How long to equilibrium? The communication complexity of uncoupled equilibrium procedures

Sergiu Hart, Yishay Mansour
2010 Games and Economic Behavior  
We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash
more » ... ilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players. This may be one reason-though not the only one-that it appears to be more difficult to converge to the former than to the latter. 2 Continuous with respect to both actions and time. 3 Dynamics of the "best-reply" variety have been studied in Bayesian setups where players possess certain probabilistic beliefs about the payoff functions of the other players (see, e.g., Jordan, 1991; Kalai and Lehrer, 1993) ; however, additional coordination between the players is needed to obtain convergence to Nash equilibria (cf. Section 4 in Jordan, 1991 and footnote 20 in Hart, 2005).
doi:10.1016/j.geb.2007.12.002 fatcat:xeqrst7hazbyrg6islro2m5epm