Symmetric Coalitional Binomial Semivalues

Abstract

We introduce here a family of mixed coalitional values. They extend the binomial semivalues to games endowed with a coalition structure, satisfy the property of symmetry in the quotient game and the quotient game property, generalize the symmetric coalitional Banzhaf value introduced by Alonso and Fiestras and link and merge the Shapley value and the binomial semivalues. A computational procedure in terms of the multilinear extension of the original game is also provided and an application to political science is sketched.

Coalitional multinomial probabilistic values

“…, n. To ease the proof of this theorem it is convenient to state a preliminary result, which holds for any profile. Remark 3.10 Theorem 3.9, where linearity could be replaced with additivity, generalizes Theorem 1 in [1] and Theorem 3.5 in [10] (i.e., Theorem 3.6 in [11]). We have checked the logical independence of the axiomatic system for Theorem 3.9 in [14].…”

“…Here we first apply the p-multinomial probabilistic value λ p in the quotient game to obtain a payoff for each union; next, we use within each union the Shapley value, denoted here by φ, to share the payoff efficiently by applying this value to a reduced game played in that union. 5 The proof is essentially the same as in [23,24,1,10,3,11] and will be omitted. Moreover, the new value coincides with one of these two values when the coalition structure is trivial.…”

“…The coalitional p-multinomial probabilistic value Λ p yields the result of a two-step bargaining procedure analogous to that used in [23,24] and also in [1,10,3,11]. Here we first apply the p-multinomial probabilistic value λ p in the quotient game to obtain a payoff for each union; next, we use within each union the Shapley value, denoted here by φ, to share the payoff efficiently by applying this value to a reduced game played in that union.…”

“…also [18,17]) and gave rise later, among others, to the coalitional q-binomial total power property [10,3,11] and the p-multinomial total power property [12,13]. It reduces to this latter if B = B n but it also extends efficiency, to which it reduces if B = B N .…”

“…In 2006, Carreras and Puente [10] (see also [11]) extended the binomial semivalues to games with a coalition structure: they used these values in the quotient game and the Shapley value within unions and obtained the symmetric coalitional binomial semivalues, a family depending on one parameter q ∈ [0, 1] that includes (when q = 1/2) the symmetric coalitional Banzhaf value introduced in 2002 by Alonso and Fiestras [1]. The only axiomatic difference between these new coalitional values and the Owen value is that the former satisfy the total power property whereas the latter satisfies efficiency.…”