Abstract
We introduce new theoretical insights into two-population asymmetric games allowing for an elegant symmetric decomposition into two single population symmetric games. Specifically, we show how an asymmetric bimatrix game (A,B) can be decomposed into its symmetric counterparts by envisioning and investigating the payoff tables (A and B) that constitute the asymmetric game, as two independent, single population, symmetric games. We reveal several surprising formal relationships between an asymmetric two-population game and its symmetric single population counterparts, which facilitate a convenient analysis of the original asymmetric game due to the dimensionality reduction of the decomposition. The main finding reveals that if (x,y) is a Nash equilibrium of an asymmetric game (A,B), this implies that y is a Nash equilibrium of the symmetric counterpart game determined by payoff table A, and x is a Nash equilibrium of the symmetric counterpart game determined by payoff table B. Also the reverse holds and combinations of Nash equilibria of the counterpart games form Nash equilibria of the asymmetric game. We illustrate how these formal relationships aid in identifying and analysing the Nash structure of asymmetric games, by examining the evolutionary dynamics of the simpler counterpart games in several canonical examples.
Highlights
We are interested in analysing the not only valuable insights into the (Nash) structure and evolutionary dynamics of strategic interactions in multi-agent systems
We first present our main findings, formally relating Nash equilibria in asymmetric 2-player games with the Nash equilibria that can be found in the corresponding counterpart games
We examine the stability properties of the corresponding rest points of the replicator dynamics in these games
Summary
The most straightforward and classical approach to asymmetric games is to treat agents as evolving separately: one population per player, where each agent in a population interacts by playing against agent(s) from the other population(s), i.e. co-evolution[21]. There are relationships between the sets of evolutionarily stable strategies and rest points of the replicator dynamics between the original and symmetrized game[19,23] This single-population model forces the players to be general: able to devise a strategy for each role, which may unnecessarily complicate algorithms that compute strategies for such players. There are two-population variants that formulate the problem slightly differently: a new matrix that encapsulates both players’ utilities assigns 0 utility to combinations of roles that are not in one-to-one correspondence with players[24]. This too, results in an unnecessarily larger (albeit sparse) matrix. We consider the original co-evolutionary interpretation, and derive new (lower-dimensional) strategy space mappings
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