Int J Game Theory (2001) 30:195±207

2001
9
99
9

The consistency principle and an axiomatization of
the a-core*
Koji Takamiya**
Graduate School of Economics and Business Administration, Hokkaido University,
Kita9 Nishi7 Sapporo 060-0809 Japan(e-mail: koji@pop.econ.hokudai.ac.jp)
Received: June 1998/Final version: 6 July 2001

Abstract. This paper examines the a-core of strategic games by means of theconsistency principle. I provide a new de®nition of a reduced game for strategic
games. And I de®ne consistency (CONS) and two forms of converse consistency (COCONS and COCONS*) under this de®nition of reduced games. Then
I axiomatize the a-core for families of strategic games with bounded payo¨functions by the axioms CONS, COCONS*, weak Pareto optimality (WPO)and one person rationality (OPR). Furthermore, I show that these four axioms
are logically independent. In proving this, I also axiomatize the a-individually
rational solution by CONS, COCONS and OPR for the same families ofgames. Here the a-individually rational solution is a natural extension of theclassical `maximin' solution.
JEL Classi®cation: C71, C72.
Key words: axiomatization, reduced game, consistency, converse consistency,
a-core0. Introduction
This paper examines the a-core of strategic games from an axiomatic point ofview. The a-core is one of several core concepts in strategic games. The fol* This paper is based on Chapter 2 of my MA thesis carried out at Otaru University of Commerce. An earlier version was reported in the Fourth International Meeting of the Social Choiceand Welfare in Vancouver July 1998.
** I am very grateful to Prof Tomoichi Shinotsuka for his great e¨ort in supervision. Also I wish
to thank an anonymous referee, Profs Dipankar Dasgupta, Tsuneyuki Namekata, Chisato Shibayama, Masaru Uzawa, Kenji Yamamoto and the participants of the above-mentioned meeting. I
am indebted to an associate editor whose comments and suggestions were very helpful in extending
the results in an earlier manuscript. All errors are my own responsibility.

196

K. Takamiya

lowing scenario describes a main idea of the concept. Consider a strategic
game. Assume that a strategy pro®le is `suggested' for play. Each coalition has
a chance to deviate from the suggested strategies. In other words, each coalition
may `block' the suggested strategies. In blocking the strategy pro®le, a coalition
must select their blocking strategies ®rstly. And secondly, the remaining players
move in response. And a coalition must `improve its position' by blocking. The
problem is how one de®nes this `improvement'. The a-core assumes that a coalition has a `pessimistic perception1' about the choices by the complementary
coalition. That is, a coalition blocks a strategy pro®le if each member of thecoalition strictly improves upon the `suggested' payo¨, no matter what strategies the remaining players choose. The a-core is the set of all strategy pro®les which are not blocked by any coalition. The a-core was introduced by
Aumann (1961). Scarf (1971) proved the existence of the a-core for a generalclass of games. Later, Kajii (1992) and Yannelis (1991) investigated the existence problem further.
This paper adopts an axiomatic approach. One of the purposes of thismethod is to compare one solution with another. Especially in this paper, theconsistency principle plays a main role. Consistency is a general property thatmany ideas appearing in social sciences have in common. It has been shownthat a considerable number of concepts from allocation rules, public ®nance,
game theory and many other ®elds in social sciences satisfy consistency. The
property gives a beautiful and strong unity across a variety of distinct subjects2. This is the reason why we call it a `principle', not merely a `property'.
Particularly in game theory, many solutions have been axiomatized by consistency. The following describes a general idea of consistency in game theory3:
Let G be a game4. Let x be an outcome which `solves' the game. Assume thatsome of the players of the game leave the scene and the remaining players,
denoted by S, play the `reduced ' game G S; x . Apply the same solution to G S; x .
The solution is said to satisfy consistency if xjS, the restriction of x to S, results
as a solution outcome of G S; x . It is not always easy to ®nd how the reduced
games G S; x should be de®ned. For coalitional form games, several de®nitionsof reduced games are available, while only one is available for strategic form
games to the best of my knowledge.
In the theory of coalitional form games5, research on the consistency principle has a long history. And a large literature has been published. On the other
hand, studies on the consistency principle for strategic games is relatively new.
As far as I know, the ®rst systematic research in this area is Peleg and Tijs
(1996)(P&T, henceforth). They de®ned reduced games in strategic form. And
they gave axiomatic characterizations to the following solutions for strategic
games: Nash equilibria (Nash, 1951), strong Nash equilibria (Aumann, 1959),
semi-strong Nash equilibria (Kaplan, 1992) and coalition-proof Nash equilibria (Bernheim, Peleg and Whinston, 1987). All these axiomatizations are done by
1 I borrowed this terminology from Ichiishi (1997).
2 For comprehensive survey, see Thomson (1990, 1996).
3 The following description on consistency is due to Aumann (1987).
4 G may take any form, TU, NTU or strategic.
5 I avoid the term `cooperative game theory' in this context on the grounds that what distinguishes between non-cooperative and cooperative games is not game forms, but the restriction on
the behavior of players. Indeed, the a-core is a cooperative solution in strategic forms. I owe thisviewpoint to Ichiishi (1997).

Axiomatization of the a-core

197

consistency together with various forms of its converse, converse consistency.
Peleg, Potters and Tijs (1996) explored some general relations between consistency and the non-emptiness axiom using a graph theoretical method.
Norde, Potters, Reijnierse and Vermeulen (1996) gave further axiomatizationsof Nash equilibria on two classes of strategic games: The class of mixed extensions of ®nite games and the class of games with continuous concave payo¨functions. It is worth noting that their characterizations do not employ any
form of converse consistency. All these studies are based on the reduced games
introduced in P&T (PT reduced games, henceforth).
Unfortunately, there are solutions to strategic games which violate consistency under PT reduced games. The a-core, de®ned on some classes of strategic
games, falls into this category. This raises the questions: How should I de®ne
the reduced game concept to restore consistency for the a-core? And then withwhat other axioms can I axiomatize the a-core by consistency? These are theproblems that this paper attempts to answer to. Precisely, I shall provide anaxiomatization of the a-core correspondence on any closed6 family of games(in strategic form) with bounded payo¨ functions by consistency under a newde®nition of a reduced game. Further, I shall show the logical independence of
the axioms used in the axiomatization. In proving this, I also axiomatize thea-individually rational solution by consistency for the same families of games
as the ones on which the a-core is axiomatized. Here the a-individually rational
solution is a natural extension of the classical `maximin' solution.
The plan of the paper is as follows: The next section introduces some preliminaries. Section 2 provides some de®nitions, including a new de®nition of
reduced games. Section 3 states the main results with some lemmas. Finally, in
Section 4, I discuss the interpretations of the new reduced games. Throughout,I will compare the notions and results from P&T with mine.
1. Preliminaries
In this section, I introduce some basic de®nitions.
(1) Strategic form game: A strategic ( form) game, is a list G :N G; fXj ; uj gj A N G . Here N G is the ®nite set of players. Xj is the(non-empty) strategy set of player j A N G. For a coalition
S, S HQ
N G7 with S 0 q, XS denotes the Cartesian product j A S Xj 8. And
uj : XN G ! R9 is the payo¨ function of player j A N G. For S H N Gwith S 0 q, xS A XS and j A N G, denote by Im uj jxS  the image
uj fxS g  XN GnS , i.e., the set fuj xS ; yNnS  j yNnS A XN GnS g.
(2) The a-core: Let a strategic game G be given. Let S H N G with S 0 q,
and x A XN G . Say that a coalition S blocks a strategy pro®le x if byS A
XS : Ej A S: inf Im uj jyS  > uj x. A strategy pro®le x is said to be in the acore of G if no coalition blocks x. Thus the a-core is the set of all strategy
pro®les which are immune to blocking. Note that my de®nition of block6 See De®nition 2-2 in Section 2.
7 Throughout the paper, inclusion `H' is weak.
8 This subscript notation applies also for a strategy pro®le, e.g. xS denotes xj j A S A XS .
9 R denotes the real line.

198

K. Takamiya

ing is slightly stronger than the `standard' de®nition: Most authors de®nethat S blocks x if byS A XS : EyNnS A XN nS : Ej A S: uj yS ; yN nS  > uj x.
Thus my a-core is slightly larger than the `standard' one. All preceding
existence results (e.g. Scarf, 1971) remain true for my de®nition. For further remark on this modi®cation, see Remark 3-4.10(3) The a-individually rational solution: A strategy pro®le x is said to be aindividually rational if no singleton coalition blocks x. The a-individually
rational solution (a-IR solution) of a strategic game G is the set of all aindividually rational strategy pro®les. The de®nition of a-individual rationality is a natural extension of the traditional `maximin' type individual
rationality. It is easy to see that a strategy pro®le x is a-individually rational
if and only if for each player j; x gives him a payo¨ at least as large as his`security level', i.e., uj x b supx j inf xNnf jg uj xj ; xN nf jg . By de®nition, thea-IR solution contains the a-core.
2. De®nitions
This section provides some de®nitions. Henceforth, without mentioning, I
refer to a strategic game simply as a game.
De®nition 2-1. Let G  N G; fXj ; uj gj A N G  be a game. Let x A XN G andS H N G with S 0 q. Then the reduced game of G with respect to S and x isa game G S; x  S; fXj ; ujS; x gj A S , where the payo¨ functions ujS; x are de®ned asfollows:
ujS; x yS   inf Im uj jyS ;

if yS 0 xS ;

 uj yS ; xN nS 

otherwise;

for any yS A XS :Some discussion on reduced games comes in length in Section 4. Here it suf®ces to mention the following points. Firstly, the reduced game may not wellde®ned for every S and x. This is not the case for PT reduced games: Anygame G has its PT reduced game with respect to any S and x. Their de®nitiondi¨ers from mine in how the payo¨ functions ujS; x are de®ned: P&T de®nes
ujS; x yS   uj yS ; xNnS  for any yS A XS . Secondly, the reduction operation may
destroy some properties that the original game possesses (e.g. continuity of thepayo¨ functions). Thus I restrict my attention to families of games within
which the reduction operation makes sense.
De®nition 2-2. A family of games G is said to be closed if
EG A G: ES H N G;S 0 q: Ex A XN G : G S; x A G:

This type of closedness appears in P&T together with some other types. LetG  denote the family of games fG j G is a game; and Ej A N G: the payoff
10 I owe this modi®cation of the blocking concept to the associate editor.

Axiomatization of the a-core

199

function uj is boundedg. The family G  is one example of closed families ofgames. In Section 3, where I state my theorems, I deal with only all closed
subfamilies of G  as the domains of the solutions to be axiomatized11. I chosethese families because in many important economic applications, payo¨ functions are assumed to be bounded.
De®nition 2-3. Let G be a family of games. A solution on G is a correspondence
j: G !! 6G A G XN G such that EG A G: j G H XN G .
De®nition 2-4. Let G be a closed family of games. Let j be a solution on G.
Then if G A G has no less than 2 players, i.e., jN Gj b 2, then de®nej@ G  fx A XN G j ES H N G; S 0 q; N G: xS A j G S; x g:
Now I am ready to introduce the axioms. Consistency comes ®rst.
De®nition 2-5. Let G be a closed family of games. A solution j on G satis®es
consistency (CONS) if
EG A G;jN Gj b 2: j G H j@ G:De®nition 2-6. Let G be a closed family of games. A solution j on G satis®esconverse consistency (COCONS) if
EG A G;jN Gj b 2: j G I j@ G:

Substituting my reduced games with PT reduced games, this axiom COCONS
would be the same as COCONS in P&T, and COCONSA in Peleg, Potters
and Tijs (1996). P&T gives an axiomatization of the Nash equilibrium correspondence with their COCONS axiom.
De®nition 2-7. Let G be a game. Then denote
WPO G  fx A XN G j sby A XN G : Ej A N G: uj y > uj xg:
WPO G is simply the set of weakly Pareto optimal strategy pro®les of gameG.
De®nition 2-8. Let G be a closed family of games. A solution j on G satis®esconverse consistency* (COCONS*) if
EG A G;jN Gj b 2: j G I j@ G X WPO G:COCONS* is a weaker version of COCONS. If I substitute my reduced
games with PT reduced games, this axiom COCONS* would be the same as
COCONS 3 in P&T, and COCONS C in Peleg, Potters and Tijs (1996). A
11 I am grateful to the associate editor for suggesting these families of games.

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K. Takamiya

similar axiom is used in axiomatizations of the Walras correspondence forgeneralized economies (Van den Nouweland, Peleg and Tijs, 1996).
De®nition 2-9. Let G be a family of games. A solution j on G satis®es weak
Pareto optimality (WPO) if
EG A G: j G H WPO G:
De®nition 2-10. Let G be a family of games. A solution j on G satis®es one
person rationality (OPR) if
EG A G;jN Gj  1: j G  WPO G:The axiom OPR simply says that the solution j solves single person optimization problems.
3. The results
This section states characterization results of the a-core and the a-IR solution.
As the domains of these two solutions, I consider all closed subfamilies of G  ,the family of games with bounded payo¨ functions. Note that G  itself is a
closed family of games. I start with the following useful lemma.
Lemma 3-1. Let G be a non-empty closed family of games. Let m be a solution
on G. Then there exists at most one solution j on G which satis®es(A) EG A G: j G H m G,(B) EG A G, jN Gj b 2: j G  m G X j@ G, and(C) EG A G, jN Gj  1: m G  j G:Remark 3-212: Lemma 3-1 above shall be applied to prove the `uniqueness'part of the axiomatic characterizations (Lemmas 3-5 and 3-13). Although j
is de®ned with reference to the reduction operation, Lemma 3-1 holds true
regardless of the type of reduced games to adopt. This is also the case forLemmas 3-5 and 3-13. Thus Lemmas 3-1, 3-5 and 3-13 all hold true even if
I alternatively adopt PT reduced games or any other.
Proof of Lemma 3-1: Let m; c1 and c2 be solutions on G. For m and eachj A fc1 ; c2 g, assume that the conditions (A), (B) and (C) in the above aresatis®ed. Suppose that c1 0 c2 . Then there must exist a game H A G for
which (i) c1 H0c2 H, and (ii) EG A G: jN Gj < jN Hj ) c1 G  c2 G.
Clearly, by (C), I have jN Hj b 2. Without loss of generality, I assume thatx A c1 H and x B c2 H. By (A), x A m H, Then by (B), I have x A c1@ H
S; x and xS B
and x B c
2 H. Thus bS H N H, S 0 q, N H: xS A c1 H
S; x
c2 H . Since jSj < jN Hj, this contradicts (ii). 8
12 I owe this remark and the proof of Lemma 3-1 to the associate editor.

Axiomatization of the a-core201

3-1. The a-core
My main result is as follows:
Theorem A. Let G be a non-empty closed subfamily of G  . A solution j on G
satis®es CONS, COCONS*, WPO and OPR if and only if j is the a-core correspondence.
To prove this theorem, I use the following two lemmas:
Lemma 3-3. Let G be a non-empty closed subfamily of G  . The a-core correspondence C on G satis®es CONS, COCONS*, WPO and OPR.
Proof: Let G A G. Denote the player set N G simply by N.
(i) (OPR): trivial.
(ii) (WPO): Let x B WPO G. Then N blocks x. Thus x B C G.
(iii) (CONS): Suppose that x A C G. Let S H N and S 0 q. Then consider the reduced game G S; x . Suppose that xS B C G S; x . Then I getbT H S H N, T 0 q: byT A XT : Ej A T: inf Im ujS; x jyT  > ujS; x xS . Thatis, bT H N, T 0 q: byT A XT : Ej A T: inf zSnT A XSnT inf zNnS A XNnS uj
yT ; zSnT ; zNnS   inf Im uj jyT  > uj x. That is, T blocks x in G. Thissays x B C G, a contradiction.
(iv) (COCONS*): Let x A WPO G with x B C G. Thus bS H N, S 0 q,
N: byS A XS : Ej A S: inf Im uj jyS  > uj x. That is, bS H N, S 0 q, N:
byS A XS : Ej A S: ujS; x yS  > ujS; x xS . Thus I have xS B C G S; x , which
implies x B C  G. Thus I obtain x A WPO G and x B C G ) x B
C  G. That is, x A WPO G X C  G ) x A C G. 8
Remark 3-4: If I alternatively adopt the `standard' blocking concept (see (2) of
Section 1) to de®ne the a-core, the axiom COCONS* is not necessarily satis®ed. (The other three axioms are all satis®ed.) However, the two blocking
concepts (the `standard' one and mine) coincide on some closed subfamilies ofG  . For such families, the `standard' a-core, which is after all the same as mya-core, is characterized by the present axioms. An example of such families isany closed family G satisfying the following condition: EG A G: ES H N G:EyS A XS : Ej A S: min Im uj jyS  exists. For example, the family of all gameswith ®nite strategy sets satis®es this condition. In general, if I adopted the`standard' de®nition of blocking, then the families of games that my axiomatization covers would be more limited.
Lemma 3-5. Let G be a non-empty closed subfamily of G  . There exists at mostone solution j on G which satis®es CONS, COCONS*, WPO and OPR.
Proof: Since CONS, COCONS* and WPO are satis®ed, it is immediate that
EG A G, jN Gj b 2: j G  WPO G X j@ G. Let m be the solution on G
such that EG A G: m G  WPO G. Then applying Lemma 3-1, I have thedesired conclusion. 8
Now Theorem A follows immediately from Lemmas 3-3 and 3-5.

202

K. Takamiya

Remark 3-6: If I replace my reduced games with PT reduced games, then thesefour axioms characterize the strong Nash equilibrium correspondence (Theorem 3.2 in P&T). In other words, the a-core and strong Nash equilibria areaxiomatized with the same combination of axioms but under di¨erent types of
reduced games.
Further, these four axioms are logically independent. The following examples verify this.
Example 3-7. Let j be a solution on G  such that
EG A G  : j G  q:
Then j satis®es CONS, COCONS* and WPO, but not OPR.
Example 3-8. Let j be a solution on G  such that
EG A G  : j G  WPO G:Then j satis®es COCONS*, OPR and WPO, but not CONS. The following
counterexample shows that CONS is not satis®ed:
s2
H1 : s1 3; 0
t1 1; 2

t2
0; 3
2; 1

Here player 1 chooses row while player 2 column. WPO(H1 ) is the whole set
f1g; s1; s2
is:of strategy pro®les. The reduced game H1
s1 3
t1 1
f1g; s1; s2

Then WPO(H1  fs1 g. Thus j violates CONS.

Example 3-9. Let j be the a-IR solution on G  .
Then j satis®es CONS, COCONS* and OPR, but not WPO. This follows from
Theorem B in the next subsection. The following counterexample shows that
WPO is not satis®ed:

H 2 : s1
t1

s2
5; 5
6; 0

t2
0; 6
2; 2

The strategy pro®le t1 ; t2  belongs to the a-IR solution of H2 but is Paretodominated by s1 ; s2 . Thus j violates WPO.

Axiomatization of the a-core203

Example 3-10. Let j be a solution on G  such that
EG A G  : j G  WPO Gq

if jN Gj  1;otherwise:

Then j satis®es CONS, OPR and WPO, but not COCONS*. The following
counterexample checks this:F:s2
s1 1; 4
t1 2; 3

t2
4; 2
3; 1

WPO F   f s1 ; s2 ; s1 ; t2 ; t1 ; s2 g. Consider the reduced games F f1g; t1; s2
and F f2g; t1; s2 .
F f1g; t1; s2 : s1
t1

1
2

F f2g; t1; s2 : s2 t2
3 1
Then t1 A WPO F f1g; t1; s2  and s2 A WPO F f2g; t1; s2 . Thus t1 ; s2  A j F .
Then I get j F  X WPO F  0 q. Thus j violates COCONS*.
3-2. The a-individually rational solution
In Example 3-9, I indicated that the a-IR solution satis®es CONS, COCONS*
and OPR. To verify this, I axiomatize the a-IR solution.
Theorem B. Let G be a non-empty closed subfamily of G  . A solution j on
G satis®es CONS, COCONS and OPR if and only if j is the a-IR solution
correspondence.
Remark 3-11: Comparing with the foregoing axiomatization of the a-core, thissystem of axioms not only drops WPO but adopts a stronger version of converse consistency (COCONS implies COCONS*).
The proof of Theorem B is similar to the proof of Theorem A.
Lemma 3-12. Let G be a non-empty closed subfamily of G  . The a-IR solution
IR on G satis®es CONS, COCONS and OPR.
Proof: Let G A G. Denote the player set N G simply by N.

204

K. Takamiya(i) (OPR): trivial.
(ii) (CONS): Let x A IR G. Suppose that bS HN; S 0q; N: xS B IR G S; x .
That is, bj A S: f jg blocks xS in G S; x . Then setting T  f jg, the same
argument as in (iii) of the proof of Lemma 3-3 applies. Then I have thatf jg blocks x in G. That is, x B IR G, a contradiction.
(iii) (COCONS): Suppose that x A XN and x B IR G. That is, b j A N: f jg
blocks x in G. Then, setting S  f jg, applying the same argument as in(iv) of the proof of Lemma 3-3, I have that f jg blocks xj in G f jg; x . Thatis, xj B IR G f jg; x . The desired conclusion follows. 8
Lemma 3-13. Let G be a non-empty closed subfamily of G  . There exists at
most one solution j on G which satis®es CONS, COCONS and OPR.
Proof: It is immediate from CONS and COCONS that EG A G; jN Gj b 2:j G  j G. Let m be the solution on G such that EG A G: jN Gj b 2 )m G  XN G , and jN Gj  1 ) m G  WPO G. Then applying Lemma
3-1, I obtain the desired conclusion. 8
Now Theorem B follows directly from Lemmas 3-12 and 3-13.
Remark 3-14: Substituting my reduced games with PT reduced games, thesethree axioms give an axiomatization of the Nash equilibrium correspondence(Theorem 2.12 in P&T). This is analogous to the fact mentioned in Remark
3-6. Readers can ®nd some discussion on this point in Section 4.
One can check the logical independence of these three axioms by the following examples:
Example 3-15. Let j be a solution on G  such that
EG A G  : j G  q.
Then j satis®es CONS and COCONS, but not OPR.
Example 3-16. Let j be a solution on G  such that
EG A G  : j G  WPO Gq

if jN Gj  1;otherwise:

Then j satis®es CONS and OPR, but not COCONS.
Example 3-17. Let j be a solution on G  such that
EG A G  : j G  WPO G XN G

if jN Gj  1;otherwise:

Then j satis®es COCONS and OPR, but not CONS.

Axiomatization of the a-core

205

4. Discussion
This section discusses the following two points: (I) On the interpretations of
reduced games. (II) An implication of the facts indicated in Remarks 3-6 and
3-14.
(I)13 Although formally de®ned, I have not yet given an intuitive interpretation of my reduced games. On the other hand, PT reduced games havea straightforward interpretation. They remarked: ``(Let a game G be given
and q 0 S H N G and x A XN G .) If it is common knowledge among themembers of S that the members of N GnS have chosen the strategies xi ;i A N GnS, then the members of S are faced with the game G S; x .'' Although
not so simple as this one, the following interpretation of my reduced games
seems to illustrate the point. Let a game G be given, and let q 0 S H N Gand x A XN G . Now split the game G into two stages. In the ®rst stage, themembers of S choose their strategies, and then, in the second stage, themembers of N GnS move in response knowing what S has chosen. Each
player belonging to S believes that (i) N GnS will play xN GnS if S plays xS ,and that (ii) if S choose yS 0 xS , then N GnS will jointly punish him bypractising the strategies that minimizes his payo¨ (given yS has been chosen).
Then, in e¨ect, the ®rst stage of the game is equivalent to the reduced game
G S; x .
To be fair, however, I have to point out the following di½culties about thisinterpretation.
(a) The members of S do not expect that, in the second stage, the players in
N GnS behave as payo¨ maximizers when S deviated from the strategy
pro®le xS .
(b) It is not always possible for N GnS to jointly choose strategies to simultaneously minimize the payo¨ of each member of S.
(c) Each player j in S believes that, when S collectively deviated from xS , he is
to be punished regardless of whether or not he deviated from his own
strategy xj .
Despite all these facts, I am assuming that the members of S believe (ii) in myinterpretation. I consider this to be an expression of the `pessimism14' found inthe `maximin' thought, which underlies both the a-IR solution and the a-core.
My scenario re¯ects the nature of the solutions to be axiomatized. This makesa clear contrast to the idea behind PT reduced games. In the PT reduced game
G S; x , a player in S is `passive', rather than `pessimistic', in a sense, to the
choices of strategies by the members of N GnS. Or one might see that in the
13 I thank the associate editor for various comments which were useful in improving this part ofthe paper.
14 This word expresses the point that both in the a-core and the a-IR solution, a coalition (a player)
is concerned with every conceivable action open to the complementary coalition regardless of its
likelihood. Scarf (1971) pointed out this as a drawback of the a-core: ``A coalition S, in attempting
to obtain an improved position for all of its members must confront the entire range of strategic
possibilities open to the players not in S, including those which lead to disastrous consequence forthe complementary coalition and would in all probability not be undertaken. This inability to discriminate among counterresponses . . . results in the inclusion of more outcomes in the solution
than might seem reasonable.''206

K. Takamiya

case of the PT reduced game G S; x , a player in S has `more information' about
the choices to be made by the players in N GnS than in the case of my reducedgame.
However, there may be an alternative path to avoid the di½culty (c). One
may de®ne reduced games as follows:
ARG ujS; x yS   uj x

if yj  xj ;

 inf Im uj jyS otherwise:

According to the de®nition (ARG) in the above, only player(s) j who deviatedfrom the strategy xj is to be punished in the corresponding scenario. Interestingly, it is not di½cult to prove that both the a-core and the a-IR solution
satisfy consistency under the de®nition (ARG) (proof in Appendix). Further,
as far as the reduced games with respect to any singleton coalitions (i.e., S is asingleton in the above) are considered, the de®nition (ARG) is the same asDe®nition 2-1. In the proof that the a-IR solution satis®es COCONS (Lemma
3-12, (iii)), I considered only one-person reduced games. Then it is immediatethat the a-IR solution satis®es COCONS also under the de®nition (ARG).
Lemma 3-13 also applies without any change (see Remark 3-2). Thus I conclude that the characterization of the a-IR solution (Theorem B) holds true
even under the de®nition of reduced games (ARG). However, at the present
time, I am not able to axiomatize the a-core using the de®nition (ARG). It is asubject of future research to see whether this is possible or not.
(II) As pointed out in Remarks 3-6 and 3-14, the two pairs of solutions,{the a-core, strong Nash equilibria} and {the a-IR solution, Nash equilibria},
are respectively axiomatized by the same system of axioms under di¨erentde®nitions of reduced games15. The a-core of a game is the set of strategy
pro®les that no coalition blocks. And the a-IR solution is the set of strategy
pro®les that no single player blocks. Similarly, a strong Nash equilibrium is a
strategy pro®le from which no coalition can deviate with each member of thecoalition improving his payo¨. And a Nash equilibrium is a strategy pro®lefrom which no single player can pro®tably deviate. One may see a similarity
between the two pairs of solutions,{the a-core, the a-IR solution} and {strong
Nash equilibria, Nash equilibria}, in the way one solution relates to the other.
The points made in Remarks 3-6 and 3-14 give a formal content to this intuitive similarity. I ®nd this rather important from the viewpoint that one of themain purposes of axiomatizations is to compare solutions.
Appendix
The following claim is established under the de®nition of reduced games (ARG)
in Section 4(I) in the place of De®nition 2-1.
15 Interestingly, a similar thing happens in coalitional form games: The Shapley value and theprenucleolus are axiomatized by the same axioms under di¨erent types of reduced games (Hartand Mas-Colell, 1989 and Sobolev, 1975). Maschler (1990) stated: ``I ®nd this a fascinating result:
It shows that the intrinsic di¨erence between the Shapley value and the prenucleolus lies in theway the subsets S of N interpret `their own game'!''Axiomatization of the a-core

207

Claim: Let G be a non-empty closed subfamily of G  . Then both the a-core
correspondence C and the a-IR solution correspondence IR on G satisfy
CONS.
Proof: (a-core): Let G  N; fXj ; uj gj A N  A G. Assume that x A C G. LetS H N with S 0 q; N. Suppose that xS B C G S; x . Then bT H S H N;
T 0 q: byT A XT : Ej A T: inf Im ujS; x jyT  > ujS; x xS . Denote by U the set
f j A T j yj 0 xj g. Clearly, U 0 q. Then by the de®nition (ARG), Ej A U:
inf zSnT A XSnT inf zNnS A XNnS uj yT ; zSnT ; zNnS  > uj x and Ej A TnU: uj x >uj x. This implies U  T. Thus I have Ej A T: inf Im uj jyT  > uj x. That is,
T blocks x in G. This says x B C G, a contradiction.
(a-IR solution): Similar (Set T  f jg in the above proof ). 8
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