Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics [article]

Jason Milionis, Christos Papadimitriou, Georgios Piliouras, Kelly Spendlove
2022 arXiv   pre-print
Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems. We apply this theory, and in particular the concepts of chain recurrence, attractors, and Conley index, to prove a general impossibility result: there exist games for which any
more » ... namics is bound to have starting points that do not end up at a Nash equilibrium. We also prove a stronger result for ϵ-approximate Nash equilibria: there are games such that no game dynamics can converge (in an appropriate sense) to ϵ-Nash equilibria, and in fact the set of such games has positive measure. Further numerical results demonstrate that this holds for any ϵ between zero and 0.09. Our results establish that, although the notions of Nash equilibria (and its computation-inspired approximations) are universally applicable in all games, they are also fundamentally incomplete as predictors of long term behavior, regardless of the choice of dynamics.
arXiv:2203.14129v1 fatcat:v3uzmg4gpfdfre2geyok3kybee