On the equal division core

On the equal division coreAuthor(s): Anindya BhattacharyaSource: Social Choice and Welfare, Vol. 22, No. 2 (April 2004), pp. 391-399Published by: SpringerStable URL: http://www.jstor.org/stable/41106583 .

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Soc Choice Welfare (2004) 22: 391-399 ~ -

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DOI: 10.1007/s00355-003-0221-2 SOClfll UI01C6 «dWdbrc

© Springer-Verlag 2004

On the equal division core

Anindya Bhattacharya

Department of Economics and Related Studies, The University of York, York YO10 5DD, United Kingdom (e-mail: [email protected])

Received: 31 March 2000/ Accepted: 23 December 2002

Abstract. The Equal Division Core (EDC) of a transferable utility cooperative game (TU game) is the set of efficient pay-off vectors for the grand coalition which are not blocked by the equal division allocation for any sub- coalition. Our objective is to provide an axiomatic characterization of the EDC as a solution of TU games.

1 Introduction

The Equal Division Core (EDC) of a transferable utility cooperative game (TU game) is the set of efficient pay-off vectors for the grand coalition which are not blocked by the equal division allocation for any sub-coalition. In other words, an efficient pay-off vector is in the equal division core if no coalition can divide its value equally among its members and in this way give more to each of the members than they receive in the pay-off vector.

Evidently, it is a solution concept related to the norm of equity. The idea of the EDC was proposed by Selten [13] to explain outcomes of experimental

This is revised version of a part of Chapt. 3 of my Ph.D. dissertation. Beginning with suggesting the problem, my supervisor Bhaskar Dutta extended his unstinting help in every step of this pursuit without which this work could not have been possible. I have also much benefitted from the comments and suggestions from the responsible editor, two anonymous referees, Hervé Moulin and seminar participants at Bilkent and Bonn Universities (especially from the detailed comments of Reinhard Selten). Of course, the shortcomings remaining are mine. Somdeb Lahiri, Anjan Mukherji and T. Yamato have been kind enough to make certain literature accessible to me.

Research fellowships from the Indian Statistical Institute and CNRS, France and hospitality at GEMMA, Universite de Caen are gratefully acknowledged.



392 A. Bhattacharya

cooperative games.1 It has been found that experimental "evidence clearly suggests that equity considerations have a strong influence on observed pay- off division" [14] in a laboratory set-up and the EDC (as defined more gen- erally by Selten) proved to be quite successful as a solution in this context [13]. Selten explains the intuitive motivation behind the solution-notions like the EDC thus:

It is unreasonable to suppose that the experimental subjects perform complicated mathematical operations in an attempt to understand the strategic structure of the situation. It seems plausible to assume that they look for easily accessible cues, such as obvious ordinal power comparisons and equitable shares, in order to form aspiration levels for their payoffs. [14]

Farell and Scotchmer [5] also cite a number of real-life examples where a coalition is observed to share its worth equally.

The EDC has a theoretical justification as well from a quite different angle. It is obtained as one solution of a game satisfying participation con- straints for the coalitions if the norm of egalitarianism is used consistently for the coalitions (see [4] for the details).

Our objective in this paper is to provide an axiomatic characterization of the EDC. The equity-related solutions of cooperative games have been subject to axiomatic analysis for some time - one early and very celebrated contribution being that of Shapley [15]. Equity-based or egalitarian solutions for bargaining problems have been extensively studied (see [17] for a survey). The analysis of the egalitarian core-like solutions for TU games in an axiomatic framework began with Dutta [2] who provided a characterization of the weak egalitarian solution (defined by Dutta and Ray [3]) on the class of convex games. Recent contributions in this area include [1], [6], [7], [10] etc.. Since the concept of EDC was proposed as an ad-hoc solution to explain experimental results, a theoretical justification of this idea from an axiomatic standpoint is a worthwhile exercise.

The plan of the paper is as follows. In Sect. 2 we give certain preliminary definitions and notation. In Sect. 3 we provide the axiomatic characterization of the EDC on the class of all TU games.

2 Preliminary definitions and notation

Let U be a set of potential players that may be finite or infinite. For a set A we shall denote the cardinality of A by 'A'. For any finite subset S of U, by R* we

1 Selten's definition of the EDC is more general than the definition used here- he does not require a pay-off in the EDC to be efficient for the grand coalition and also, in his framework every coalition may not be permissible. The class of games he studied was a proper subclass of all TU games.



Equal division core 393

denote the set of all functions from S to R, the set of real numbers. We would think of elements of Rs as 'S' -dimensional vectors whose coordinates are indexed by the members of 5.

Definition 1. A Transferable Utility Cooperative Game (TU game) is a pair (jV, v) where N is a finite subset of U and v is a function that associates a real number v(S) with each subset S of N. We assume t;(0) = 0.

A non-empty subset of N is called a coalition. Henceforth, we shall often simply use game for a TU game with no possibility of confusion.

Given a game {N,v), we use the following notation. The set of efficient pay-off vectors for the grand coalition, X(N,v) = {x e RN' HieNxi = v(N)i- Fot SCN,S¿ 0,fl(5,u) = v(S)/'Sl the average worth of the coalition S. For S Ç N, S / 0, the equal division allocation for 5, e(S, v)9 is the vector x e Rs such that jc, = a(S, v) for all i e S. Given two vectors x,y € R", x » y means that jc, > y¡ ? for / = 1, . . . n. Given a vector x € RN we denote the restriction of x on a coalition S by xs. The set of TU games is denoted by F. We use the notation c for indicating the proper set inclusion.

Definition 2. For a game (N, r), the Equal Division Core (EDC) of (N, v),L(N, v) = {xe X(N, v)' there is no coalition S for which e(S, v) > xs}.

We also include the following definition for later use.

Definition 3. Given a game (N, v) we call a coalition S an Equity Coalition of (N,v)if

a(S, v) > a(T, v) for all T C S.

3 Characterization of the EDC on F

First, we define the notion of a solution on F.

Definition 4. A solution on F is a mapping a which associates with each game (N, v) e r a subset a(N, v) ofX(N, v).

In this section we show that the EDC is the unique solution on F that satisfies a certain set of properties.

We begin with a property akin to continuity. Continuity is quite a desirable feature for a solution. Intuitively, if two games are quite close to each other (in a certain sense) then the solutions should also be close to each other. We introduce the following axiom.



394 A. Bhattacharya

Weak upper hemicontinuity (WUHC). Let {{N,if)} be a sequence of games such that V*, ¿(S) = v{S) for S c N and vfi(N) - ► v(N). Let {**} be a sequence such that jc* G a(W, i/) for all k and x* - ► x. Then jc € a(W, v).

This axiom states that if two games in F, {N,v]) and {N,v2), are close enough - the differences in the worth of the grand coalitions being small and the worth of the other coalitions being equal - then an allocation in the solution of (Ni vl ) will be close enough to one allocation in the solution of (N,v2). Similar axioms are quite prevalent in the literature (e.g., [11], [16], [17]).

Next, we introduce another axiom that has been useful in studying the core as a solution.

Antimonotonicity (AM). Let (N, v') be such that vf(S) < v{S) for all S c N and v'(N) = v(N). Then, a{N, v) Ç a(tf , i/).

The intuition is that if the coalitions get impoverished then the pay-off vectors in the solution of the original game remain in the solution of the new game and additionally some more pay-off vectors feasible for the grand coalition may qualify as solution vectors. Keiding [9] introduced this axiom in the literature.

Consistency is another property of a solution that has been widely regarded to be quite desirable (see [18] for a survey). Consistency implies that if a pay-off vector is in the solution of a situation involving a set of players (the situation may not necessarily be a TU game) then the restriction of the pay-off vector to a coalition should be in the solution of the substitution involving that coalition. Our next axiom is one of consistency.

Usually the consistency properties of a solution on the class of TU games are defined in terms of reduced games (see [18]). Given a game (N, v) and a pay-off vector x e X(N, v) a reduced game on a coalition S describes the pay-offs available to the different subcoalitions of S. Different assumptions on the behavior of the players in S and the possibility of their cooperating with the players outside S generate different reduced games.

Let us define the following reduced game.

Definition 5. Let x G X{N, v). The secession reduced game on S CN,(S / 0) with respect to x, (S, i£) is given by:

tfs{S) = v(N) - £ *,, ieN's

ifs(T) = v(T) for TCS. The intuition is that if the players in N'S leave, no cooperation with them is possible any more. However, the commitment to their pay-offs has to be honored by the grand coalition S in the reduced game. Moreover, since no




cooperation with the players in N'S is possible, the worth of each T C S in the reduced game remains what it was in the original game.

Nagahisa and Yamato [12] have used a similar reduced game in their characterization of the core.

Secession consistency (SC). If xea(N,v) then for any coalition S, xs€<r(5,t§).

Since the EDC is a solution related to the norm of equity, it should have some egalitarian properties. Our next axiom, a condition on subgames, is one such property.

Definition 6. Given a game (N, v), the subgame on a coalition 5, (S, vs), is given by:

vs(T) = v{T) for all TÇS.

Weak internal stability for proximal coalitions (WISPC). Let x e (t(N,v). Consider any S c N such that |S| = 'N' - 1. Then for all y € g (S, vs),

maxjes */ > miitjes yj.

Suppose for a coalition S proximal to TV (obtained by dropping only one player) even the worst-paid player in a pay-off vector y in the solution of the subgame on S gets more than that is given to any player of S in an allocation x for the grand coalition. Then, this axiom specifies that if x is so bad for a large fraction of the players and is skewed in favour of only one player, then x should not be in the solution of the whole game.

The next and final property is also in terms of subgames. First, we have the following definition.

Definition 7. Call W Ç R" weakly symmetric if there exists x € W such that any vector obtained by permuting the components of x is also in W.

Irrelevance of a-asymmetric coalitions (IRAC). Suppose for no non-singleton and non-empty S cN ii holds that <r(S, vs) is weakly symmetric. In that case, if there exists x 6 X(N, v) such that x¡ > v({i}) for all i € N, then a(N,v)¿9.

The explanation of this axiom is that if a coalition is to affect the solution set for the grand coalition, then the solution set for itself should have some symmetry. The axiom IRAC cannot be justified as an intuitively desirable property for a solution. It is a weak condition for non-emptiness of a solution. It is weaker than properties like "restricted non-emptiness", an axiom used by Voorneveld and van den Nouweland [19] for the axiomatization of the core.



396 A. Bhattacharya

Theorem. There is a unique solution on F that satisfies WUHC, AM, SC, WISPC and IRAC and it is the EDC.

It is obvious that EDC satisfies IRAC, AM, SC and WISPC on F. We prove the rest of the theorem with the help of the following lemmata.

Lemma 1. The EDC, /,(.), satisfies WUHC on F.

Proof. For a game (N, v) G F, define the following corresponding non- transferable utility game (NTU game) (N, V):

V(N) = {x € RN'x < y for some y G X(N, v)}; for each proper coalition S of N, V(S) = {xe Rs'xi < a(S, v) for every / G S}; F(0) = 0.

Note that L(N, v) is the core of (N, V) (for formal definitions of an NTU game and the core of such a game, see [8]). Now, the core is a closed-valued correspondence for variations in V(N) at constant F(5); S ^ N (see, e.g., the remark in [9], p. 114). Hence, the lemma follows.2 ■

Lemma 2. If a solution a(.) satisfies IRAC, SC and WISPC on T then for any (N, v) e r,jc e o(N, v) implies that for all i G N, x¡ > v({i}).

Proof Note that by IRAC, for any single-player game (N, v), o(N, v) ̂ 0. So, by the definition of a solution (in particular, the property that the pay-off vectors in a solution must be efficient), <r({i}, v^) = v({i}). Now, let 'N' > 1 and Xi < v({i}) for some i G N. If 'N' = 2, then <x(.) clearly violates WISPC. If |A^| > 2, then picky G N'{/} and construct ({/,y}, tf^ J, the secession reduced game on {ij} with respect to*. Then by SC, (x,-,jc/) G <r{{ij}, ^{¡j})- But then again, a(.) violates WISPC. ■

Lemma 3. If a solution <r(.) satisfies IRAC, WUHC, AM, SC and WISPC on r then for any (N,v) G r,L(N,v) Ç a(N,v).

Proof Take (A^, v) e Y and let x G L(N, v). Fix e > 0 and construct the game (N,ve) as follows:

ve(N) = v(N) + e, and for S c N,

if(S) = v(S). Construct the vector jc% given by x] = x¡ -h e/'N' Vi G N.

Now, further construct (N.v^) G F for which v€^(S) = v((S) for every non-singleton coalition S and ve^({i}) =x].

2 I am indebted to the responsible editor for pointing out this connection between the EDC of a game and the core of the corresponding NTU game.




We claim that for any S CN such that |5| > 1, (7(5, v*/) cannot be weakly symmetric. Suppose otherwise. Fix ScN, 'S' > 1, such that g(S, v€/) is weakly symmetric. Let y e a(S, v^*) be such that any vector obtained by permuting the components of y is also in <x(S, v6/). Since x € L(N, v) it must be true that there exist ij'eS such that *,>»• Suppose not. Then, maxkes *k < minkes yk- Then, by the construction of (N, vtiX) and the definition of a solution (in particular the fact that a vector in the solution must be efficient), the following string of inequalities is true:

maxk€S Xk < mink£S yk < a{S, v€/) = a(S, v).

But this contradicts the fact that x e L(N, v). Thus, there exist ij e S such that Xj > yj. This implies that x) > yj. Now, let y be a vector obtained by permuting the components of y such that v, = yj. By our supposition that a(S, v^) is weakly symmetric, y e ^(S.v^). But then Lemma 2 is violated since vCiX({i}) =x] by construction. Hence, the claim is true.

By the claim and using IRAC we find that a(N, Ve*) / 0. Then, by the definition of a solution and Lemma 2, o(N,vtiX) = {xe}. Therefore, by AM, x€ e o(N,tf). Now, take a decreasing sequence of positive numbers {e*} such that e1 = e and ek - > 0. For each k, construct a game (N.v^) such that:

/(#) = »(#) + €*, and for S c N,

v<k(S) = v(S).

Let Xe* be the vector given by xf =x¡ + ¿/'N' VieN. By our argument above, for each h, Xe* is in a{N, i/). Then for the sequence {(N, t'6*)}, Ve* (N) - ♦ v(N) and xé - -> x. Then by WUHC, x e <f{N, v). ■

Lemma 4. If a solution a(.) satisfies IRAC, WUHC, AM, SC and WISPC on r then for any (N,v) E T, <r(A^,i;) C L(N,v).

Proof Take (N,v) e T and let x € a(^V,i;)'I(JV,i?). Then there is an equity coalition S for which e(S, v) > xs. Notice that by Lemma 3, e(S, v) e o*(5, vs). Therefore, if |S| = 'N' - 1, then <) violates WISPC. Suppose l^l < 'N' - 1. Then pick j G N'S and let T be S U {j}. By SC, at G cr(7' 4). But then once again a(.) violates WISPC. ■

This completes the proof of the theorem. Below we show that every axiom is independent of the others. If 'U' > 2,

then for each of the above axioms we show that there is a solution on F which satisfies the other four but fails to satisfy it.

IRAC. For a game {N,v), let <r(N,v) be the core of (Nyv). Then a{.) satisfies WUHC, AM, SC and WISPC on T but not IRAC as the following example shows.



398 A. Bhattacharya

Lettf = {l,2,3}, i;({l,2,3}) = 5, i;({l,2}) = 5, i>({2,3})=4, 17({1,3}) = 1, t?({l}) = u({3}) = 1 and t>({2}) = 3. In this game the core of none of the subgames on the doubleton coalitions is weakly symmetric. However, the core of this game is empty even though the vector (1, 3, 1) is individually rational.

WUHC. Fix N' CU such that AT' = {1,2,3}. Fix (W, vf) e F such that

a(N' v') = a(S, v') for all coalitions S C N.

Define cr(.) as follows.

(/) If for a game (#,»), N' C N and v(S) > v'(S) for every ScNf then a(N, v) = {xe L(N, v)'xN, ± e(N' v')}.

(ii) If for a game (N,v), N = N',v(N) = v'(N') and v(S) > v'(S) for every coalition S of N, then <r(N, v) = 0.

(iii) The solution a(.) = L(.) otherwise. Then a(.) satisfies IRAC, AM, SC and WISPC on T but not WUHC.

AM. Fix N' CU such that TV' = {1,2, 3}. Fix (W;, i/) G V such that

(a) v'{{3}) > a(S, i/) for all S Ç AT' for which |S| > 2 and player 3 G 5; (¿7) *({l,2},i/) > fl(Ä,i/) for all 5 c {1,2}; (^¿(AT',!/)^^ Define a(.) as follows.

(i) If for a game (tf,t>), JV' C N, v{S) = i/(S) for SC{1,2} and p(JV) ̂ v'(N') then a(AT,i;) = 0;

(ii) If for a game (N, v), N = AT7, u(5) = i/(S) for all coalitions 5; 5 ^ {3} and u({3}) > a(S,v) for all coalitions S for which 'S' > 2 and player 3 € S then ff(JV,r) = {x e L{N,v)' x3 = »({3})};

(ïiî) The solution o(.) = Z,(.) otherwise. Then <y(.) satisfies IRAC, WUHC, SC and WISPC on F but not AM.

SC. Define a(.) on F as follows. For any (N,v) e F and any coalition T of N, a(T,vT) = {xeX(T,üT)' there does not exist ScT such that 'S' = |r| - 1 and e(S,vT) » xs}. Then <r(.) satisfies IRAC, WUHC, AM and WISPC on F but not SC.

WISPC. Define <r(.) on F as follows. For any (N, v) € F and any coalition T of N, a(T,vT) = {x €X(T,vT)' there does not exist ScT such that |S| < 'T' - 2 and e(S, vT) » xs}. Then <y(.) satisfies IRAC, WUHC, AM and SC on F but not WISPC.

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On the equal division core