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Probabilistic values for games. (English) Zbl 0707.90100
The Shapley value. Essays in honor of Lloyd S. Shapley, 101-119 (1989).
[For the entire collection see Zbl 0694.00032.]
A game here is a set of players, \(N=\{1,2,...,n\}\), and a function v that assigns to each coalition \(S\subset N\) a number v(S), interpreted as the payoff the coalition S gets if it forms. The paper examines values of such games, namely functions \(\phi_ i(v)\), that measure for the player i, his prospects from participation in the game v. The paper concentrates on probabilistic values, namely functions \(\phi_ i\) obtained by a formula \[ \phi_ i(v)=\sum_{T}p_ T\cdot (v(T\cup \{i\})-v(T)), \] where the summation is over all coalitions T in N such that \(i\not\in T\), and where \(p_ T\) is a probability distribution. The paper investigates in detail the relationship between properties and forms of probabilistic values, and the basic axioms for values, e.g. linearity, monotonicity, dummy, etc.
Reviewer: Z.Artstein

91A12 Cooperative games