Exactly and biexactly stable game forms

268 J . Abdou Mathematical Social Sciences 39 2000 263 –275

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 5 hB , B [ PA 3 PAuB B ± 5, 1 2 1 2 B B 5 A j. 1 2 G Definition 3.1. The exact effectivity function is the mapping E : PN → PP A where j G for any S [ PN and B [ P A, B [ E S if and only if: j ;a [ B, x [ X , a [ gx , X , B c S S S S G The biexact effectivity function is the mapping E : PN → P where for any 2j G S [ PN and B , B [ , B , B [ E S if and only if: 1 2 1 2 2j ;a [ B B , x [ X : gx 5 a and gx , X , B , gx , X , B c c 1 2 N N N S S 1 S S 2 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: PN → PPA where for any S [ PN : G E S ua 5 hB , Aux [ X , a [ gx , X , Bj c S S S S G The bieffectivity function of G at a is the mapping E ? ua: PN → P where for 2 any S [ PN : G E S ua 5 hB , B [ ux [ X : gx 5 a and 2 1 2 N N N gx , X , B , gx , X , B j c c S S 1 S S 2 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 [ PN : N c G c P a, S, R [ E S ua. N ii a [ A is in the biexact-core of G, R if and only if ;S [ PN : N c c c G P a, S , R , P a, S, R [ E S ua. N N 2 G G It follows from Propositions 2.6 and 3.3 that E ? ua and E ? ua are to the exact core 2 G 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, G a [ A resp. E ? ua, a [ A. In the sequel we shall drop the reference to G in the 2 G G G G notations of E , E ? ua, E , E etc . . . . Some of the relations between the objects just j 2j defined are easy to state: Fact 3.4. For every S [ PN one has the following: J . Abdou Mathematical Social Sciences 39 2000 263 –275 269 i E S 5 hB [ PAu5 ± Y , X : gY ,X 5 Bj c j S S S S ii E S 5 hB [ P Au;a [ B: B [ ESuaj j iii ;a [ A: ES ua 5 hB [ PAuC [ 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 [ PN : i ES 5 ES ua a [ A ii ES 5 hB [ PAuB9 [ E S: B9 , Bj j iii E S 5 hB [ PAuB,A [ E Sj j 2j 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: E 5 E j G is bi-exact if for all S [ PN : c E S 5 ES 3 E S 2j 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 E satisfies the following properties: 2j 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, ? 5 CG, ? 2j ii G is exact if and only if C G, ? 5 CG, ? j 4 In this reference biexactness was called joint exactness. 270 J . Abdou Mathematical Social Sciences 39 2000 263 –275 Proof. Assume that G is biexact and let a [ CG, R . For any coalition S, put N c c c B 5 P a, S , R and B 5 P a, S, R . In view of Proposition 2.6 we have B [ ES , 1 N 2 N 1 c B [ ES and B B 5 A so that by biexactness B , B [ E S and since a [ B 2 1 2 1 2 2j 1 B it follows that B , B [ E S ua so that by Proposition 3.3ii, a [ C G, R . 2 1 2 2 2j N c Conversely, assume that C G, ? 5 CG, ? . Let S [ PN , B , B [ ES 3 ES 2j 1 2 such that B B 5 A. For every a [ B B let Q , Q [ LA 3 LA satisfy: 1 2 1 2 1 2 c c Q : B Q aQ B B \ hajQ B 1 2 1 1 1 2 1 1 c c Q : B Q aQ B B \ hajQ B . 2 1 2 2 1 2 2 2 Define a preference profile R 5 R , R such that R 5 Q ;i [ S and R 5 Q c N S S i 1 j 2 c ; j [ S . Clearly haj 5 CG, R and by assumption the latter is equal to C G, R so N 2j N that B , B [ E S ua and since this is true for every a [ B B it follows that B , 1 2 2 1 2 1 B [ E S . The proof of ii is left to the reader. h 2 2j 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. Proof. i Assume that G is not biexact. We can find S, B , B ,a such that a [ B B , 1 2 1 2 c B B 5 A, B [ ES , B [ ES and B , B [ ⁄ E S ua. Take the profile defined in 1 2 1 2 1 2 2 c the proof of Proposition 3.7. We have C G, R , CG, R 5 haj. Moreover P a, S, 2j N N c c R 5 B and P a,S , R 5 B so that a [ ⁄ C G, R Proposition 3.3ii. It follows N 2 N 1 2j N that C G, R is empty. The proof of ii is similar. h 2j N The following definition can be found in Li 1991: Definition 3.9. G is exactly tight if: ;S [ PN , ;B [ PA: 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 [ ES , then B [ ⁄ ES hence B [ ⁄ E S . By exact tightness B [ E S. It j j follows that G is exact, moreover since E 5 E , G is tight. h j 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, ii G is stable and biexact resp. exact, J . Abdou Mathematical Social Sciences 39 2000 263 –275 271 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: LX → 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: SOG, ? 5 CE, ? ; 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 D with the same effectivity function as G: E 5 E . 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 that uXu 2 and 0 [ ⁄ X and let A 5 X h0j. The n-player unanimity game form on X is defined by setting: X 5 X, i 5 1, . . . , n and i gx , . . . , x 5 x if x 5 ? ? ? 5 x 5 x 1 n 1 n gx , ? ? ? ,x 5 otherwise. 1 n One can compute the following: ES 5 hB [ PAuuBu 2, 0 [ Bj if uSu 5 1 ES 5 hB [ PAu0 [ Bj if 2 uSu , n EN 5 P A Let a [ X: E S ua 5 hB , B [ uh0, aj , B B j if 1 uSu n 2 1 2 1 2 1 2 E S u0 5 hB , B [ u0 [ B B j if 2 uSu n 2 2 2 1 2 1 2 E S u0 5 hB , B [ u0 [ B B , uB u 2j if uSu 5 1, n 3 2 1 2 1 2 1 E S u0 5 hB , B [ ua , a [ X, a ± a , h0, a j , B , h0, a j , B j if uSu 5 1, n 5 2 2 1 2 1 2 1 2 1 1 2 2 E S 5 hB , B [ u0 [ B B , uB u 2j if uSu 5 1, n 3 2j 1 2 1 2 1 E S 5 hB , B [ u0 [ B B j if 2 uSu n 2 2 2j 1 2 1 2 E S 5 hB , B [ ua , a [ X, a ± a , h0, a j , B , h0, a j , B j if uSu 5 1, n 5 2 2j 1 2 1 2 1 2 1 1 2 2 272 J . Abdou Mathematical Social Sciences 39 2000 263 –275 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