A Stochastic Solution Concept for n-Person Games

Let X be a set of outcomes among which a set of N players, each having a preference relation on X, must choose. Let v: 2^N\/Ø} → 2^X be a game in generalized characteristic function form, where v(C) denotes the set of outcomes that a coalition C can guarantee its members regardless of actions by players outside of C. By letting c(x, y) denote the number of minimal coalitions via which y is directly accessible from x, a Markov chain model of outcome selection is developed. By establishing convergence results for both finite and spatial (X ⊆ R^m) outcome cases, a probability measure on X is obtained and referred to as the stochastic solution of the game. After presenting some initial results on the dependence of the stochastic solution on the initial outcome distribution, it is shown under reasonable assumptions that a (strong) core, if it exists, must occur with probability one. Generally, the results obtained have natural interpretations and proofs using the language and theory of Markov chains. Finally, some examples and previous experimental results are considered in terms of the model. Stochastic solution values obtained and their agreement with available experimental values appear to be very encouraging.