The Shapley value in totally convex multichoice games

-In this paper, we introduce a class of totally convex multichoice cooperative games a.nd prove that the Shapley value of such games is always in the core. @ 2000 Elsevier Science I~td. All rights reserved. K e y w o r d s M u l t i c h o i c e game, Shapley value, Total convexity, Core, Coalition. 1. I N T R O D U C T I O N Hsiao and R a g h a v a n [1] i n t roduced a class of mul t ichoice coopera t ive games and tbund its Shap ley value using an ax iomat i c approach. Later , Nouweland et al. [2] de t e rmined the Shap ley value ~br mul t ichoice coopera t ive games following its p robabi l i s t i c in t e rp re ta t ion . Howewu. the w~lues o b t a i n e d by these two me thods are qui te different. In our paper , while avoiding the p rob lem of incons is tency of the Shap ley value between Hs iao-Raghavan and Nouweland, we consider a necessary and sufficient condi t ion for the Shap ley vahle by Nouweland to be in the core of a mul t ichoice coopera t ive game. I t is well known t h a t in the class of coopera t ive games in t i le charac te r i s t i c funct ion form, the Shap ley value is in the core if the charac te r i s t i c funct ion is e i ther convex [3], aw~rage convex [4], or t o t a l l y convex [5]. The l a t t e r pape r shows t h a t the class of t o t a l l y convex games inchnles tha.t of average convex games. We discuss condi t ions for the Shap ley value to be in the core tot the ('lass of mul t ichoice coopera t ive games. 2. M U L T I C H O I C E C O O P E R A T I V E G A M E Fi r s t of all, we descr ibe the mult ichoice coopera t ive game (MCG) in t roduced ill [1]. Let N -= {1, 2 , . . . , 'n} be the set of players, M~ = {0, 1, 2 . . . . ,m4} the set of ac t iv i ty levels of player i ¢ N . Vv'e assume t h a t mi = L, L ¢ R +, for all i c N as in [1]. A coal i t ion ill M C G is d e n o t e d by a vec tor s = ( S l , . . . , s n ) , where s, ¢ M i , i c N . shows ac t iv i ty level of p layer i in t he coal i t ion s. If a p layer does not coopera te , his level of ac t iv i ty is set a t zero. Hence, 0893-9659/00/$ see fl'ont matter @ 2000 Elsevier Science Ltd. All rights reserved. Typeset 1)y ,4.~S-'I~F~\ PII: S0893-9659(99)00216-5 9 6 D . A . A Y O S H I N A N D T . T A N A K A the coali t ion in which no player par t ic ipa tes is specified by the zero vector 0 = ( 0 , . . . , 0). We denote the set of all coalit ions by M = M1 x . . x Mn. T h r o u g h o u t this paper , a coali t ion s A t = (min{s l , t l} , min{s2, t 2 } , . . . , min{s~, t~}) is considered as the intersection of coali t ions s and t, and a coali t ion s Vt = (max{s1, t l} , max{s2, t 2 } , . . . , max{sn, tn}) is admi t t ed as the union of s and t. A superaddi t ive function v: M --* R 1 with v(O) = 0 is called a character is t ic funct ion of an MCG. I t is easily seen t ha t v(m), m = ( m l , . . . , ran), is the max imal value of the character is t ic function. We denote M C G by G(v, N). Consider an (m + 1) x n-dimensional payoff ma t r ix ~ = (~ji) d is t r ibut ing v(m) among all players and their ac t iv i ty levels. A componen t ~ji shows the increase in payoff to player i when he changes his ac t iv i ty f rom level j 1 to level j . I t is said t h a t the payoff ma t r ix ~ is efficient if ~-~i=1 m i _ . , . , ~j£o ~ji = v(m) and it is level increase rational if ~ = 0 ~ji > v ( ( 0 , . . 0, si, 0 , . . 0)), where i E N , s~ E M+ An efficient and level increase rat ional payoff ma t r ix is called imputation and considered as a solution of G(v, N). Let I(v, N) be the set of all impu ta t ions in G(v, N). We shall say t h a t the set C(v, N) {f E I(v, N) { ~ s~#o ~ = ~'~jLO ~ji > V(S) for all s E M } is the core of G(v, N). 3. T O T A L C O N V E X I T Y In [2], the following procedure of cons t ruc t ion was proposed for the Shapley value. Suppose t h a t a given coalit ion s E M is formed s tep-by-s tep , s ta r t ing from the zero coali t ion 0 = ( 0 , . . . , 0). On each s tage of the procedure , one of the players has to increase his act iv i ty level by 1. Thus, the coali t ion s E M will be created after k(s) = ~ i : ~ # 0 si steps, i.e., each player i E N will reach his level of ac t iv i ty s~ in s. Define M + = { ( i , j ) ] i E N , j E M~ \ {0}}. An admissible order is a bijection w:M + --~ { 1 , 2 , . . . ,~-~ieNmi} satisfying w((i,j)) < w((i , j + 1)) for all i c~N and j E {1, 2 , . . . , mi 1}. The number of the admissible orders for G(v, N) is