We introduce a new class of succinct games, called weighted boolean formula games. Here, each player has a set of boolean formulas he wants to get satisfied. The boolean formulas of all players involve a ground set of boolean variables, and every player controls some of these variables. The payoff of a player is the weighted sum of the values of his boolean formulas. For these games, we consider pure Nash equilibria [42] and their well-studied refinement of payoff-dominant equilibria [30], where every player is no worse-off than in any other pure Nash equilibrium. We study both structural and complexity properties for both decision and search problems with respect to the two concepts: • We consider a subclass of weighted boolean formula games, called mutual weighted boolean formula games, which make a natural mutuality assumption on the payoffs of distinct players. We present a very simple exact potential for mutual weighted boolean formula games. We also prove that each weighted, linear-affine (network) congestion game with player-specific constants is polynomial, sound Nash-Harsanyi-Selten homomorphic to a mutual weighted boolean formula game. In a general way, we prove that each weighted, linear-affine (network)

Abstract. Every finite noncooperative game can be presented as a weighted network congestion game, and also as a network congestion game with player-specific costs. In the first presentation, different players may contribute differently to congestion, and in the second, they are differently (negatively) affected by it. This paper shows that the topology of the underlying (undirected two-terminal) network provides information about the existence of pure-strategy Nash equilibrium in the game. For some networks, but not for others, every corresponding game has at least one such equilibrium. For the weighted presentation, a complete characterization of the networks with this property is given. The necessary and sufficient condition is that the (undirected) network does not have four routes with the property that no two of them transverse any edge in the opposite directions, or it consists of several such networks connected in series. The corresponding problem for player-specific costs remains open. Keywords: Congestion games, network topology, existence of equilibrium. 1

Abstract. In this paper we consider Nash Equilibria for the selfish routing model proposed in [12], where a set of n users with tasks of different size try to access m parallel links with different speeds. In this model, a player can use a mixed strategy (where he uses different links with a positive probability); then he is indifferent between the different link choices. This means that the player may well deviate to a different strategy over time. We propose the concept of evolutionary stable strategies (ESS) as a criterion for stable Nash Equilibria, i.e. Equilibria where no player is likely to deviate from his strategy. An ESS is a steady state that can be reached by a user community via evolutionary processes in which more successful strategies spread over time. The concept has been used widely in biology and economics to analyze the dynamics of strategic interactions. We establish that the ESS is uniquely determined for a symmetric Bayesian parallel links game (when it exists). Thus evolutionary stability places strong constraints on the assignment of tasks to links.

Players in a congestion game may differ from one another in their intrinsic preferences (e.g., the benefit they get from using a specific resource), their contribution to congestion, or both. In many cases of interest, intrinsic preferences and the negative effect of congestion are (additively or multiplicatively) separable. This paper considers the implications of separability for the existence of pure-strategy Nash equilibrium and the prospects of spontaneous convergence to equilibrium. It is shown that these properties may or may not be guaranteed, depending on the exact nature of player heterogeneity.

Network users can choose among different security solutions to protect their data. Those solutions are offered by competing providers, with possibly different performance and price levels. In this paper, we model the interactions among users as a noncooperative game, with a negative externality coming from the fact that attackers target popular systems to maximize their expected gain. Using a nonatomic weighted congestion game model for user interactions, we prove the existence and uniqueness of a user equilibrium, compute the corresponding Price of Anarchy, that is the loss of efficiency due to user selfishness, and investigate some consequences for the (higher-level) pricing game played by security providers.