A Representation for Many Player Generalized Divide the Dollar Games

Divide the dollar is a simplified version of a two player bargaining problem game devised by John Nash. The generalized divide the dollar game has n \u3e 2 players. Evolutionary algorithms can be used to evolve individual players for this generalized game but representation—i.e., a genome plus a move or search operator(s)—must be carefully chosen since it affects the search process. This paper proposes an entirely new representation called a demand matrix. Each individual in the evolving population now represents a collection of n players rather than just an individual player. Players use previous outcomes to decide their choices (bids) in the current round. The representation scales linearly with the number of players and the move operator is a variant of an evolution strategy. The results indicate that this proposed representation for the generalized divide the dollar game permits the efficient evolution of large player populations with high payoffs and fair demand sets

Modeling Undependable Subsidies with Three-player Generalized Divide the Dollar

Divide the dollar is a two-player simultaneous game derived from a game invented by John Nash because its strategy space contains an entire subspace of Nash equilibria. This study applies a family of generalizations of divide the dollar, called set-based divide the dollar, to the problem of understanding the impact of undependable subsidies. Set based divide the dollar defines a family of games with easily controlled properties making it ideal for this modeling task. These subsidies are intended to encourage deal making but, if abruptly discontinued or funded unreliably, may have different effects from those intended. The study also demonstrates the generalization of the game to three players, something that the set-based formalism makes easy. Agents are encoded using a finite state representation that conditions its transitions on the result of deals. These results fall into three categories, the agent obtains the highest amount, the agent receives a lesser amount, or the agents fail to make a deal. This study compares a situation with no subsidies with dependable and undependable subsidies, using the rate at which deals are made as a assessment statistic. Two sorts of undependability are studied, abrupt cessation of the subsidy and unreliable funding