264 J
. Abdou Mathematical Social Sciences 39 2000 263 –275
tightness. None of the mentioned conditions is sufficient for strong solvability. However for the two-player case necessary and sufficient conditions for strong solvability have
been established in Abdou 1995. Note that in that case, a new notion is needed in order to accomplish the characterization: joint exactness. The present paper carries out a
similar analysis for the general case. We introduce two new solution concepts called respectively exact core and biexact core which both refine the core and contain the
strong equilibrium outcomes. Therefore new necessary conditions for strong solvability are exact stability and biexact stability. It turns out that exactly stable game forms are
those which are subadditive and exactly tight. Thus our results imply that of Li 1991. The new ingredient between stability and exact stability or equivalently between
tightness and exact tightness is a property that we call exactness. Though the latter appears in this study as coupled with tightness, it has its own interpretation: a game form
is exact if and only if the exact core correspondence coincides with the b-core correspondence. Biexact stability is more stringent. A game form is biexactly stable if
and only if it is subadditive, tight and biexact. This in its turn extends the result of Abdou 1995 where joint exactness appears as a particular case of biexactness. The
second contribution of this paper is a characterization of exact and rectangular game forms.The class of rectangular game forms has been introduced by Gurvich Gurvich,
1975, 1989. A game form is rectangular if the inverse image of any alternative is a cartesian product of strategy subsets. Any free extensive game form, i.e. with the
endpoints of the underlying tree as outcomes, is rectangular. Thus the class of rectangular game forms is quite large. If we add exactness then this class shrinks
drastically: We prove that an exact rectangular game form is essentially a one-player game form. Note that we do not even need tightness for this ‘‘impossibility result’’. In
particular any strongly solvable rectangular game form is essentially a one-player game form. This generalizes a result of Ichiishi 1985 where the same is stated for free
extensive game forms.
2. Definitions and preliminaries
Throughout this paper we shall consider a game form G 5 X , . . . , X , A, g. The set
1 n
N 5 h1, . . . , nj where n 2 is the set of players, X is the strategy set of player i,
i
i [ N , A is the set of alternatives and g:
P
X →
A is the outcome function. We
i [N i
shall assume that A is finite and g is onto. In the sequel we shall adopt the following:
Notation 2.1. For every set X, we denote by PX resp. P X the set of all resp.
non-empty subsets of X and by LX the set of all linear orders on X. Set difference is denoted by the symbol ‘‘ 2 ’’. For every S [ PN 2
h5j, the product
P
X is denoted
i [S i
c
X by convention X
is the singleton h5j and N 2 S is denoted S . Similarly if
S 5
c
B [ PA, A 2 B is denoted B . If x [ X the notation gx , X stands for
hgx ,
c
N N
S S
S N
y uy [ X j if S ± 5 and for gX if S 5 5. For every preference profile R [ LA
c c
c
S S
S N
N i
c
and S [ P N we put Pa, S, R 5 hb [ AubR a ;i [ Sj and P a, S, R 5 A 2 Pa, S,
N N
c
R . By convention Pa, 5, R 5 A 2 haj and P a, 5, R 5 haj. Finally we denote by
N N
N
G, R the game in strategic form derived from G and R .
N N
J . Abdou Mathematical Social Sciences 39 2000 263 –275
265
In what follows we shall introduce two new solution concepts for game forms called the exact core and the biexact core which both are refinements of the usual b -core. For
completeness we recall also more classical definitions: Let G be a game form and let
N
R [ LA .
N
• x [ X is a strong equilibrium of G,R if the following does not hold:
N N
N
S [ PN, y [ X : gx , y [ P gx , S, R .
c
S S
S S
N N
• a [ A is a strong outcome of G, R if for some strong equilibrium x [ X one has:
N N
N
gx 5 a.
N
• a [ A is dominated in G, R if there exists S [ PN such that:
N
;x [ X , y [ X : g y , x [ Pa, S, R
c c
c
S S
S S
S S
N
• a [ A is exactly dominated in G, R if there exists S [ PN such that:
N
;x [ X : gx 5 a, y [ X : g y , x [ Pa, S, R
c
N N
N S
S S
S N
• a [ A is biexactly dominated in G, R if there exists S [ PN such that:
N
;x [ X : gx 5 a
N N
N
either y [ X : g y , x [ Pa, S, R
c
S S
S S
N
or
c
y [ X : g y , x [ Pa, S , R
c c
c
S S
S S
N
In words, an alternative a is exactly dominated whenever some fixed coalition S can improve upon a by deviating each time a is proposed whereas a is dominated if some
c
fixed coalition S can improve upon a whatever is the behaviour of S and a fails to be a strong equilibrium outcome if each time that a is proposed, some coalition, depending
on the strategy vector which yields a, can be better off by deviating. An alternative a is
c
biexactly dominated whenever some fixed splitting of the players hS, S j exists for which
c
each time a is proposed, either S or S can improve upon that alternative by deviating. In the core resp. exact core, resp. biexact core logic the objecting coalition forms when
some alternative is proposed whereas in the strong equilibrium logic the objecting coalition forms only when a strategy profile is proposed.
2
The core resp. exact core, resp. biexact core of G, R is the set of undominated
N
resp. exactly undominated, resp. biexactly undominated alternatives. They are denoted respectively by CG, R , C G, R and C G, R . The set of strong equilibrium
N j
N 2j
N
outcomes is denoted SOG, R .
N
2
Called the b-core elsewhere.
266 J
. Abdou Mathematical Social Sciences 39 2000 263 –275
G is said to be strongly solvable resp. stable, resp. exactly stable, resp. biexactly stable if the set SOG, R resp. CG, R , resp. C G, R , resp. C G, R is
N N
j N
2j N
N
non-empty for any R [ LA . The following is clear from the definitions:
N N
Proposition 2.2. i For any game form G and any preference assignment R [ LA
N
we have the following inclusions: SOG, R , C G, R , C G, R , CG, R . ii
N 2j
N j
N N
For n 5 2 one has: SOG, R 5 C G, R
N 2j
N
The inclusions of the proposition may all be strict as shown in the following:
Example 2.3. The core can be strictly larger than the exact core. Consider the following two-player game form where player one plays rows and player two plays columns:
Take the preference assignment R 5 R , R where bR cR a and cR bR a. An easy
1 2
1 1
2 2
computation shows that CG, R 5 hb, cj, C G, R 5 C G, R 5 SOG, R 5 hcj.
j 2j
Alternative b is in the core because it is not Pareto dominated and it is the top alternative for player one; moreover Player two cannot improve upon b if x is played by player
1
one. However b fails to be in the exact-core since player two can improve upon b each time b is the proposed outcome that is if y , x is to be played.
1 2
Example 2.4. The exact core can be strictly larger than the biexact core. Consider the following two-player game form:
Take the preference assignment P 5 P , P where cP aP bP d and dP aP bP c. One
1 2
1 1
1 2
2 2
has: CG, P 5 C G, P 5 haj, C G, P 5 SOG, P 5 5. Alternative a is not a strong
j 2j
outcome because each time that a is proposed one of the players can improve upon it but it is in the exact-core since no fixed player can by himself improve upon it in all
situations where it occurs namely z , x and x , z .
1 2
1 2
Example 2.5. The biexact core can be strictly larger than the strong outcome set when n 3. Consider the following three-player game form where player 1 plays rows x , y ,
1 1
player 2 plays columns x , y , z and player 3 plays matrices x , y :
2 2
2 3
3
J . Abdou Mathematical Social Sciences 39 2000 263 –275
267
The preference assignment R is as follows: R :
c R
a R
d R
b
1 1
1 1
R : d
R a
R c
R b
2 2
2 2
R : d
R c
R a
R b
3 3
3 3
One can verify that a [ C G, R but a [ ⁄ SOG, R .
2j N
N
It follows from Proposition 2.2 that necessary conditions for stability, exact stability and biexact stability provide necessary conditions for strong solvability. Our aim is
therefore to obtain structural characterizations for each of these properties. We begin by stability and for that purpose we recall some useful definitions related to the core.
3 G
The effectivity function is the mapping E : PN →
PPA where for any S [ PN :
G
E S 5 hB , Aux [ X , gx , X , Bj
c
S S
S S
The main reason to introduce the effectivity function is that it captures precisely the power of coalitions needed to define the core. Indeed it is easy to verify the following:
N
Proposition 2.6. For any R [ LA and a [ A, a is in the core of G, R if and only if
N N
c c
;S [ PN: P a, S, R [ ES .
N
Moreover the core correspondence and the effectivity function carry the same data about the underlying game form. Actually it is not difficult to see that one can compute
one given the other and vice versa. The following definitions related to the effectivity function allow an elegant characterization of stability.
• G is said to be tight if: ;S [ PN ,;B [ PA:
G c
G c
B [ ⁄ E S
⇒ B [ E S
• G is said to be subadditive if: ;S [ PN , ;S [ PN , ;B [ PA, ;B [ PA:
1 2
1 2
G G
G
B B 5 5, B [ E S , B [ E S ⇒
B B [ E S S
1 2
1 1
2 2
1 2
1 2
We end this section by stating a characterization of stability which can be easily deduced from Abdou 1982 and Peleg 1984, theorem 6.A.9.
Theorem 2.7. G is stable if and only if it is tight and subadditive.
Tightness and subadditivity are thus necessary conditions for strong solvability but are far from being sufficient even for n 5 2 see Abdou, 1995. We are thus lead to
investigate exact and biexact stability in order to obtain stronger necessary conditions.
3
Called the a-effectivity function elsewhere.
268 J
. Abdou Mathematical Social Sciences 39 2000 263 –275
3. Exactly and biexactly stable game forms
