Values for Markovian coalition processes

Abstract

Time series of coalitions (so-called scenarios) are studied that describe processes of coalition formation where several players may enter or leave the current coalition at any point in (discrete) time and convergence to the grand coalition is not necessarily prescribed. Transitions from one coalition to the next are assumed to be random and to yield a Markov chain. Three examples of such processes (the Shapley-Weber process, the Metropolis process, and an example of a voting situation) and their properties are presented. A main contribution includes notions of value for such series, \emph{i.e.}, schemes for the evaluation of the expected contribution of a player to the coalition process relative to a given cooperative game. Particular processes permit to recover the classical Shapley value. This methodology's power is illustrated with well-known examples from exchange economies due to Shafer (1980) and Scafuri and Yannelis (1984), where the classical Shapley value leads to counterintuitive allocations. The Markovian process value avoids these drawbacks and provides plausible results.