Learning and Efficiency in Games with Dynamic Population

We study the quality of outcomes in games when the population of players is dynamically changing, and where participants have to adapt to the dynamic environment. Price of Anarchy has originally been introduced to study the Nash equilibria of one-shot games. The Price of Total Anarchy extends this notion to learning outcomes in a repeated setting, assuming all players use a form of no-regret learning to choose their strategies, and the environment as well as the player population is stable. In this paper we consider repeated games in dynamically changing environments. We show that in large classes of games, if players use a form of adaptive learning to adapt their strategy choices to the changing environment, this guarantees high social welfare, even under very frequent changes. An important feature of our analysis is the connection to differential privacy. The optimal solution to the welfare optimization problem can alter drastically even with a single change in the parameters of one participant. To take advantage of the adaptive learning in the analysis, we need to show that there is benchmark solution with which we compare the regret of a player, that is both close to optimal and relatively stable over time. Differential privacy offers just the right tool. It requires that the proposed solution does not alter much with the change of one person’s input. We show that the existence of a differentially private algorithm for the welfare optimization problem implies high welfare in the dynamic game even under very frequent changes. We demonstrate our techniques by focusing on two classes of games as examples: independent item auctions and congestion games. In both applications we show that adaptive learning guarantees high social welfare even with surprisingly high churn in the player population.