Information Aggregationin Spatial Committee Games

Vincent Anesi ∗GREMAQ, University of Toulouse

March 2005

AbstractThis paper introduces information aggregation into the standard

spatial committee game. We assume that committee members mustagree on a decision rule to aggregate their private information on apolicy-relevant state of the world. We derive sucient conditions forthe ex ante incentive compatible core to be nonempty, and providesome characterization results for incentive compatible core decisionrules, called durable decision rules. In particular, core points ofthe underlying complete-information game are shown to be constant,durable decision rules of the game with incomplete information if theysatisfy some robustness property. Moreover, we show that durable de-cision rules exist whenever information is Pareto-improving relative tothe core of the underlying complete-information game, provided thatvoters' private signals are weakly informative.

1 IntroductionThe spatial model is one of the most employed by political scientists whostudy preference aggregation and collective choice. That model, introducedby Davis and Hinich in 1966 in the case of majority rule, portrays a nitegroup of voters having continuous, strictly convex preferences over a set ofpolicy alternatives which is modelled as a convex subset of Euclidean space.Given a voting rule, these individual preferences are directly aggregated into

∗Address: GREMAQ, Université de Toulouse I, Manufacture des Tabacs, Bât. F, 21Allée de Brienne, 31000 Toulouse, FRANCE . Email: vincent.anesi@univ-tlse1.fr.

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a collective preference relation. Social-choice theorists have devoted consid-erable attention to the existence and the characterization of the maximalelements of this relation, known as the core of the voting rule under con-sideration. For expositional convenience, the present paper adopts the coop-erative game theory approach to these issues but it is worth noting that allthe results stated below could be reformulated in the standard framework ofsocial choice theory 1.

Thus, we turn to the core of spatial simple games. A simple game, orcommittee, is dened as a set of voters and a list of coalitions, called win-ning coalitions. These coalitions are all-powerful in the sense that they canenforce any policy irrespective of the other voters' behavior. For instance,if the committee makes its decision by simple majority rule, the set of win-ning coalitions consists of all majority coalitions. Alternatively, suppose thecommittee is a dictatorship game, then all coalitions involving the dicta-tor are winning. The core of a simple game is then dened as the set ofpolicies that cannot be improved upon by any winning coalition. Generalcore-characterization results are provided by McKelvey and Wendell (1976),Matthews (1980), and McKelvey and Schoeld (1987).

This paper introduces information aggregation into the standard spatialcommittee game. We assume that committee members are uncertain about apolicy-relevant state of the world. This parameter uncertainty causes uncer-tain policy preferences in members of the committee. However, each memberreceives a private signal that is correlated with the true state of the world,and then makes a report to an independent authority, which uses this infor-mation to select a policy. That is, the independent authority follows somepredetermined decision rule that associates a policy outcome to any proleof signals reported by individuals. It is however important to keep in mindthat signals are private, and are not veriable by the independent authority.Conicting interests may then lead voters to misrepresent their private in-formation. As a result, only self-enforcing decision rules, which induce eachvoter to truthfully reveal his or her information, can be enforced by thisauthority.

For committees in which some coalitions of voters are all-powerful, incen-tive compatibility appears to be only a minimal test that any decision ruleshould pass. Indeed, committee members have induced preferences over thedecision rules that depend on their preferences over policy outcomes and theinformation service they have access to. As a consequence, a decision rule can

1The non-cooperative approach to collective choice, more focused on strategic issues,is left out of the analysis. We refer the reader to Austen-Smith and Banks (1998) for adiscussion of the dierences between the two approaches.

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durably be used to aggregate information in a committee only if the winningcoalitions of this committee would never approve a change from this rule toanother one. Equivalently put, any mechanism that we wish to implementin a committee must not only be incentive compatible but must also havethe core property that assures that it cannot be blocked by any winningcoalition. Putting together incentive compatibility and this core property,this paper is concerned with the ex ante incentive compatible core of spatialcommittee games with incomplete information. For brevity, in what followswe will refer to elements of the ex ante incentive compatible core as durabledecision rules. Thus, a decision rule is durable if it is incentive compatibleand it cannot be blocked via another incentive compatible decision rule.

These assumptions may cause existence problems related to the dimen-sionality of the model. Indeed, in such a setting, the policies coalitions blockare no longer points in a subspace of a Euclidean space but decision rules,or functions. Thus, incomplete information necessarily leads to voting overmultidimensional policies, even if the initial policy space has a unique di-mension. Schoeld (1984) shows that, for any non-collegial simple game, thecore may be empty if the dimension of the policy space is suciently large2. Furthermore, Rubinstein (1979), Cox (1984), and Le Breton (1987) provethat the core of non-collegial simple games with high-dimension issue spacesis generically empty. In view of these results, one might expect non-collegialsimple games with incomplete information to have an empty core in a many-signal environment, what would considerably limit the room for informationaggregation through durable decision rules. To belabor the point, McKelvey(1976, 1979) shows that, for most non-collegial simple games, the top-cycleset coincides with the entire policy space when the core is empty.

Our main concern is therefore with nding conditions that guarantee theexistence of durable decision rules in non-collegial committees. We thus ask apositive, not a normative, question. Imagine however that the committee hasaccess to an information service that is Pareto-improving. That is, there existtruthfully implementable decision rules that make all committee membersbetter o relative to what they could expect in the absence of information.The ability of the committee to collectively adopt such rules goes beyond apositive perspective, and becomes normatively signicant. In addition, whendurable decision rules actually exist, what can we say about such rules?

To address these issues, we rst present the underlying complete-informa-tion simple game, which is simply the game that would be played by votersif they did not receive any information about the state of the world. Using

2A simple game is called collegial if some members of the committee belong to allwinning coalitions. Note that core existence is not a major issue in those games.

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familiar equilibrium concepts from existing literature, we dene a collectionof choice sets for this game. In particular, core points of the complete-information game are shown to be constant, durable decision rules of thegame with incomplete information if they satisfy some robustness property(Condition (R) and Theorem 1). Moreover, we prove in Theorem 2 thatthe incentive compatible core is nonempty whenever information is Pareto-improving relative to the core of the underlying complete-information game,provided that voters' private signals are weakly informative. In such a case,incentive ecient decision rules that Pareto-improve the core of the complete-information game are durable. Theorem 3 extends these conclusions to sit-uations in which the complete-information game has no core but possessesother choice sets. Theorem 4 further characterizes durable decision rules. Itshows that, under a durable decision rule, the risk borne ex ante by winningcoalitions is required to be low relative to the degree of informativeness ofvoters' signals.

Related Literature

We have borrowed the term durable from Holmström and Myerson(1983) who dene durable decision rules as incentive compatible rules thatare not rejected unanimously by the agents in an economy. In their paper,the voting game is non-cooperative and played at the interim stage. Despitethese important dierences, we adopt their terminology which appears to bemore telling and quite shorter than ex ante incentive compatible core de-cision rule. Holmström and Myerson prove the existence of decision rulesthat are both durable and incentive ecient. Implementation in Bayesianequilibrium has already received a great deal of attention (see Palfrey, 1992),but we introduce an additional requirement here: the core property.

The theory of information aggregation in committees has traditionally be-longed to the realm of Condorcet's Jury Theorem (1785). According to thistheorem, increasing the number of informed committee members raises theprobability of selecting the best of two alternatives when there is uncertaintyabout which of the two alternatives is in fact preferred. This conclusionis nevertheless based on simplifying assumptions. Austen-Smith and Banks(1996) rst questioned the implicit assumption of sincere voting, showingthat it can easily fail to be true in a Nash equilibrium. There is actually alarge body of game-theoretic literature that extends Condorcet's model tosituations in which voters have heterogenous preferences and may misrepre-sent their private information. We refer the reader to Gerling et al.(2003)for an exhaustive review of this literature. Chwe (1999) takes an approachsimilar to the approach in the present paper, focusing on the properties of

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decision rules used in committees. He studies an information aggregationframework à la Condorcet, in which voters report their private informationto a central computer, which chooses an alternative according to some in-centive compatible decision rule. He shows that the decision rule preferredby the majority may allow the minority to enforce its favored alternativeeven when overruled by a majority. A maintained assumption of almost allof those papers is that the set of alternatives facing the committee containsonly two elements. But there are many situations in which this assumptionappears to be inappropriate. In this paper, we consider information aggrega-tion and committee decision making when the set of feasible policies is somesubset of Euclidean space.

Most previous research on core existence in the presence of informationalasymmetries has focused on pure exchange economies. This literature, sur-veyed by Forges, Minelli, and Vohra (2002), presents conditions for incentivecompatible cores to be nonempty. Although a pure exchange economy anda spatial committee game are two very dierent settings, we should explainour paper's relationship to McLean' and Postlewaite's (2003) which uses themost closely related framework. We will use the notation and terminology ofthat paper whenever possible. They consider a complete-information Arrow-Debreu economy in which they introduce private signals. They show thatthe incentive compatible core of the economy with incomplete information isnonempty whenever the two following conditions hold: (i) the strict core ofthe underlying complete-information economy is nonempty, and (ii) all agentsare informationally small. With regards to the rst condition, our rst exis-tence result also makes use of the core of the underlying complete-informationsimple game. While the strict core of the Arrow-Debreu economy is shownto be generically nonempty, the generic emptiness of the core of non-collegialsimple games leads us to extend our results to other solution concepts 3. Anagent is said to be informationally small if, given the other agents' informa-tion, it is very likely that the given agent's information will have a smalleect on the probability distribution over states. There is a close relation-ship between the weakness of signals' informativeness and the informationalsize. Suppose that, in a simple game with incomplete-information, signalsare weakly informative for all winning coalitions. This implies that the sig-nals received by the members of the grand coalition are weakly informative,and then have a small eect on the probability distribution over the truestate. However, the two concepts play quite dierent roles in their respective

3In a framework similar to McLean and Postlewaite (2003), Krasa and Shafer (2001)show that every core allocation of the underlying complete-information economy is thelimit point of incentive compatible core allocations when the prior over the signals andthe true state converge to the complete information prior.

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models.The paper is organized as follows. In Section 2, we present the basic

notations and dene the incentive compatible core of a simple game. InSection 3, we examine the properties of durable decision rules, and establishexistence results. Finally, Section 4 is devoted to some open problems andconcluding remarks. The proofs of theoretical results are relegated to theAppendix.

2 Notation and Denitions2.1 Decision Rules and Simple GamesLet N = 1, 2, ..., n, denote the set of committee members (or voters),indexed by i, and let N ≡ 2N \ ∅. Each member of N is referred to as acoalition. Every voter i ∈ N has preferences over the set of feasible policiesX ⊆ Rk, assumed to be compact and convex. These preferences depend onthe state of the world in which the policies are implemented, but this stateis still unknown when a policy is chosen. Letting Ω be the nite state space,we posit that individual i's preferences are represented by a utility functionui : Ω × X → R+. For each x ∈ X, ω ∈ Ω, ui(ω, x) represents the utilityagent i derives from policy x in state ω. For each i ∈ N , ui is assumed to becontinuous and strictly concave. For expositional clarity, we assume that:

• min ui(ω, x) : i ∈ N, x ∈ X,ω ∈ Ω ≥ 0,

• max ui(ω, x) : i ∈ N, x ∈ X, ω ∈ Ω = 1,

which is simply a matter of normalization.Although voters do not observe ω when the policy is chosen, they each

receive a private signal on the possible realizations of ω. Thus, every individ-ual i ∈ N receives a signal ti ∈ Ti, where Ti is a nite set. For each nonemptyS ⊆ N , let TS ≡

∏i∈S Ti. A typical element of TS is denoted tS. We will

write TN simply as T .We also introduce the following notation for the set of probability distri-

butions over a nite set A:

∆A ≡ π ∈ R|A| | π(a) ≥ 0,∑a∈A

π(a) = 1.

where |A| denotes the cardinality of A.Common prior beliefs over ω are represented by σ ∈ ∆Ω. We assume that

individuals' private signals t = (t1, . . . , tn) are stochastically related to ω in a

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manner described by a function p : Ω×T → R+, which satises p(ω, ·) ∈ ∆T

for each ω ∈ Ω. In words, p(ω, t) denotes the probability of receiving signalst when the true state of the world is ω. The probability of receiving t is thendened as

p(t) ≡∑ω∈Ω

σ(ω)p(ω, t).

With the terminology of information theory, we say that (T, p) is an infor-mation service for Ω.

A policy in X is chosen through some decision rule at the interim stage(the stage at which each voter i knows only her own private informationti). A decision rule, or mechanism, is dened as a function f : T 7−→ X.This must be interpreted as follows: f(t) ∈ X is the policy chosen whenindividuals report the prole of signals t = (t1, . . . , tn). We denote by XT

the set of functions from T to X. Agent i's ex ante preferences on XT canbe represented by a utility function Ui dened as:

Ui(f) ≡∑ω∈Ω

∑t∈T

ui(ω, f(t))σ(ω)p(ω, t),

for all f ∈ XT . If f ∈ XT prescribes the same policy in all states of the world,that is, if f(t) = x for any t ∈ T , then it is called constant and we writef = x. Also, with an abuse of notation, we shall adopt throughout the paperthe convention of using the symbol Y instead of

f ∈ XT : f = y, y ∈ Y

.

At the ex ante stage, a simple game is played to choose the decision ruleused at the interim stage. To describe this game, we have to dene the setof winning coalitions W ⊆ N , which satises the following monotonicitycondition: if S ∈ W and S ⊂ S ′, then S ′ ∈ W . A coalition S ∈ N isa winning coalition if, for any pair of decision rules (f, g) ∈ XT × XT , fdefeats g in a pairwise vote whenever all members of S vote for f against g4. Under voting by quota q, for instance, all coalitions S with |S| ≥ q arewinning. The pair (N,W) constitutes a committee and we refer to the gamewith incomplete information as Γ(T,p) ≡ ((N,W), X, uii∈N , Ω, σ, (T, p)).

2.2 Incentive Compatibility and the CoreIn presenting the cooperative framework we adopt in this paper, we needthe notions of incentive compatible agreements and core that were used, forexample, in McLean and Postlewaite (2003). We will recall these concepts

4Such a denition assumes that we restrict our attention to proper voting games, thatis: if S ∈ W, then N \ S /∈ W. It would not make much sense to allow separate winningcoalitions to take dierent alternatives.

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below, but refer the reader to that paper for an in-depth discussion of theconcepts.

Voters are assumed to be able to form coalitions at the ex ante stage, thatis, before receiving their private signals 5. A coalition is an agreement amongseveral voters to coordinate their voting behavior. Why should they do so?Suppose f ∈ XT is expected to be the decision rule. If some coalition S ofvoters is able to induce a dierent rule g ∈ XT and Ui(g) > Ui(f) , for alli ∈ S, then it is natural to expect these voters to coalesce so as to induce g.In such a situation, we say that coalition S ex ante blocks f .

Blocking in the presence of incomplete information is however a moreproblematic concept. Indeed, the set of decision rules over which voterscan vote contains an important restriction. The signal received by eachvoter being private, the decision rules must be incentive compatible (IC forshort); that is, they must be designed so as to induce each voter to truthfullyreveal her own signal to the other members of the committee. Formally, letpσ

i (·|ti) ∈ ∆Ω×T−ibe the conditional probabilities that would be obtained by

Bayes' rule given ti ∈ Ti. pσi (ω, t−i|ti) is interpreted as the probability for

voter i that the true state is ω and the others receive signals t−i when shereceives signal ti. Hence, a decision rule g ∈ XT is incentive compatible if:∑ω∈Ω

∑t−i∈T−i

ui (ω, g(t−i, ti)) pσi (ω, t−i|ti) ≥

∑ω∈Ω

∑t−i∈T−i

ui (ω, g (t−i, t′i)) pσ

i (ω, t−i|ti)

for each ti, t′i ∈ Ti and i ∈ N .

We denote by A∗(T, p) the set of incentive compatible policies when thecommittee has access to an information service (T, p) 6. Furthermore, wedenote by E∗(T, p) the set of incentive ecient policies:

E∗(T, p) ≡ f ∈ A∗(T, p) | @g ∈ A∗(T, p) : Ui(g) > Ui(f),∀i ∈ N.

Thus a coalition S can ex ante incentive compatible block (ICX-block) adecision rule f if the two following requirements are satised : rst, S is awinning coalition and, consequently, it is able to induce any g ∈ A∗(T, p) asthe policy outcome irrespective of the other voters' behavior; second, thereexists a policy g ∈ A∗(T, p) such that all members of S ex ante prefer g to f .

With these observations in mind, we are now in a position to dene thecore of a simple game with incomplete information.

5See Section 4 for a discussion of this assumption.6We use ∗ to indicate incentive compatibility, as most authors do.

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Denition 1 Suppose the committee (N,W) has access to an informationservice (T, p). The ex ante incentive compatible core of of the committeegame is dened as

C∗ (T, p) ≡⋂

S∈Wf ∈ A∗(T, p) | @g ∈ A∗(T, p) : Ui(g) > Ui(f),∀i ∈ S.

Elements of C∗ (T, p) are said to be durable in (N,W).

In words, rule f is durable if it cannot be ICX-blocked by any winningcoalition.

A word of caution may be in order before we proceed any further. In thecontext of an exchange economy, the blocking opportunities of a coalition Swould be restricted to decision rules that depend only on what its membersobserve at the interim stage: their own signals tS. As we mentioned earlier,winning coalitions are all-powerful in a committee, so that they can enforceany decision rule that is incentive compatible.

2.3 The Complete-Information GameBefore getting to the core of the matter, we need to introduce an auxiliaryvoting game which will play an important role in what follows. To a simplegame with incomplete information Γ(T,p) we associate a complete-informationsimple game in which voters do not receive any information about the truestate ω. The complete-information simple game induced by Γ(T,p) is thusdened by ∂Γ ≡ (N,W , X, uii∈N) where:

ui(x) ≡∑ω∈Ω

ui(ω, x)σ(ω).

The core of ∂Γ will be denoted by C.The core seems to be a natural concept to predict the outcome of spatial

committee games. As we mentioned above, most non-collegial voting gameswith multidimensional policy spaces have however an empty core. Politicalscientists then resort to other equilibrium concepts to predict the outcome ofthose games. Two of them will play an important role in what follows. Wedevote the rest of this subsection to a brief reminder of their denition.

First, the set of Von Neumann-Morgenstern stable sets of ∂Γ, V , is denedas follows: a set V ⊆ X is a stable set if it satises both internal stability (nopolicy in V can be blocked via another policy in V ) and external stability(any policy in X \V is blocked via some policy in V ). Unlike the core, stablesets have no obvious interpretation in terms of voting. They are, however,

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useful in deriving conditions that are sucient for a nonempty IC core, aswe shall see in the next section. In general, existence of stable sets is nota major diculty, for most spatial games have stable sets, and even manystable sets (Ordeshook, 1986).

The last equilibrium concept we use for the auxiliary game is related toelectoral competition. Consider the standard symmetric, zero-sum game inwhich two oce-motivated candidates compete in an election. Their respec-tive payos are

π(x, y) =

1 if ∃S ∈ W : ui(x) > ui(y),∀i ∈ S,−1 if ∃T ∈ W : ui(y) > ui(x),∀i ∈ T,0 otherwise,

and −π(x, y), where x and y in X are the policies they announce to thecommittee (N,W). Dene B as the set of supports of mixed-strategy Nashequilibria of this non-cooperative game. Banks, Duggan and Le Breton (1999,2002) oer a thorough discussion of this solution.

The complete-information game induced by Γ(T,p) must be thought of asthe cooperative game that would be played by the committee if voters did notreceive private signals about ω. It is then important to predict the possibleoutcomes of that game so as to investigate the impact of private informationin Γ(T,p). For future reference, let E ≡ (2C \∅)∪V∪B∪X. In what follows,we call any element of E a choice set of ∂Γ in the sense that such a set maybe used to predict the possible choices of the committee in ∂Γ.

3 Durable Decision RulesWe now move to the analysis of durable decision rules. Notice that if sig-nals were publicly observable we would have a standard simple game with a(k|T |)-dimensional policy space. As we mentioned earlier, a famed result ofvoting theory states that the core of simple games with high-dimension issuespaces is generically empty. Does the same conclusion apply to simple gameswith incomplete information ? Before going on to formal results, it is worthemphasizing that core existence in collegial committees is not a major issue.Indeed in such committees, some coalition of voters, the collegium, belongsto all winning coalitions. The core is then nonempty and comprises the setof IC decision rules that cannot be Pareto-improved by the collegium. Toavoid such trivialities, hereafter assume the committee is not collegial.

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3.1 Signals' InformativenessMost of our results depend on the informativeness of the signals received bycommittee members. Let (Tp, p) and (Tq, q) be two information services forΩ. (Tp, p) is said to be more informative than (Tq, q) if, and only if, thereexists a map θ : Tp × Tq → R+ such that

(i)∑

τ∈Tqθ(t, τ) = 1 for each t ∈ Tp, and

(ii) q(ω, t) =∑

τ∈Tpp(ω, τ)θ(τ, t) for each (ω, t) ∈ Ω× Tq.

An information service (T, p) for Ω is uninformative if p(ω, t) = p(ω′, t) forall (ω, ω′) ∈ Ω2, and all t ∈ T .

For our purposes, it will turn out that a useful measure of the informa-tiveness of the signals received by committee members when they have accessto an information service (T, p) is:

ψ(T, p) ≡∑ω∈Ω

σ(ω)

[∑t∈T

|p(t)− p(ω, t)|]

.

Let (T,p) and (T ′, p′) be two information services for Ω. If (T, p) is moreinformative than (T ′, p′), then ψ(T, p) ≥ ψ(T ′, p′). And if ψ(T, p) ≥ ψ(T ′, p′),then (T, p) is not less informative then (T ′, p′).

3.2 Existence of Durable Decision RulesThe aim of this section is to derive conditions that guarantee the existenceof durable decision rules. Sucient conditions are provided in the two sub-sections immediately following. The principle approach for guaranteeing theexistence of such rules will be to identify information services with whichcommittee decision making allows a well-dened collective choice.

3.2.1 Constant Durable Rules and the Core of the Complete-Information Game

Our rst step is to relate the IC core of Γ(T,p) to core points of the cor-responding complete-information simple game ∂Γ. More precisely, we askunder what conditions core points of ∂Γ are also members of the IC core ofΓ(T,p) as constant decision rules. By denition, core points of ∂Γ cannot beblocked as long as voters do not receive signals. The question is then: Underwhat conditions are they robust to the introduction of incomplete informa-tion in the game? The next denition suggests candidates to this sort ofrobustness.

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Condition (R) : Let (T, p0) be an uninformative information service. Apolicy x ∈ X satises condition (R) if there is no f ∈ A∗(T, p0) \ Xsuch that

∑t∈T p0(t)f(t) = x.

The intuitive idea behind Condition (R) can be put as follows: Supposethe committee has access to an information service (T, p), and consider somewinning coalition's opportunity of ICX-blocking a core point x that satisesthis condition. This coalition, say S, must use a non-constant decision rulef ∈ A∗(T, p). Even if the constant rule δf ≡

∑t∈T p(t)f(t) cannot by deni-

tion be used by S to block x, f might be. However, the eventual informationgain of using f rather than δf is small whenever p is close to p0, for thislatter is uninformative. Since δf 6= x by Condition (R), there must be anindividual in S who prefer x to δf , and consequently to f whenever p isweakly informative. Thus, x is robust in the sense that it remains in the coredespite the introduction of weakly informative information. This intuition isconrmed by the next result.

Theorem 1 If x ∈ C satises condition (R), then there exist P ⊂ RΩ×T

and ℘ ∈ RΩ×T such that:(i) x ∈ C∗ (T, p) whenever p ∈ P , and(ii) p /∈ P whenever (T, p) is more informative than (T, ℘).

Thus, if members of a committee agree on a policy that satises Condition(R) in the absence of information, there exist information services that donot alter the collective choice of that committee. The following exampleillustrates this for simple majority rule.

Example 1: A Simple Majority Game

To facilitate the exposition, we consider a majority game that involves onevoter receiving information, say voter 1, and totally uninformed voters. Theinformational structure is described in Table 1 7. In this table, π and ρ are twoparameters in (0, 1). The parameter ρ is a measure of the correlation betweenω and t. It can then be interpreted as a measure of signals' informativenessabout the true state of the world. All information contained in this table iscommon knowledge.

Let n = 3, and let voters' preferences be represented by

ui(x, θ) = 1− 1

2(x− iθ)2, i = 1, 2, 3,

7This table is borrowed from Bhaskar and van Damme (2002).

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Prob(t = t1) Prob(t = t2)∑

Prob(ω = ω1) (1− π)2 + ρπ(1− π) (1− ρ)π(1− π) 1− πProb(ω = ω2) (1− ρ)π(1− π) π2 + ρπ(1− π π∑

1− π π 1

Table 1: Example 1

where x ∈ X = [0, 1], and ω ∈ Ω = 0, 1/2. The unique core pointof the complete-information game is π = arg maxx∈X u2(x). Some tediouscomputations show that, when signals are uninformative (ρ = 0), a non-constant IC decision rule f must satisfy: f(t1) + f(t2) = π. This impliesthat (1 − π)f(t1) + πf(t2) < π, so that π satises condition (R). Moreover,one can show that, for any π ∈ (0, 1), there exist ρπ ∈ [0, 1) such that theconstant decision rule g, dened as g(t1) = g(t2) = π, is durable if and onlyif ρ ≤ ρπ.

¤

Theorem 1 provides sucient conditions under which the incentive com-patible core is nonempty. We could raise, however, an immediate objectionto this seemingly positive result: Although the decision rules described inthis theorem are durably implementable in the committee, they are obvi-ously not attractive because they do not use the information received bycommittee members. In some sense, information is irrelevant. This lastobservation prompts the following question: What decision rules can be re-garded as attractive for the committee? This is the subject matter of thenext subsection.

3.2.2 Pareto-Improving Information ServicesIn this subsection, we focus on the special, but much relevant, class of en-vironments in which information is Pareto-improving. Let us begin withclarifying what we mean by Pareto-improving information service.

Denition 2 Let Y be a nonempty subset of X. We say that an informationservice (T, p) is Pareto-improving relative to Y if there exists a decision rulef ∈ A∗(T, p) such that, for all y ∈ Y , Ui(f) ≥ Ui(y) for every i ∈ S with atleast one strict inequality.

Denition 2 says that an information service is Pareto-improving relativeto some choice set Y if, when only policies in Y are expected to be selectedin the absence of information, it gives the committee the opportunity of

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adopting an IC decision rule that makes some voters strictly better o, andno voters worse o. By doing so, this information service also raises thedimension of the policy space, and then makes it more dicult for committeemembers to reach an agreement. This fact motivates the next result.

Theorem 2 Suppose X \ C is not dense in X. There exists η > 0 suchthat the incentive compatible core of Γ(T,p) is nonempty whenever (T, p) isPareto-improving relative to C and ψ(T, p) ≤ η. In such a case, the set ofincentive ecient decision rules that Pareto-improve C are durable.

Besides insuring existence of durable decision rules in a committee gamewith incomplete information, this theorem oers a partial characterizationof the IC core when signals are weakly informative. In particular, it showsthat it may be the case that committee members agree on a decision rulethat raises the ex ante expected utility of its members relative to what theycould expect in the absence of information service. It can also be viewed asproviding a prescription of what sorts of informational structures should beused if one is concerned about the existence of IC core outcomes.

Theorem 2 does not extend to situations in which the game ∂Γ has nocore point. With this as motivation, we now look for a more general ex-istence result. The next theorem is more delicate than the rst, and itsvalidity requires an additional assumption: When blocking a decision rule,each member of the blocking coalition must incur an innitesimal cost ε.Formally, a coalition S ex ante IC ε-blocks a decision rule f if there existsg ∈ A∗(T, p) such that Ui(g) − ε > Ui(f) for every i ∈ S. This naturallyallows us to dene the ex ante IC ε-core of the committee game, C∗

ε (T, p), asthe set of IC decision rules that cannot be ICX-ε-blocked.

We are now ready to establish our next result which is easily seen to bea generalization of Theorem 2.

Theorem 3 For any ε > 0, the incentive compatible ε-core of Γ(T,p) isnonempty whenever information is Pareto-improving relative to some Y ∈ Eand ψ(T, p) ≤ ε. In such a case,

f ∈ E∗(T, p) : Ui(f) > Ui(y),∀y ∈ Y, ∀i ∈ N ⊆ C∗ε (T, p).

Theorem 3 thus gives sucient conditions analogous to those identiedin Theorem 2. Although these theorems evade the restriction of Theorem 1while still establishing the existence of durable decision rules under the weak-informativeness condition, they do so by imposing a new condition: informa-tion must be Pareto-improving relative to some choice set. This naturally

14

raises the question of whether an information service may be both Pareto-improving and weakly informative, thereby guaranteeing from Theorems 2and 3 the existence of durable decision rules. These seemingly contradic-tory conditions can be rendered consistent by suciently congruent voters'interests.

In the remainder of this section, we provide an example that that bothillustrates the above result and shows that an information service can indeedbe both weakly informative and Pareto-improving.

Example 2: A Majority Game with Converging Preferences

We consider a majority game with a two-dimensional policy space, X ⊂ R2.There are three voters (n = 3) with the following Euclidean preferences overX:

u1(ω, x, y) = 1− 1

2(x− x(ω))2 − 1

2(y − y(ω))2 ,

um2 (ω, x, y) = 1− 1

2

[x−

(x(ω) +

1

m

)]2

− 1

2(y − y(ω))2 ,

um3 (ω, x, y) = 1− 1

2

[x−

(x(ω) +

1

2m

)]2

− 1

2

[y −

(y(ω) +

√3

2m

)]2

,

where m ∈ N. Note that um2 and um

3 both converge to u1 uniformly. Thismeans that voters' preferences get closer as m increases. However, for allm ∈ N, the core of the corresponding complete-information game is empty.

Let z = (1−π) (x(ω1), y(ω1))+π (x(ω2), y(ω2)), and let V m = zm1 , zm

2 , zm3 ,

where zm1 = z + (1/2m, 0), zm

2 = z + (1/4m,√

3/4m), and zm3 = z +

(3/4m,√

3/4m) are policies in X. One can check that V m is a Von Neumann-Morgenstern stable set of the complete-information game for every m ∈ N .

Suppose that voter 1 has access to some information service described byTable 1 above. Let f ∗ : t1, t2 → X be dened as:

f ∗(t) ≡

((1− π)2 + ρπ(1− π), (1− ρ)π(1− π)) · (x(ω1), x(ω2)) if t = t1,((1− ρ)π(1− π), π2 + ρπ(1− π)) · (x(ω1), x(ω2)) if t = t2.

One can show the following

Observation 1 For any π ∈ (0, 1), there exists a decreasing function M :(0, 1) → N, such that

i) Ui (f∗) > Ui

(zm

j

), i = 1, 2, 3, j = 1, 2, 3, and

ii) f ∗ is an ε-durable decision rule,whenever m ≥ M(ρ).

15

An immediate consequence of the above result is that the IC approximatecore may be nonempty even when the complete-information game has nocore point. Of particular importance here is this: When voters' interests arecongruent enough, there must exist an ε-durable rule that is strictly preferredby all voters to any policy in the stable set V . If policies in V are the onlyexpected outcomes of the vote when voters do not receive signals, informationis well Pareto-improving in the sense that it allows for an IC decision rulewhich makes all voters strictly better-o. Note that information may bePareto-improving even if it is weakly informative, for the above result holdsfor any ρ > 0. However, when ρ is low, preferences have to be more congruentfor this result to still hold. These conclusions are fully consistent with thoseof Theorems 2 and 3 above.

¤

3.3 Risk Aversion and Signals' InformativenessWe nally state an important property of IC core decision rules. It impliesthat, under an IC core rule, the risk borne ex ante by winning coalitions isrequired to be low relative to the degree of informativeness of voters' signals.The reason is obvious: the only benet from using a state-contingent rulef is that it allows voters to adapt their voting behavior to the informationthey receive; if signals are little informative, however, this benet is smalland is outweighed by the cost of taking a risk ex ante. Due to the concavityof utility functions in each state of the world, the state-independent decisionrule

∑t p(t)f(t), which is by denition IC for all winning coalitions, becomes

indeed a serious blocking opportunity for risk-averse voters.To examine this issue formally, we need a number of denitions. A state-

contingent policy f is said to be measurable with respect to the informationavailable to S if

f(t) = f(t′),∀t, t′ ∈ T : tS = t′S.

We denote by XTS the set of policies measurable with respect to the infor-mation available to S ∈ N . For clarity, we often write f(tS) rather than f(t)whenever f ∈ XTS .

Two further denitions will be useful in the sequel. First, for any f ∈ XT

and S ∈ N , we dene the ex ante risk-aversion cost of f for coalition S as

r(S, f) ≡ mini∈S

∑ω∈Ω

σ(ω)

[ui

(ω,

∑t∈T

p(t)f(t)

)−

∑t∈T

ui(ω, f(t))p(t)

].

16

Second, we dene the following index for the informativeness of the signalsreceived by a coalition S as

ψS(T, p) ≡∑ω∈Ω

σ(ω)

[ ∑tS∈TS

|pS(tS)− pS(ω, tS)|]

,

where pS(ω, ·) denotes the conditional distribution on TS given ω ∈ Ω, andpS(·) the distribution on TS induced by p.

The following theorem establishes a necessary condition for an IC decisionrule to belong to the IC core of Γ(T,p).

Theorem 4 Let f be measurable with respect to the information available tocoalition S. If f ∈ C∗(Γ), then maxS∈W r(S, f) ≤ ψS(T, p).

Proof: Let f ∈ C∗(Γ), and suppose there exists S ∈ W such thatr(S, f) > ψS(T, p). Then, for each i ∈ S, the utility gain, ∆ui, followinga switch from policy f to policy δf ≡

∑t∈T p(t)f(t) satises

∆ui =∑ω∈Ω

σ(ω)

[ui(ω, δf )−

∑tS∈TS

ui(ω, f(tS))pS(ω, tS)

]

=∑ω∈Ω

σ(ω)

[ui(ω, δf )−

∑tS∈TS

ui(ω, f(tS))pS(ω, tS)

]

+∑ω∈Ω

∑tS∈TS

ui(ω, f(tS))σ(ω) [pS(tS)− pS(ω, tS)]

≥ r(S, f)− ψS(T, p) > 0.

The last inequality implies that coalition S can ICX-block f with δf ; acontradiction.

¤

As we expected, a durable decision rule requires that the risk-aversioncost borne by winning coalitions be limited relative to the informativeness ofvoters' signals. Note that for a decision rule measurable with respect to in-formation available to some coalition S, we only consider the informativenessof signals received by members of that coalition.

The intuition of this result is clearer when we assume that all votershave Euclidean preferences with state-dependent bliss points: ui(ω, x) =1 − (x− xi(ω))2, where xi is a function from Ω to X, for every i ∈ N . In

17

this case, the ex ante risk-aversion cost of a given f ∈ XT is the same for allvoters, and is equal to the variance of f . That is,

r(S, f) = var(f),∀S ∈ N ,

where var(f) ≡ ∑t∈T

[(f(t)−∑

t∈T p(t)f(t)) · (f(t)−∑t∈T p(t)f(t)

]p(t).

Thus, a direct implication of Theorem 3 is that the variance of durable deci-sion rules cannot exceed signals' informativeness.

4 DiscussionThis paper makes four main points. First, we argue that a decision ruleis durably implementable in a committee only if it has the IC core prop-erty. Second, core points of the complete-information game may be constant,durable decision rules of the committee game with incomplete informationif they are robust in the sense of Condition (R). Third, if information isPareto-improving relative to some choice set of the corresponding complete-information game, the IC core is nonempty whenever voters' private signalsare weakly informative. In this case, the IC core comprises all incentive e-cient rules that Pareto-improve the choice set under consideration. Finally,durable decision rules have a notable property: the ex ante risk borne by vot-ers at a durable decision rule is limited relative to signals' informativeness.

But the arguments we employ are limited in that we do not provide atighter characterization of durable decision rules. If we are to say somethingmore about their properties, we must introduce additional assumptions. Forinstance, we could posit a specic functional form for the ui's, or focus on aparticular set of winning coalitions. Conceptually, the assurance of existenceof outcomes nevertheless takes precedence over the characterization of theseoutcomes. With regards to further exploration of the non-emptiness of theex ante IC core, allowing coalitions to randomize over state-dependent rulesmight be a relevant approach. In the context of an exchange economy, Allen(2003) establishes by doing so the non-emptiness of the modied IC corewithout any restriction on private signals' informativeness.

Another aspect of our analysis to which attention should be drawn isthat winning coalitions block decision rules at the ex ante stage, and not atthe interim stage. Some papers have studied the interim IC core in settingsdierent from voting. In particular, Bahçeci (2003) analyzes the interim ICcore of a pure exchange economy in which agents receive noisy signals asin the present framework. He shows that the interim IC core is nonemptywhen the strict cores of the underlying ex post economies are nonempty. We

18

regard his ndings as a useful direction for follow-up work on the analysis ofcommittees with asymmetric information.

Finally, we should note that our existence results are related to socialchoice theory. Indeed, if we think of a simple game with incomplete infor-mation as a collective choice rule, the non-emptiness of the game's core isequivalent to the acyclicity of social preference over decision rules.

AppendixThroughout this appendix, we use the following notation:

δ(T,p)(f) ≡∑t∈T

p(t)f(t),

for any information service (T, p). Moreover, for any set A, we denote by Athe closure of A. If A is a subset of the k-dimensional Euclidean space, thenNε(a) stands for the ε-neighborhood of a ∈ A in Rk.

Proof of Theorem 1For notational ease, we dene δ : RΩ×T ⇒ X as

δ(p) ≡ δ(T,p) (A∗(T, p) \X) ,

for any information service (T, p).We begin with establishing a useful Lemma.

Lemma 1 Suppose x is a core point of ∂Γ that satises Condition (R). Thenthere exists P ⊂ RΩ×T and b : RΩ×T → R such that

• b(p) > ψ(T, p) ≥ 0, and

• if S ∈ W, @y ∈ δ(p) : ui(y) > ui(x)− b(p), ∀i ∈ S,

for any p ∈ P .

Proof: The map δ(T,p) can be seen as a projection of vectors in RT ontothe diagonal of RT . Since A∗(T, p0) is compact, the only elements on theboundary of A∗(T, p0) \X that do not belong to A∗(T, p0) \X are constantrules that do not satisfy Condition (R). This implies that x /∈ δ(p0). There-fore, there exists, by continuity of δ, an ε1 > 0 such that x /∈ δ(p) for allp ∈ Nε1(p0).

19

Now, for all S ∈ W , dene

αS(y) ≡ maxi∈S

[ui(x)− ui(y)] , and

αS(p) ≡ miny∈δ(p)

αS(y)

which is continuous in p by the Maximum Theorem.Since x is a core point of ∂Γ, the two following conditions must hold for

all S ∈ W , and all p ∈ Nε1(p0): αS(p) > 0, and

@y ∈ δ(p) : ui(y) > ui(x)− αS(p),∀i ∈ S.

Next, dene b(p) ≡ minS∈W αS(p) > 0 for all p ∈ Nε1(p0). By denition,we have b(p0) > 0 = ψ(T, p0). Since b(p) − ψ(T, p) is continuous in p, thereexists ε2 ∈ (0, ε1] such that b(p) > ψ(T, p) for all p ∈ Nε2(p0). Dening P asthe ε2-neighborhood of p0, we obtain the Lemma.

¤

Now, suppose contrary to the statement of the theorem, that there is awinning coalition S and a policy g ∈ A∗ such that:

∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω)p(ω, t) > ui(x)

for all i ∈ S. This implies that g ∈ A∗ \X. Then, using Jensen's inequality,we obtain, for all i ∈ S,

ui

(δ(T,p)(g)

)=

∑ω∈Ω

ui

(ω,

∑t∈T

p(t)g(t)

)σ(ω)

≥∑ω∈Ω

∑t∈T

ui(ω, g(t))p(t)σ(ω)

=∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω)p(ω, t)

+∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω) [p(t)− p(ω, t)]

≥∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω)p(ω, t)− ψ(T, p)

> ui(x)− b(p).

This contradicts Lemma 1, and then x must be a durable decision rule.To see the second part of the theorem, just pick ℘ such that supp∈P ψ(p) =

ψ(℘). Thus, if (T, p) is more informative than (T, ℘), we have ψ(p) > ψ(℘),and then p /∈ P .

20

Proof of Theorem 2To prove the theorem, we shall need the following results.

Lemma 2 Suppose X \C is not dense in X. If x is an interior point of C,then there exists η1 > 0 such that, for all S ∈ W and all y ∈ X \ C,

ui(x)− η1 ≥ ui(y),

for some i ∈ S.

Proof: Let x be an interior point of C, and let B stand for the closureof X \ C. For all S ∈ W , let αS ≡ miny∈B maxi∈S [ui(x)− ui(y)]. Then, bysetting η1 ≡ minS∈W αS, we obtain the lemma.

¤

Lemma 3 Suppose X \C is not dense in X. There exists η2 > 0 such that,for all S ∈ W and all y ∈ C,

ui(y)− η2 ≥ ui(y),

for some y ∈ C and some i ∈ S.

Proof: Let Nε(y) be the ε-neighborhood of y in X. Since X \ C is notdense, there exist ε > 0 and nitely many points θ1, . . . , θK in C such thatK ≥ 2, θk /∈ Nε(θl) when k 6= l, and

C ⊆K⋃

k=1

Nε(θk).

Next, dene βS(θk, x) ≡ maxi∈S [ui(θk)− ui(x)]. By strict concavity ofthe ui's, βS(θk, x) > 0, for all x ∈ C. Hence,

β(θk) ≡ minS∈W

minx∈Nθl

βS(θk, x) > 0.

We obtain the result by setting η2 ≡ min β(θk) : k = 1, . . . , K. Indeed,for any y ∈ C, we know that y belongs to the neighborhood of some θl ∈ C,so that there is θk, k 6= l, such that

ui(θk)− η2 ≥ ui(θk)− β(θk) ≥ ui(y).

¤

21

We now prove Theorem 2. To do so, we set η ≡ minη1, η2.Let x be an interior point of C. Since information is Pareto-improving

relative to C, there exists a signal-contingent decision rule f ∈ E∗(T, p) suchthat ∑

ω∈Ω

∑t∈T

ui(ω, f(t))σ(ω)p(ω, t) ≥ ui(x),

for every i ∈ N . Let us show that f is durable.Suppose there exists S ∈ W and g ∈ A∗(T, p) such that Ui(g) > Ui(f)

for every i ∈ S. Under the conditions of the theorem, we then have

ui

(δ(T,p)(g)

)> Ui(g)− ψ(T, p) ≥ Ui(g)− η > Ui(f)− η,

for every i ∈ S. This last inequality implies that δ(T,p)(g) /∈ C. To see this,suppose that δ(T,p)(g) ∈ C. Since f Pareto-improves C, we have

ui

(δ(T,p)(g)

)> Ui(f)− η ≥ ui

(δ(T,p)(g)

)− η,

for every i ∈ S. But this contradicts Lemma 3.As a consequence, δ(T,p)(g) /∈ C and

ui

(δ(T,p)(g)

)> Ui(f)− η ≥ ui(x)− η,

for every i ∈ S. This in turn contradicts Lemma 2.

Proof of Theorem 3By assumption, there exists f ∈ A∗(T, p) such that Ui(f) > Ui(y), for anyy ∈ Y . Since A∗(T, p) is compact and the ui's are all continuous, there existsa state-contingent decision rule f ∈ E∗(T, p) such that

∑ω∈Ω

∑t∈T

ui(ω, f(t))σ(ω)p(ω, t) ≥∑ω∈Ω

∑t∈T

ui(ω, f(t))σ(ω)p(ω, t), (1)

for every i ∈ N .Now, suppose there is a winning coalition S 6= N and a decision rule

g ∈ A∗(T, p) such that:∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω)p(ω, t)− ε >∑ω∈Ω

∑t∈T

ui(ω, f(t))σ(ω)p(ω, t), (2)

for all i ∈ S.

22

We thus obtain under the conditions of the theorem:

ui

(δ(T,p)(g)

) ≥∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω)p(ω, t)

+∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω) [p(t)− p(ω, t)]

≥∑ω∈Ω

∑t∈T

ui(ω, g(t))p(ω, t)− ψ(T, p)

≥∑ω∈Ω

∑t∈T

ui(ω, g(t))p(ω, t)− ε. (3)

Combining inequalities (1), (2), and (3), we obtain, for every i ∈ S andall y ∈ V ,

ui

(δ(T,p)(g)

) ≥∑ω∈Ω

∑t∈T

ui(ω, g(t))σ(ω)p(ω, t)− ε

>∑ω∈Ω

∑t∈T

ui(ω, f(t))σ(ω)p(ω, t)

≥ ui(y).

Therefore, δ(T,p)(g) defeats any policy in the choice set Y , a contradic-tion. Suppose indeed that Y is a Von Neumann-Morgenstern stable set. Ifδ(T,p)(g) ∈ Y , the internal stability condition is violated. If δ(T,p)(g) /∈ Y , theexternal stability condition is violated. The reasoning is similar if Y ⊆ C,Y ∈ B, or Y = X, instead.

¤

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