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Player representation, Bargaining games, Divide the dollar, N-player mathematical games, Evolution strategies


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 > 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.


© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (

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