core,” Mathematical Social Sciences, 50, 148–165.
[16] Einy, E., Moreno, D., Shitovitz, B., (2000a). “Rational Expectations Equilibria
and the Ex-Post Core of an Economy with Asymmetric Information,” Journal
of Mathematical Economics 34, 527–535.
[17] Einy, E., Moreno, D., Shitovitz, B., (2000b). “On the Core of an Economy with
Differential Information,” Journal of Economic Theory 94, 262–270.
[18] Forges, F., Heifetz, A., Minelli, E., (2001). “Incentive Compatible Core and
Competitive Equilibria in Differential Information Economies”, Economic Theory 18, 349–365.
[19] Forges, F., Mertens, J.-F., Vohra. R., (2002). “The Ex Ante Incentive Compatible Core in the Absence of Wealth Effects,” Econometrica, 70, 1865–1892.
[20] Forges, F., Minelli, E. (1997). “Self-Fulfilling Mechanisms and Rational Expectations,” Journal of Economic Theory, 75, 388–406.
[21] Forges, F., Minelli, E., Vohra, R., (2002). “Incentives and the Core of an
Exchange Economy: a Survey,” Journal of Mathematical Economics, 38, 1–
41.
[22] Foster, D. F., Viswanathan, S. (1996). “Strategic Trading When Agents Forecast the Forecast of Others,” Journal of Finance 51, 1437–1478.
[23] Gottardi, P., Serrano, R. (2005). “Market Power and Information Revelation
in Dynamic Trading,” Journal of the European Economic Association 3, 1279–
1317.
[24] Gul, F., Postlewaite, A., (1992). “Asymptotic Efficiency in Large Exchange
Economies with Asymmetric Information,” Econometrica 60, 1273–1292.
[25] Hart, S. and A. Mas-Colell (1989). “Potential, Value and Consistency,” Econometrica 57, 589–614.
[26] Holden, C. W., Subrahmanyam, A. (1992). “Long-Lived Private Information
and Imperfect Competition,” Journal of Finance, 47, 247–270.
[27] Holmström, B., Myerson, R. B., (1983). “Efficient and Durable Decision Rules
with Incomplete Information,” Econometrica 51, 1799–1819.

99
[28] Kreps, D., (1977). “A Note on Fulfilled Expectations Equilibria,” Journal of
Economic Theory 14, 32–43.
[29] Kyle, A. (1985). “Continuous Auctions and Insider Trading,” Econometrica
53, 1315–1335.
[30] Laffont, J.-J., (1985). “On the Welfare Analysis of Rational Expectations Equilibria with Asymmetric Information,” Econometrica 53, 1–29.
[31] Macho-Stadler, I., D. Pérez-Castrillo and D. Wettstein (2007). “Sharing the
Surplus: An Extension of the Shapley Value for Environments with Externalities,” Journal of Economic Theory 135, 339–356.
[32] McLean, R. P., Postlewaite, A., (2002). “Informational Size and Incentive Compatibility,” Econometrica, 70, 2421–2453.
[33] McLean, R. P., Postlewaite, A., (2003). “Informational Size, Incentive Compatibility, and the Core of a Game with Incomplete Information,” Games and
Economic Behavior, 45, 222–241.
[34] McLean, R., Postlewaite, A., (2005). “Core Convergence with Asymmetric
Information,” Games and Economic Behavior 50, 58–78.
[35] Milgrom, P. (1981). “Rational Expectations, Information Acquisition, and
Competitive Bidding,” Econometrica 49, 921–943.
[36] Myerson, R. B., (1991). Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, MA.
[37] Myerson, R. B., (2007). “Virtual Utility and the Core for Games with Incomplete Information,” Journal of Economic Theory 136, 260–285.
[38] Pesendorfer, W., Swinkels, J. (1997). “The Loser’s Curse and Information Aggregation in Common Values Auctions,” Econometrica 65, 1247–1281.
[39] Peters, M., Severinov, S. (2008). “An Ascending Double Auction,” Economic
Theory 37, 281–306.
[40] Postlewaite, A., Schmeidler, D., (1986). “Implementation in Differential Information Economies,” Journal of Economic Theory 39, 14–33.
[41] Qin, C.-Z., (1993). “The Inner Core and the Strictly Inhibitive Set,” Journal
of Economic Theory 59, 431–444.
[42] Radner, R., (1979). “Rational Expectations Equilibrium: Generic Existence
and the Information Revealed by Prices,” Econometrica 47, 655–678.
[43] Reny, P., Perry, M. (2006). “Toward a Strategic Foundation for Rational Expectations Equilibrium,” Econometrica 74, 1231–1269.
[44] Scarf, H. (1967). “The Core of an N-Person Game,” Econometrica 38, 50–69.
[45] Serrano, R. (2009). “Cooperaive Games: Core and Shapley Value,” in Encyclopedia of Complexity and Systems Science, Springer, Berlin.

100
[46] Serrano, R., Vohra, R., (2007). “Information Transmission in Coalitional Voting Games,” Journal of Economic Theory 134, 117–137.
[47] Serrano, R., Vohra, R., Volij, O., (2001). “On the Failure of Core Convergence
in Economies with Asymmetric Information,” Econometrica 69, 1685–1696.
[48] Shapley, L. S. (1953). “A Value for n-Person Games,” in Contributions to the
Theory of Games II, A. W. Tucker and R. D. Luce (ed.), Princeton University
Press, 307–317.
[49] Shapley, L. S. (1967). “On Balanced Sets and Cores,” Naval Research Logistics
Quarterly 14, 453–460.
[50] Shapley, L., Shubik, M. (1977). “Trade Using One Commodity as Means of
Payment,” Journal of Political Economy, 85, 937–968.
[51] Thrall, R. M. and W. F. Lucas (1963). “n-Person Games in Partition Function
Form,” Naval Research Logistics Quarterly 10, 281–298.
[52] Vohra, R., (1999). “Incomplete Information, Incentive Compatibility and the
Core,” Journal of Economic Theory 86, 123–147.
[53] Wilson, R., (1978). “Information, Efficiency and the Core of an Economy,”
Econometrica 46, 807–816.
[54] Winter, E. (2002). “The Shapley Value,” in Handbook of Game Theory, Volume
3, Aumann, R., Hart, S. (ed.), North-Holland, Amsterdam, 2025–2054.
[55] Wolinsky, A. (1990). “Information Revelation in a Market with Pairwise Meetings,” Econometrica 58, 1–23.