We argue that learning equilibrium is an appropriate generalization to multi-agent systems of the concept of learning to optimize in singleagent setting. We further define and discuss the concept of weak learning equilibrium. 1

We consider a resource selection game with incomplete information about the resource-cost functions. All the players know is the set of players, an upper bound on the possible costs, and that the cost functions are positive and nondecreasing. The game is played repeatedly and after every stage each player observes her cost, and the actions of all players. For every ɛ> 0 we prove the existence of a learning ɛ-equilibrium, which is a profile of algorithms, one for each player such that a unilateral deviation of a player is, up to ɛ not beneficial for her regardless of the actual cost functions. Furthermore, the learning equilibrium yields an optimal social cost. 1.

Abstract. We introduce the first near-optimal polynomial algorithm for obtaining the mixed safety level value of an initially unknown multi-stage game, played in a hostile environment, under imperfect monitoring. In an imperfect monitoring setting all that an agent can observe is the current state and its own actions and payoffs, but it can not observe other agents ’ actions. Our result holds for any multistage generic game with a “reset ” action. 1

While the class of congestion games has been thoroughly studied in the multi-agent systems literature, settings with incomplete information have received relatively little attention. In this paper we consider a setting in which the cost functions of resources in the congestion game are initially unknown. The agents gather information about these cost functions through repeated interaction, and observations of costs they incur. In this context we consider the following requirement: the agents ’ algorithms should themselves be in equilibrium, regardless of the actual cost functions and should lead to an efficient outcome. We prove that this requirement is achievable for a broad class of games: repeated symmetric congestion games. Our results are applicable even when agents are somewhat limited in their capacity to monitor the actions of their counterparts, or when they are unable to determine the exact cost they incur from every resource. On the other hand, we show that there exist asymmetric congestion games for which no such equilibrium can be found, not even an inefficient one. Finally we consider equilibria with resistance to the deviation of more than one player and show that these do not exist even in repeated resource selection games.

Under many protocols — in computerized settings and in economics settings — participants repeatedly “best respond ” to each others ’ actions until the system “converges ” to an equilibrium point. We ask when such myopic “local rationality ” implies “global rationality”, i.e., when is it best for a player, given that the others are repeatedly best-responding, to also repeatedly best-respond? We exhibit a class of games where this is indeed the case. We identify several environments of interest that fall within our class: models of the Border Gateway Protocol (BGP) [9], that handles routing on the Internet, and of the Transmission Control Protocol Protocol (TCP) [7], and also stable-roommates [5] and cost-sharing [12, 13], that have been extensively studied in economic theory.