Resumen de An Adaptive Model of Demand Adjustment in Cooperative Games
Alex Possajennikov, María Montero
The paper presents a simple best reply adaptive model of demand adjustment in transferable utility cooperative games. The process is designed to model competition among players. Players set demands for their participation in a coalition. The demands of �scarce� players will be driven up by competition, and the demands of �abundant� players will be driven down. The process operates in the following way. A player is chosen randomly and looks for a coalition that leaves the highest surplus provided that the demands of other players in the coalition are satisfied. The player then proposes this coalition and adjusts the demand to coincide with the surplus left. This process has been shown to have aspiration vectors as stationary states, and to converge to the set of aspirations.
We introduce a particular type of mutations in this process. With small probability, each player changes coalitions only if offered a strictly larger amount than he is currently getting. This can be interpreted as small transaction costs of changing a coalition, a small friction. If the current demand vector is an aspiration but not a partnered one, then a demand of a strictly larger amount by a �scarce� player leads to a strictly lower demand by another player, which amounts to a transfer from one player to another. The vector of demands resulting from the transfer may not be an aspiration but the base process will then lead to an aspiration vector.
We show that a sequence of transfers, together with subsequent adjustment, leads to a partnered aspiration. Considering the base process of adjustment together with rare mutations, we show that in a discretized model with sufficiently small grid size the stochastically stable states of the process lie in the set of partnered aspirations. For simple majority games it implies that only minimal winning coalitions form, and in the apex game the payoff division is proportional to the number of votes. Further selection inside the set of partnered aspiration can also be considered, ultimately leading to (the interior of) the aspiration core, or the set of balanced aspirations.
