An axiomatization of the core of cooperative games without side payments

Journal of Mathematical Economics 14 (1985) 203-214. North-Holland

AN AXIOMATIZATION OF THE CORE OF COOPERATIVE GAMES WITHOUT SIDE PAYMENTS

Bezalel PELEG*

The Hebrew Uniuersity of Jerusalem, Jerusalem 91904, Israel

Let M be a set of m players, mz3, and let r be the set of all (finite) games (without side payments) that have a non-empty core. When M is finite, the following four (independent) axioms fully characterize the core on l? (i) non-emptiness, (ii) individual rationality, (iii) the reduced game property, and (iv) the converse reduced game property. If M is infinite, then the converse reduced game property is redundant.

1. Introduction

The core is, perhaps, the most intuitive solution concept in game theory.

However, there exist quite a few counterintuitive examples for the core [see, e.g., Maschler (1976) and Aumann (1985b,c)]. Thus, an (intuitively acceptable) axiom system for the core might reinforce its position as the most ‘natural’ solution (provided, of course, that it is not empty). But, in our opinion, an axiomatization of the core may serve two other, more important, goals. First, by obtaining axioms for the core, we single out those (important)

properties of solutions (to games) that determine the most stable solution in game theory. Secondly, we may compare the system of axioms for the core with systems of other solutions, and thereby obtain more information on solutions whose definition is not simple or ‘natural’. In particular, we may consider a solution to be ‘acceptable’ if its axiomatization is very similar to that of the core. As we shall remark in the following paragraph, there are, indeed, examples of this kind.

The main result of this work is that the core of non-transferable-utility (NTU) games is characterized by the following three (independent) properties: (i) individual rationality (IR), (ii) the reduced game property (RGP),

and (iii) the converse reduced game property (CRGP) (see section 5). Roughly, (RGP) says that the core is self-consistent in the following sense: The restriction of a payoff vector in the core to a subset of players is in the core of the reduced game. (CRGP) is a sort of converse property: If every restriction of a payoff vector to a two-person coalition is in the core of the (two-person) reduced game, then that vector is in the core. Moreover, every

*I am grateful to R.J. Aumann for several valuable discussions, and to M.A. Perles for a helpful remark. Also, I am indebted to the referees of this paper for their comments.

03044068/85/$3.30 0 198.5, Elsevier Science Publishers B.V. (North-Holland)

204 B. P&g, An axiomatization of the core of cooperative games

solution that satisfies the foregoing three properties coincides with the core (see Theorem 5.5). We remark that (RGP) replaces Independence of Irrelevant Alternatives in Lensberg’s (1981) axiomatization of Nash’s solution to the bargaining problem. Also, (RGP) and (CRGP), together with the usual axioms of Pareto-optimality, equal treatment, and covariance, characterize the prekernel of transferable utility (TU) games [see Peleg (1984b)].

We now review briefly the contents of the paper. Section 2 contains a list of properties of NTU games that we use. Reduced games are introduced and discussed in section 3. In section 4 we prove that the core satisfies (RGP) and (CRGP). The main result is formulated and discussed in section 5, and proved in section 6. Various questions on the domain and independence of the axioms are answered in section 7.

2. NTU games

Let M be a finite set of m players, m 2 3. A coalition is a non-empty subset of M. Let N be a coalition and let R be the set of real numbers. A payoff vector for N is a function x:N+R. Thus, RN is the set of all payoff vectors for N. If x E RN and i E N, then we will write xi for x(i). Let x, y E RN. x 2 y if x’2y’ for all ieN; x>y if xzy and x#y; x>>y if x’>y’ for all iEN. We denote RN,={xER~Ix~O}. Let A c RN. A is comprehensive if x E A and x2 y imply y E A. The boundary of A is denoted by 8A. If x E RN then x + A= {x+alaEA}.

Definition 2.1. An NTU game (or simply game) is a pair (N, V) where N is a coalition and V is a function that assigns to each coalition SC N a subset V(S) of RS, such that

V(S) is non-empty and comprehensive, (I)

V(S) n(xS + RT) is bounded for every xs E RS, (2)

V(S) is closed, (3)

if xs, ys E aV(S) and xs 2 ys, then xs = ys. (4)

Conditions (1) and (3) are standard. (2) guarantees that V(S) is a proper subset of RS. It is a very weak requirement of boundedness. For example, it is implied by Aumann’s definition of NTU games [see Aumann (1985a)]. Condition (4) is the familiar non-levelness property [see, e.g., Aumann (1985a), Hart (1983), and Asscher (1977)]. In section 7.5 we will show how it may be weakened. Indeed, one may drop (3) and require (4) only for S= N. However, the analysis of the resulting larger class of games is less intuitive.

B. Peleg, An axiomatization of the core of cooperative games 205

3. Reduced games

Let M be a finite set of m players, rnz 3, and let N and S be coalitions, S c N. If x E RN then we denote by xs the restriction of x to S.

Definition 3.1. Let (N, I’) be an NTU game, let XE V(N), and let SC N, S # 0. The reduced game with respect to S and x is the game (S, V,), where

T/,(S) = {y"l(ys, xN-? E VW)), (5)

T/,(T)= u {yT~(yT,xO)~V(TuQ)} if TcS, T#S. (6) QcN-S

In the reduced game, the players of S are allowed to choose only payoff vectors ys that are compatible with x N-S, the fixed payoff distribution to the members of N-S. On the other hand, proper subcoalitions T of S may count on the cooperation of subsets Q of N-S, provided that in the resulting payoff vectors for Tu Q each member of Q gets exactly xi. (Hence, T has to ensure the feasibility of xQ but not that of xNes.) Thus, the reduced game (S, V,) describes the following situation. Suppose that all the members of N agree that the members of N-S will get xN-‘. Further, assume that the members of N-S continue to cooperate with the members of S (subject to the foregoing agreement). Then V, describes the possible payoffs that various coalitions of members of S may obtain. However, it is assumed that S will choose some payoff vector in V,(S). Thus, the sets V,(T), Tc S, T# S, serve only to determine the final choice in V’JS).

Remark 3.2. Reduced games of NTU games were first used in Greenberg (1982). The present definition is due to Peleg (1984a) (see also section 7.5). However, it is interesting to notice that the idea of considering reduced games of NTU games may be traced back to Harsanyi (1959). Harsanyi extended Nash’s solution to multi-person pure bargaining games by considering reduced games with respect to pairs of players. Finally, Lensberg (1981) and Thomson (1982) have recently used reduced games in their axiomatization of various solutions of the bargaining problem. Reduced games of TU games (i.e., coalitional games with side payments) were introduced in Davis and Maschler (1965). For a brief survey of reduced games of TU games see Peleg (1984b).

Lemma 3.3. Let (N, V) be an NTU game, let XEV(N), and let ScN, S#(b. Then the reduced game (S, V,) is a game [i.e., it satisfies (Z)-(4)].

Proof: The verification of (1) (2) and (3) is straightforward. Thus, it remains to prove (4). Let Tc S, T#~#J, and let y’, Z’E aVx(T). Assume, on the

206 B. Peleg, An axiomatization of the core of cooperative games

contrary, that yT > zT. There exists Q c N- S such that (y’, xQ) E V( Tu Q). By (l), (zT, xQ) E V(7’uQ). By (4) there exists wTvQ~ V(TuQ) such that wTUQ>>(zT,xQ). By (l), ( wT,xQ) E V(Tu Q). Hence, by (5) or (6), WOE V’(T). Because wT>>zT, z’# aV,( T), and the desired contradiction has been obtained.

4. The core of reduced games

In this section we shall investigate the relationship between the core of a game and its reduced games. Let M be a finite set of m players, m 2 3.

Definition 4.1. Let (N, V) be an NTU game and let XE V(N). A coalition SC N can improve upon x if there exists y’~ V(S) such that ys>>xs. x is in the core of (N, V), C(N, V), if no coalition can improve upon x.

We denote

r={(N, V)/)IC(N, I’?#@. (7)

Definition 4.2. Let (N, V) be an NTU game and let XE V(N). x is Pareto- optimal if there is no y E V(N) such that y > x.

Remark 4.3. Let (N, V’) be an NTU game. Because of (1) and (4), XE V(N) is Pareto-optimal if and only if XE aI/( Clearly, C(N, V) caV(N). Hence, every payoff vector in the core is Pareto-optimal.

Lemma 4.4 Let (N, V) be an NTU game, let XE V(N), and let SC N, S+O. Then x is Pareto-optimal if and only if xs is Pareto-optimal in the reduced

game (S, Y-J.

Proof If xs is not Pareto-optimal then there exists yse VI(S) such that ys> xs. Hence, (y”, xN-‘) E V(N) and (y”, xNpS) > x. Therefore x is not Pareto- optimal. Similarly, if x is not Pareto-optimal, then there exists YE V(N) such

that y>>x. Hence, (y’, x”~‘)E V(N). Thus, y’~ I/,(S) and ys>>xs. Therefore, xs is not Pareto-optimal.

Lemma 4.5. Let IN, V) be an NTU game, let x E V(N), and let S c N, S # 0. If x E C(N, V), then xs E C(S, V,).

Proof Assume, on the contrary, that there exist Tc S, T # 0, and yT E V,(T), such that yT>>xT. By Definition 3.1 there exists Q c N-S such that (y’, xQ) E V(Tu Q). Clearly, (y’, xQ) >xTuQ. Hence, by (4), there exists z’“~EV(TUQ) such that zTUQ>>x T”Q Because x is in the core, the desired . contradiction has been obtained.


The following converse of Lemma 4.5 is true for the core. Let N be a coalition. We denote

n(N)={{i, j}Ii,jEN, i#jb (8)

Lemma 4.6. Let (N, V) be an NTU game and let x E V(N). If for every S E n(N) xs E C(S, V,), then x E C(N, V).

Proof: Let S’en(N). By Remark 4.3, xs is Pareto-optimal in (S, V,). Hence, by lemma 4.4, x is Pareto-optimal in (N, V). Thus, N cannot improve upon x. Now let Tc N, T# 8, N. Choose i E T, j $ ?: and let S = {i, j}. Assume, on the contrary, that there exists yr~ V(T) such that yT>>xT. Then in the game (S, V’) y’ E V’({ i>) and y’ > xi. Thus, xs q! C(S, V,), and the desired contradiction has been obtained.

The following definition will be used in section 5:

Notation 4.7. Let (N, V) be an NTU game and let iE N. We denote

~~=sup{x'~x'~ V({i})}. (9)

By (1) and (2) vi is well defined (see also section 7.5).

Definition 4.8. Let (N, V) be an NTU game and let x E V(N). x is individually rational if xi 2 vi for all i E N.

Remark 4.9. Let (N, V) be an NTU game. If x E C(N, V), then x is individually rational.

5. Axioms for the core

A solution on r [see (7)] is a function cr that assigns to each NTU game (N, v) or a subset a(N, v) of V(N). We shall consider the following properties of solutions:

Definition 5.1. Let 0 be a solution on I’. rr satisfies non-emptiness (NE) if a(N, V)#@ for every (N, v) EF. (T satisfies Pareto-optimality (PO) if a(N, V) c aV(N) for every (N, V) E r (see Remark 4.3). Finally, [T satisfies individual rationality (IR) if for every (N, V) or, every payoff vector in a(N, V) is individually rational (see Definition 4.8).

Properties (PO) and (IR) are well known and widely acceptable. Indeed, many theories of TU and NTU games consider only payoffs that are Pareto- optimal and individually rational [see, e.g., Owen (1982, p. 146) and Section 8.6 of Lute and Raiffa (1957)].


Definition 5.2. A solution o on F has the reduced game property (RGP) if it satisfies the following condition: If (N, V) EF, ScN, S#@ and x~a(N, V), then (5, V’) or and x~Eo(S, V,) (see Definition 3.1 and Lemma 3.3).

The (RGP) axiom is a condition of self-consistency: If (N, v) or and x~o(N, V’), that is, x is prescribed as a solution to (N, V), then for every S c N, S# 8, xs is a solution for the restricted game (S, V,). The reader is referred to Section 3 of Aumann and Maschler (1985) and to Section 5 of Thomson (1983) for discussions of the (RGP). The (RGP) has been used in the axiomatization of the prenucleolus [Sobolev (1975)] and the prekernel [Peleg (1984b)] of TU games, and in the axiomatic characterization of the Nash solution [Lensberg (1981)] and the egalitarian solution [Thomson (1982)] of pure bargaining games.

Definition 5.3. A solution (T on r has the converse reduced game property (CRGP) if it satisfies the following condition: If (N, v) E r, x E V(N), and for every S E R(N) [see (8)] (S, Vx) E r and xs E a($ V,), then x E o(N, v).

The (CRGP) focuses on pairs of players: If at a given payoff distribution x every pair of players is ‘in equilibrium’, then x is in the solution. The (CRGP) has been used in the axiomatization of the prekernel of TU games [Peleg (1984b)]. Also, Harsanyi (1959) has used the (CRGP) as the basis for his extension of Nash’s solution to multi-person pure bargaining games.

The foregoing axioms are not independent. Indeed, the following lemma is true:

Lemma 5.4. Zf a solution 0 satisfies (IR) and (RGP), then it also satisfies

(PO).

Proof Let (N, V) or and let XE o(N, V). Assume, on the contrary, that x is not Pareto-optimal. Then there exists YE V(N) such that y> x. Choose ie N such that y’> xi. Because o satisfies (IR), xi $ o( {i}, V,) [see (1) and (5)]. Thus, 0 violates the (RGP), and the desired contradiction has been obtained.

The main result of this paper is the following theorem:

Theorem 5.5. There is a unique solution on r that satisfies (NE), (IR), (RGP), and (CRGP), and it is the core.

6. Proof of Theorem 5.5

Let M be a finite set of m players, m 2 3. First we notice that, by definition, C(N, V)#g for every (N,V)er. Thus, the core satisfies (NE). By Remark 4.9 the core satisfies (IR). Finally, by Lemmata 4.5 and 4.6 the core has the


(RGP) and the (CRGP). Now let CJ be a solution on r. First we prove the following lemma:

Lemma 6.1. Zj CJ satisfies (IR) and (RGP), then o(N, V)cC(N, V) for every (N, V) E r.

ProoJ By Lemma 5.4 c satisfies (PO). Let (N, V) E r be an n-person game. If n 52 then it follows from (PO) and (IR) that a(N, I’) c C(N, I$ Thus, let n 2 3 and let x E a(N, I’). By (RGP), X’E a(S, V,) for every S in rc(N) [see (S)]. Hence, xs E C(S, I’,) for every SE z(N). Because the core has the (CRGP), x is in C(N, V).

We also need the following lemma:

Lemma 6.2. Let (N,V)ef, N#M, qEM-N, N*=Nu{q), and let x,, E C(N, V). Then there exists a game (N*, W) E r such that

C(N*, W)={z> where 2=(x0,0), (10)

(N, w,) = W, VI. (11)

We postpone the proof of Lemma 6.2 and complete the proof of the theorem. First, we notice the following corollary:

Corollary 6.3. Let o be a solution on T that satisfies (NE), (ZR), and (RGP), and let (N, V) E r, N # M. Then o(N, V) = C(N, V).

Proof: Let x,, E C(N, V), let q E M - N, and let N* = N u {q}. By Lemma 6.2 there exists a game (N*, W) that satisfies (10) and (11). By Lemma 6.1, a(N*, W)c C(N*, W). Because a(N*, W)#& a(N*, W)={z}. By (RGP), zN~o(N, W,). By (10) and (ll), x0 E a(N, V). Thus, o(N, V) 2 C(N, V). By lemma 6.1, C(N, V) 3 a(N, V).

The proof of Theorem 5.5 now follows. Ley 0 be a solution on r that satisfies (NE), (IR), (RGP), and (CRGP), and let (N, V)E~. If N # M then, by Corollary 6.3, g(N, V) =C(N, V). Thus, let N = M, and let XEC(N, V). Then xs E C(S, VX) for every S E rc(N). Because m 2 3, a(& VI) = C(S, V,) for every SEX(N). Thus, by the (CRGP), XE o(N, V). We have proved that a(N, V) 2 C(N, V). By Lemma 6.1, o(N, V) c C(N, V). Hence, c(N, V) = C(N, V).

Proof of Lemma 6.2. Let S c N*, S+@ We distinguish the following possibilities, numbered (12), (13), (16), and (17):

q$S and S#N. (12)


In this case we define W(S) = V(S).

S=N. (13)

For x E V(N) and irz N let

h(xi)=xi+max(xi-xb,O)/{l+max(x’-xi, O)}.

Then f;(x’) is a continuous and strictly increasing function of xi. We now define

W(N)={y[y’=f;(x’), ieN, for some XE V(N)}.

We have to show that W(N) satisfies (1)<4). Clearly, W(N) # 8. Now assume that YE W(N) and us y. There exists x E V(N) such that yi=f;(xi) zui, ie N. Hence, there exists x1 S x such that f;:(xi) = ui, i E N. By (l), x1 E V(N). Hence UE W(N). Thus, W(N) is comprehensive. The proof of (2) and (3) for W(N) is straightforward. To prove (4), let y,, y, E BW(N) and let y, zyz. There exist x1,x2~ V(N) such that yi=fi(xj), iE N and j= 1,2. Because f;(.), iEN, is strictly increasing, x1, x2 E al’(N) and x1 2x2. Hence, x1 =x2. Thus, y, =y,.

We shall use the following properties of W(N):

x,, E SV(N). (14)

Indeed, if YE W(N) then y’=f;(x’), ie N, where XE V(N). Hence there exists jE N such that xjsx’,. Thus, #=xjSx’,.

If x~al/(N) and x#xO, then x#BW(N). (19

Indeed, there exists je N such that xj>x& Hence, there exist t > 0 and x1 E V(N) such that (i) xjl =xi--t, (ii) fj(x’,)>xj, and (iii) xi >xk for kEN-{j}. Therefore, if y E W(N) is given by y’ = fi(xi), i E N, then y X-X.

We now consider the remaining possibilities:

S={d. (16)

We define W({q))={xq~xqSO}.

s= Tu{q}, TcN, Tf8. (17)

Let as E RS be defined by: a’= - 1 if ic I: and aq= 1. We denote by L the line {tas 1 t E R}. Now we define W(S) as the (vectorial) sum

W(S) = (V( T) x (0)) + L. (1%

B. P&g, An axiomatization of the core of cooperative games 211

As the reader can easily verify, W(S) satisfies (1) and (3). We now prove (2). Let bSE RS and let yse W(S), ys=(xT, 0)+ tas, where xTe V(T) and TV R. If yS~bSthenxi~bi+t~bi+bqfori~~Letci=bi+b~,i~~By(2),V(T)n(cT+R~) is bounded. Hence W(S) n(bS+ RT) is bounded. Finally, we prove (4). Let yf, v’, E W(S) and let 8 >v’,. We have to show that there exists 2 E W(S) such that y$ >>& $ = (XT, 0) + tjUs, where XT E V(T), j = 1,2. Clearly, t, 2 t,. If t, = t, then xT>xI. Hence, by (4), there exists xT~ V(T) such that xT>>xT. If 0<6<x’j-xi, ie T, t,= t, +6, and 2=(x:, O)+t$, then f13 E W(S) and y:>>$. Now, if t, > t,, let t, > t,> t,, and let $=(x~,O)+t+?. Then y! E W(S) and 2 >>$.

We shall use the following property of W(S): .

{x’l(x’, 0) E W(S)} = V(T). (19)

Now we prove (10) and (11). Let ye C(N*, IV). Then y =(xN, 0) + taN*, where xN~ V(N). Because W({q})={x~)xq~O}, t?O. If t>O then xN>>yN and xN~ W(N). Thus, t =O. Hence, it follows from (14), (15), (18) and our assumption that x0 e C(N, V), that y=( x0, 0) = z. Thus, (10) has been verified. (11) follows from (19).

7. Concluding remarks

7.1. The number of players

Let A4 be an infinite set of players. Again, let

r={(N, qv . 1s a finite subset of M and C(N, I’)#@}.

Then the core is the unique solution on r that satisfies (NE), (IR), and (RGP) (see Corollary 6.3).

On the other hand, if M contains only two players (i.e., m= 2), then Theorem 5.5 is not true. Indeed, for (a two-person game) (N, V) or let o(N, V) be the egalitarian solution of the two-person pure bargaining game

(F(N), (v,, Q)), where N = (1,2} C see (9)]. Then r~ satisfies (NE) and (IR). Also, because m =2, CJ satisfies (RGP) and (CRGP). Clearly, c~ is different from the core.

7.2. TU games

Let Tt be the set of all TU games with non-empty core. For (N, V) E Tr let o(N, V)=C(N, V)nPrK(N, V), where PrK(N, I’) is the prekernel of (N, V) [see Peleg (1984b)]. Then 0 satisfies (NE), (PO), (IR), (RGP), and (CRGP) on T1. However, 0 is different from the core (on r,). An axiomatization of the core of TU games is contained in Peleg (1984b).


7.3. Balanced games

Scarf (1967) has proved that a balanced (NTU) game has a non-empty core. Thus, the set B of all balanced games is contained F. However, Example 7.1 shows that a reduced game of a game in B, may not be balanced. Thus, B may not replace F in Theorem 5.5.

Example 7.1. Let N = { 1,2,3,4} and let V be defined by

V(N)={xN(x1+x2+x3+x4j7},

I/(S)={x~~x’~2, ieS} if S={l,2),{1,3), or {2,3},

V(S)={xsIxi~O, ins}, otherwise.

As the reader can easily verify, (N, V) is balanced. Now, x =(5/2, 5/2, 0, 2) E C(N, V). However, ({ 1,2,3), V,) is not balanced.

Remark 7.2. The game of Example 7.1 does not satisfy (4). However, it may be approximated by games that satisfy (4).

7.4. Independence of the axioms

Example 7.3. Let a(N, V) =8 for every (N, V) in F. Then d satisfies (IR), (RGP), and (CRGP). Clearly, 0 violates (NE).

Example 7.4. Let c be the Pareto correspondence on F; that is, a(N, V) = aV(N) for every (N, V) E r (see Remark 4.3). Clearly, rr satisfies (NE). By Lemma 4.4, 0 satisfies (RGP) and (CRGP). However, cr violates (IR).

Example 7.5. For (N, v) E r let o(N, V) be defined by

a(N, V)={x(x~al’(N) and xizUi for all iEN}.

That is, a(N, V) is the set of all Pareto-optimal and individually rational payoff vectors [for the game (N, l’)]. Then, as the reader can easily verify, 0 satisfies (NE), (IR), (PO), and (CRGP). However, by Lemma 6.1, c violates (RGP).

Example 7.6. We shall call a game (N, V)/)E~ pure bargaining game if for every S c N, S # N, 8, V(S) is given by

V(S)={Xs~&xiS&ui) bee P)l.

B. Peleg, An axiomatization of the core of cooperatioe games 213

Now let cr be defined on r by the following rules. If (M, V) or (where M is the set of all players), is a pure bargaining game then let o(M, V) be the egalitarian solution of (M, V) [with respect to the disagreement payoff vector (Ui)i.M]. Otherwise, let o(N, v) = C(N, V).

Then, as the reader can verify, 0 satisfies (NE), (IR), and (RGP). Clearly, CJ violates (CRGP).

7.5. The non-levelness condition

It is possible to drop (3) and replace (4) by the following weaker condition:

if x,y~dV(N) and xzy, thenx=y. (20)

The foregoing change necessitates only the following modification of Detintion 3.1: (6) should be replaced by

V,(T)= u {yTl(yT,yQ)~V(TuQ) for some yQ>>xQ} QcN-S

if TcS, T#S. (21)

All our results remain true under (20)-(21). However, some of the proofs should be modified.

Condition (20) is essential for Lemma 4.5. This is shown in the following example:

Example 7.7. Let N={l,2}, let V(N)={xN)xNZ(l, l)}, and let V((i>)= {xi\xi~O}, i= 1,2. Then y=(O,l) is in C(N,V) and VY({l})={x’~x’~l). Thus, Y’ B CC{ I>, v,).

7.6. Convex-valued games

An NTU game (N, V) is convex-valued if

V(S) is convex for every S c N. (22)

(22) is a standard assumption [see, e.g., Asscher (1977), Aumann (1985a), and Hart (1983)]. Unfortunately, a reduced game of a convex-valued game may not satisfy (22). Thus, it is impossible to use (RGP) in the axiomatization of the core of convex-valued games. The class of TU games is a class of convex- valued games that contains all the reduced games of its members. I do not know whether there is any larger class of convex-valued games with the foregoing property.


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An axiomatization of the core of cooperative games without side payments