In the context of pre-Bayesian games we analyze resource selection games with unknown number of players. We prove the existence and uniqueness of a symmetric safety-level equilibrium in such games and show that in a game with strictly increasing linear cost functions every player benefits from the common ignorance about the number of players. In order to perform the analysis we define safety-level equilibrium for pre-Bayesian games, and prove that it exists in a compact-continuous-concave setup; in particular it exists in a finite setup. 1

We present a distribution-free model of incomplete-information games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our “robust game” model relaxes the assumptions of Harsanyi’s Bayesian game model, and provides an alternative distribution-free equilibrium concept, which we call “robust-optimization equilibrium, ” to that of the ex post equilibrium. We prove that the robust-optimization equilibria of an incomplete-information game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robust-optimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results.

It is standard in multiagent settings to assume that agents will adopt Nash equilibrium strategies. However, studies in experimental economics demonstrate that Nash equilibrium is a poor description of human players ’ actual behaviour. In this study, we consider a wide range of widely-studied models from behavioural game theory. For what we believe is the first time, we evaluate each of these models in a meta-analysis, taking as our data set large-scale and publicly-available experimental data from the literature. We then propose a modified model that we believe is more suitable for practical prediction of human behaviour. ii Table of Contents Abstract................................... ii

A mediator is a reliable entity which plays on behalf of the players who give her the right of play. The mediator is guaranteed to behave in a pre-specified way based on messages received from the agents. However, a mediator cannot enforce behavior; that is, agents can play in the game directly without the mediator’s help. A mediator generates a new game for the players, the mediated game. The outcome in the original game of an equilibrium in the mediated game is called a mediated equilibrium. Monderer and Tennenholtz introduced a theory of mediators for games with complete information. We extend the theory of mediators to games with incomplete information, and use the new theory to study position auctions, a central topic in practical and theoretical electronic commerce. We provide a minimal set of conditions on position auctions, which is sufficient to guarantee that the VCG outcome function is a mediated equilibrium in these auctions.

For some well-known games, such as the Traveler’s Dilemma or the Centipede Game, traditional gametheoretic solution concepts—most notably Nash equilibrium—predict outcomes that are not consistent with empirical observations. We introduce a new solution concept, iterated regret minimization, which exhibits the same qualitative behavior as that observed in experiments in many games of interest, including Traveler’s Dilemma, the Centipede Game, Nash bargaining, and Bertrand competition. As the name suggests, iterated regret minimization involves the iterated deletion of strategies that do not minimize regret. 1

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

Abstract For some well-known games, such as the Traveler's Dilemma or the Centipede Game, traditional game-theoretic solution concepts--and most notably Nash equilibrium--predict outcomes that are not consistent with empirical observations. In this paper, we introduce a new solution concept, iterated regret minimization, which exhibits the same qualitative behavior as that observed in experiments in many games of interest, including Traveler's Dilemma, the Centipede Game, Nash bargaining, and Bertrand competition. As the name suggests, iterated regret minimization involves the iterated deletion of strategies that do not minimize regret.

In the standard mechanism design setting, the type (e.g., utility function) of an agent is not known by other agents, nor is it known by the mechanism designer. When this uncertainty is quantified probabilistically, a mechanism induces a game of incomplete information among the agents. However, in many settings, uncertainty over utility functions cannot easily be quantified. We consider the problem of incomplete information games in which type uncertainty is strict or unquantified. We propose the use of minimax regret as a decision criterion in such games, a robust approach for dealing with type uncertainty. We define minimax-regret equilibria and prove that these exist in mixed strategies for finite games. We also briefly discuss mechanism design in this framework, with minimax regret as an optimization criterion for the designer itself, and the automated optimization of such mechanisms. 1 Minimax Regret Minimax regret [10, 2] is a common criterion for decision making when uncertainty over consequences of decisions is not quantified probabilistically, a case we refer to as strict uncertainty. 1 Minimax regret is usually advocated when agents are assumed to behave in a non-Bayesian way, either by choice or because prior information is unavailable or too expensive to construct. In this context, we view minimax regret

We show how solution concepts in games such as Nash equilibrium, correlated equilibrium, rationalizability, and sequential equilibrium can be given a uniform definition in terms of knowledge-based programs. Intuitively, all solution concepts are implementations of two knowledge-based programs, one appropriate for games represented in normal form, the other for games represented in extensive form. These knowledge-based programs can be viewed as embodying rationality. The representation works even if (a) information sets do not capture an agent’s knowledge, (b) uncertainty is not represented by probability, or (c) the underlying game is not common knowledge. 1

We initiate the study of scenarios that combine online decision making with interaction between non-cooperative agents. To this end we introduce online games that model such scenarios as non-cooperative games, and lay the foundations for studying this model. Roughly speaking, an online game captures systems in which independent agents serve requests in a common environment. The requests arrive in an online fashion and each is designated to be served by a different agent. The cost incurred by serving a request is paid for by the serving agent, and naturally, the agents seek to minimize the total cost they pay. Since the agents are independent, it is unlikely that some central authority can enforce a policy or an algorithm (centralized or distributed) on them, and thus, the agents can be viewed as selfish players in a non-cooperative game. In this game, the players have to choose as a strategy an online algorithm according to which requests are served. To further facilitate the game theoretic approach, we suggest the measure of competitive analysis as the players ’ decision criterion. As the expected result of noncooperative games is an equilibrium, the question of finding the equilibria of a game is of central importance, and thus, it is the central issue we concentrate on in this paper. We study some natural examples for online games; in order to obtain general insights and develop generic techniques, we present an abstract model for the study of online games generalizing metrical task systems. We suggest a method for constructing equilibria in this model and further devise techniques for implementing it.