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Exact stability and its applications to strong solvability
* J. Abdou
´ ´
CERMSEM, Universite de Paris 1, 12, Place du Pantheon, 75005 Paris, France Received 1 October 1997; received in revised form 1 January 1999; accepted 1 April 1999
Abstract
We introduce the exact core and the biexact core of a strategic game form. Those are solutions which lie between the usual b-core and the set of strong equilibrium outcomes. We define the corresponding notion of exact and biexact stability. We prove that a game form is exactly stable if and only if it is exact, tight and subadditive and that it is biexactly stable if and only if in addition it is biexact. As an application, we study the exactness of rectangular game forms. We prove that an exact rectangular game form is essentially a one-player game form. In particular any strongly solvable rectangular game form is essentially a one-player game form. This generalizes a result of Ichiishi, 1985. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
This paper contributes to the study of strongly solvable finite game forms via two new solution concepts namely the exact core and the biexact core and the related notions of 1 exact and biexact stability. A game form (henceforth GF) is said to be strongly solvable if for any preference assignment to the players, the resulting game has a pure strong equilibrium. A complete and useful characterization of such game forms seems to be out of reach for the time being. But some results are already available. Necessary conditions for strong solvability, as a by-product of strong implementation theory, can be found in Peleg (1984) and Abdou and Keiding (1991). Since a strong equilibrium outcome is in theb-core then a necessary condition for strong solvability is stability and the latter can be characterized by two structural properties: subadditivity and tightness. Recently Li (1991) has made a step further: tightness is replaced by a stronger property: exact
*Tel.: 133-1-46-33-3448; fax:133-1-46-33-3448. E-mail address: [email protected] (J. Abdou)
1
Or strongly consistent: but since the words ‘‘consistence’’ and ‘‘consistency’’ are awfully polysemic we prefer ‘‘solvable’’, ‘‘solvability’’ which belong to the same paradigm as ‘‘solution’’.
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.
shall assume that A is finite and g is onto. In the sequel we shall adopt the following:
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 usualb-core. For completeness we recall also more classical definitions: Let G be a game form and let
N
R [L(A) .
N
• x [X is a strong equilibrium of (G,R ) if the following does not hold:
N N 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 playershS, S jexists 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 undominatedN
(resp. exactly undominated, resp. biexactly undominated) alternatives. They are denoted respectively by C(G, R ), C (G, R ) and C (G, R ). The set of strong equilibriumN j N 2j N
outcomes is denoted SO(G, R ).N
2
G is said to be strongly solvable (resp. stable, resp. exactly stable, resp. biexactly
stable) if the set SO(G, R ) (resp. C(G, R ), resp. C (G, R ), resp. C (G, R )) isN N j N 2j N
N
non-empty for any R [L(A) . The following is clear from the definitions:
N
N
Proposition 2.2. (i ) For any game form G and any preference assignment R [L(A)
N
we have the following inclusions: SO(G, R ),C (G, R ),C (G, R ),C(G, R ). (ii )
N 2j N j N N
For n52 one has: SO(G, R )N 5C (G, R )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 R5(R , R ) where bR cR a and cR bR a. An easy1 2 1 1 2 2 computation shows that C(G, R)5hb, cj, C (G, R)j 5C (G, R)2j 5SO(G, R)5hcj. 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 player1 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 P5(P , P ) where cP aP bP d and dP aP bP c. One1 2 1 1 1 2 2 2 has: C(G, P)5C (G, P)j 5haj, C (G, P)2j 5SO(G, P)55. Alternative a is not a strong 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
The preference assignment R is as follows:
R :1 c R1 a R1 d R1 b
R :2 d R2 a R2 c R2 b
R :3 d R3 c R3 a R3 b
One can verify that a[C (G, R ) but a[⁄ SO(G, 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 : P(N )→P(P(A)) where for any S[P(N ):
G
E (S )5hB,Au'xS[X , g(x , X )S S Sc ,Bj
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 [L(A) and a[A, a is in the core of (G, R ) if and only if
N N
c c
;S[P(N ): P (a, S, R )[E(S ).
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[P(N ),;B[P(A):
G c G c
B[⁄ E (S)⇒B [E (S )
• G is said to be subadditive if:;S [P(N ), ;S [P(N ),;B [P(A),;B [P(A):
1 2 1 2
G G G
B >B 55, 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 n52 (see Abdou, 1995). We are thus lead to investigate exact and biexact stability in order to obtain stronger necessary conditions.
3
3. Exactly and biexactly stable game forms
The aim of this section is to formulate necessary and sufficient conditions for exact and biexact stability. It turns out that the relevant tools in this context are in the same spirit as the effectivity function. We put $5h(B , B )1 2 [P(A)3P(A)uB1>B2±5,
B <B 5Aj.
1 2
G
Definition 3.1. The exact effectivity function is the mapping E : P(N )j →P(P (A)) where0
G
for any S[P(N ) and B[P (A), B[E (S ) if and only if:
0 j
;a[B,'xS[X , aS [g(x , X )S Sc ,B
G
The biexact effectivity function is the mapping E : P(N )→P($) where for any 2j
G
S[P(N ) and (B , B )[$, (B , B )[E (S ) if and only if:
1 2 1 2 2j
;a[B1>B ,2 'xN[X : g(x )N N 5a and g(x , X )S Sc ,B , g(x , X )1 Sc S ,B2
The following definitions provide variants of the same concepts:
G
Definition 3.2. Let a[A. The effectivity function of G at a is the mapping E (?ua):
P(N )→P(P(A) where for any S[P(N ):
Such definitions are highly motivated by the following:
Proposition 3.3. (i ) a[A is in the exact-core of (G, R ) if and only if ;S[P(N ):
and the biexact core respectively what E is to the core. Moreover the exact core
G
correspondence (resp. the biexact core correspondence) carry the same data as (E (?ua),
(i ) E (S )j 5hB[P(A)u'5±YS,X : g(Y ,X )S S Sc 5Bj (ii ) E (S )5hB[P (A)u;a[B: B[E(Sua)j
j 0
(iii );a[A: E(Sua)5hB[P(A)u'C[E (S ): a[C,Bj
j
We see from Fact 3.4 that actually, the collection (E(?ua), a[A) carry the same data
as E . It is easy to see that E can be computed from (E (?ua), a[A), the converse
j 2j 2
being probably false. The relations between E , E and E are stated in the following:2j j
Fact 3.5. For any S[P(N ):
This shows that the biexact effectivity function is finer than the exact effectivity function which in its turn is finer than the effectivity function. One can compute the latter from the former but not vice versa.
The following definitions are based on the exact and biexact effectivity functions and 4
have been first formulated in Abdou (1995) for two-player game forms.
Definition 3.6. G is exact if:
It follows that G is exact if and only if E is monotonic in the following sense:j
B[E (S ), B9.B⇒B9[E (S ),
j j
and that G is biexact if and only if E2j satisfies the following properties:
(i) (C , C )[$, C .B , C .B ,(B , B )[E (S )⇒(C , C )[E (S )
1 2 1 1 2 2 1 2 2j 1 2 2j
c
(ii) (B , B )[$, (B , A)[E (S ), (B , A)[E (S )⇒(B , B )[E (S )
1 2 1 2j 2 2j 1 2 2j
The following provides a useful and strategically meaningful characterization of exactness and biexactness:
Proposition 3.7. Let G be a GF. Then:
(i ) G is biexact if and only if : C (G,2j ?)5C(G, ?) (ii ) G is exact if and only if C (G,j ?)5C(G, ?)
4
Proof. Assume that G is biexact and let a[C(G, R ). For any coalition S, put
We have the following necessary conditions for exact and biexact stability:
Proposition 3.8. Let G be a GF. Then: (i ) If G is biexactly stable then G is biexact, (ii ) If G is exactly stable then G is exact.
The following definition can be found in Li (1991):
Definition 3.9. G is exactly tight if: ;S[P(N ), ;B[P(A):
c c
B[⁄ E (S)⇒B [E (S )
j j
Proposition 3.10. G is exactly tight if and only if G is both tight and exact.
Proof. If G is exact then exact tightness and tightness are equivalent. If G is exactly
c c c c
tight, let B[E(S ), then B [⁄ E(S ) hence B [⁄ E (S ). By exact tightness B[E (S ). It
j j
follows that G is exact, moreover since E5E , G is tight.j h
By combining Proposition 3.7 and Theorem 2.7, we have a characterization of exact and biexact stability which provides a set of necessary conditions for strong solvability:
Theorem 3.11. Let G be a game form. The following statements are equivalent: (i ) is biexactly (resp. exactly) stable,
(iii ) G is tight, subadditive and biexact (resp. exact).
It follows that our results imply that of Li (1991). We end this section by a couple of remarks and an example:
Remark 3.12. Theorem 3.11 and the literature on strong implementation allow us to
N
answer the following question: Let H: L(X ) →A be a social choice correspondence (non-empty valued). Under which conditions H is the biexact core (resp. exact core) correspondence of some game form? The answer is that this is the case if and only if H is the core correspondence of some tight and stable (abstract) effectivity function (Moulin and Peleg, 1982): Clearly if H is the biexact core (rep. exact core) corre-spondence of some game form G then by Proposition 3.7 and Theorem 3.11 this is also
G
the core of the effectivity function E which is stable and tight. Conversely if H is the core correspondence of some stable and tight effectivity function E, then the core correspondence of E is strongly implementable – say by G: SO(G, ?)5C(E, ?); by Proposition 2.2, H is also the biexact core and the exact core correspondence of G.
Remark 3.13. We have the following general result the proof of which is constructive but cannot be included here: to any game form G one can associate an exact game form
D G
Dwith the same effectivity function as G: E 5E . We do not know whether a similar result is true for biexactness. However by the preceding remark if G is stable then one can find a biexact game form which has the same effectivity as G.
Example 3.14. Let X be a finite set such thatuXu$2 and 0[⁄ X and let A5X<h0j. The
n-player unanimity game form on X is defined by setting: Xi5X, i51, . . . , n and
g(x , . . . , x )1 n 5 x if x15 ? ? ? 5xn5x
g(x ,1 ? ? ?,x )n 5 0 otherwise.
One can compute the following:
E(S )5hB[P(A)uuBu$2, 0[Bj if uSu51
E(S )5hB[P(A)u0[Bj if 2#uSu,n
E(N )5P (A)0
Let a[X:
E (Sua)5h(B , B )[$uh0, aj,B >Bj if 1#uSu#n21
2 1 2 1 2
E (S2 u0)5h(B , B )1 2 [$u0[B1>B2j if 2#uSu#n22
E (S2 u0)5h(B , B )1 2 [$u0[B1>B ,2 uB1u$2j if uSu51, n$3
E (Su0)5h(B , B )[$u'a , a [X, a ±a ,h0, aj,B ,h0, aj,Bj if uSu51, n52
2 1 2 1 2 1 2 1 1 2 2
E (S)5h(B , B )[$u0[B >B ,uBu$2j if uSu51, n$3
2j 1 2 1 2 1
E (S)5h(B , B )[$u0[B >Bj if 2#uSu#n22
2j 1 2 1 2
It follows that a unanimity game form is biexact for n$2. Note that a unanimity game form is not tight hence not stable.
4. Rectangular game forms
This section is devoted to the applications of the results of the preceding section to rectangular game forms. A game form G is said to be rectangular if for every a[A the
inverse image of a by g is a direct product. This class includes all free extensive game forms. We shall prove that a rectangular game form is exact if and only if it is essentially a one-player game form. As a consequence every strongly solvable rectangular game form hence every strongly solvable free extensive game form is essentially a one-player game form. This generalizes a result of Ichiishi (1985) and Kolpin (1988).
Definition 4.1. G is rectangular if for every a[A there exist Y ,X , . . . , Y ,X such
1 1 n n
21
that: g (a)5
P
i[N Y .iFact 4.2. Each of the following properties is equivalent to rectangularity:
(i );S[P(N ),;xS[X ,S ;yS[X ,S ;xSc[X ,Sc ;ySc[X : g( y , x )Sc S Sc 5g(x , y )S Sc 5
An extensive game form is defined like the standard notion of extensive game with the difference that a path leads to an abstract outcome. An extensive game form is free when distinct paths lead to distinct outcomes. Formally an extensive game form is defined by the following array:
˜ G 5[N,7,8, (A , u[8), (8, i[N ), A, f ]
u i
where N is the set of players, 75(Z, r,u) is a finite oriented tree. Precisely Z is a finite set the elements of which are called nodes, r is a distinguished node called the root of the tree and u: Z2hrj→Z is a mapping which satisfies the following property:
k
;x[Z2hrj,'k$1 such thatu (x)5r.u(x) is called the predecessor of x. We denote 21
by Z the set of terminal nodes i.e. the set of x[Z such thatu (x)55.8 is a partition
T
of Z2Z with the following property: to every u[8 is attached a set A which, for
T u
21
every x[u, is mapped bijectively tou (x). An element of8 is called an information
set and A is called the set of actions available at u.8 itself is a disjoint union of (8,
u i
i[N ) where8 is the set of information sets of player i. Note that neither perfect recall
i
nor even linearity is assumed. Let V be the set of paths of G. f is a mapping fromV G
˜
where the strategy set of player i[N is X 5
P
A . For every x5(x , . . . ,Proposition 4.3. If G is a free extensive game form then G is rectangular.
G
Proof. Rectangularity of G is equivalent to rectangularity of each of the game forms of the family (X , X , A, g), (TT Tc [P(N )) (Fact 4.2(ii)). Therefore it suffices to prove the
property for two-player free extensive game forms the normal form of which can be G
and y prescribe the same action a [A at r, namely the unique action that makes the
i0 i0 u0
transition from r to z . The same action a1 i0 will be taken by the same player i at x in0 0
9
the strategy vector (x , y ). Since this action leads to z , we have: z1 2 1 15z .1 h
It is not true that every rectangular game form is the normal form of a free extensive game form as we can see in the following:
Example 4.4. The following two-player game form in which player one chooses a row and player two a column is rectangular but cannot be the (strategic) game form of a free extensive game form:
If such an extensive game form were to exist, then there would exist a node which has more than one successor. Let u be the closest node to the root r satisfying this property and let i be the player who decides at u. Clearly player i has a nontrivial partition of his strategy set which induces a nontrivial partition on the outcome set, which is clearly not true.
Definition 4.5. Let G5(X , . . . , X , A, g) be a game form. Player i is said to be a1 n
dummy player if:
;x , y [X ,;x [X : g(x , x )5g( y , x ).
i i i 2i 2i i 2i i 2i
A player who is not a dummy player is said to be an active player.
In Gurvich (1978) the following is proved: A strategic game form G is equivalent to that of a free extensive game form with perfect information if and only if it is
result combined with that of Ichiishi (1985) provides a characterization of stable rectangular game forms and that of strongly solvable rectangular game forms. In the latter case we have the following: A rectangular game form is strongly solvable if and
only if it has at most one active player.
Remark 4.6. In Li (1991) Theorem 3.3 the author states that a two-player rectangular and exactly tight game form is necessarily strongly solvable. In what follows we shall prove that actually such a game form has at most one active player!
More generally we shall prove the following:
Theorem 4.7. A rectangular game form G is exact if and only if it has at most one active player.
Corollary 4.8. A rectangular game form G is exactly stable if and only if it has at most
one active player.
Corollary 4.9. A rectangular game form G is strongly solvable if and only if it has at
most one active player.
Corollary 4.10. A free extensive game form G is strongly solvable if and only if it has
at most one active player.
Corollary 4.10 has been proved in Ichiishi (1985), Remark 2, p. 169. Actually Corollary 4.9 can be deduced from a result of Gurvich (1989) and Ichiishi (1985).
Proof of Theorem 4.7. A game form with no active player is trivial, that isuAu51. A game form with one active player is necessarily rectangular and exact. We shall prove the ‘‘only if ’’ part by first treating the case n52.
This ends the proof of the claim.
follows that y [Y and g( y , X )5hbjwhich again by rectangularity implies that g(x ,
1 1 1 2 1
X )5hbj. Since x [Y is arbitrary we proved that Player 2 is dummy.
2 1 1
If Y155, Y255then A5hajand G is trivial. This ends the proof of the case n52.
Case n$3. The proof is by induction on n. Assume that an exact rectangular
(n21)-player game form has at most one active player and let G5(X , . . . , X , A, g)1 n
be exact and rectangular and uAu$2. Let G be the two-player game form (X , Xn n 2n, A,
g). One player of G is player n of G and the other player is the coalition Nn 2hnj.
Clearly G is rectangular and exact so that it has only one active player. If player n is then
active player of G then n is the only active player in G. If Nn 2hnjis the active player of ¯
G then all strategies of player n are duplicates of each other in G. The game form Gn
obtained from G by eliminating all but one strategy of player n is an (n21)-player ¯ game form which is obviously rectangular and exact. By the induction hypothesis G has only one active player then so does G itself. h
Acknowledgements
I would like to thank an anonymous referee for his helpful comments.
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´
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