Perfect Regular Equilibrium

We propose a revised version of the perfect Bayesian equilibrium in general multi-period games with observed actions. In finite games, perfect Bayesian equilibria are weakly consistent and subgame perfect Nash equilibria. In general games that allow a continuum of types and strategies, however, perfect Bayesian equilibria might not satisfy these criteria of rational solution concepts. To solve this problem, we revise the definition of the perfect Bayesian equilibrium by replacing Bayes' rule with a regular conditional probability. We call this revised solution concept a perfect regular equilibrium. Perfect regular equilibria are always weakly consistent and subgame perfect Nash equilibria in general games. In addition, perfect regular equilibria are equivalent to simplified perfect Bayesian equilibria in finite games. Therefore, the perfect regular equilibrium is an extended and simple version of the perfect Bayesian equilibrium in general multi-period games with observed actions

Incentive Stackelberg Mean-payoff Games

We introduce and study incentive equilibria for multi-player meanpayoff games. Incentive equilibria generalise well-studied solution concepts such as Nash equilibria and leader equilibria (also known as Stackelberg equilibria). Recall that a strategy profile is a Nash equilibrium if no player can improve his payoff by changing his strategy unilaterally. In the setting of incentive and leader equilibria, there is a distinguished player called the leader who can assign strategies to all other players, referred to as her followers. A strategy profile is a leader strategy profile if no player, except for the leader, can improve his payoff by changing his strategy unilaterally, and a leader equilibrium is a leader strategy profile with a maximal return for the leader. In the proposed case of incentive equilibria, the leader can additionally influence the behaviour of her followers by transferring parts of her payoff to her followers. The ability to incentivise her followers provides the leader with more freedom in selecting strategy profiles, and we show that this can indeed improve the payoff for the leader in such games. The key fundamental result of the paper is the existence of incentive equilibria in mean-payoff games. We further show that the decision problem related to constructing incentive equilibria is NP-complete. On a positive note, we show that, when the number of players is fixed, the complexity of the problem falls in the same class as two-player mean-payoff games. We also present an implementation of the proposed algorithms, and discuss experimental results that demonstrate the feasibility of the analysis of medium sized games.Comment: 15 pages, references, appendix, 5 figure

Belief-Invariant and Quantum Equilibria in Games of Incomplete Information

Drawing on ideas from game theory and quantum physics, we investigate nonlocal correlations from the point of view of equilibria in games of incomplete information. These equilibria can be classified in decreasing power as general communication equilibria, belief-invariant equilibria and correlated equilibria, all of which contain the familiar Nash equilibria. The notion of belief-invariant equilibrium has appeared in game theory before, in the 1990s. However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 1980s in the foundations of quantum mechanics. Here, we explain and unify these two origins of the idea and study the above classes of equilibria, and furthermore quantum correlated equilibria, using tools from quantum information but the language of game theory. We present a general framework of belief-invariant communication equilibria, which contains (quantum) correlated equilibria as special cases. It also contains the theory of Bell inequalities, a question of intense interest in quantum mechanics, and quantum games where players have conflicting interests, a recent topic in physics. We then use our framework to show new results related to social welfare. Namely, we exhibit a game where belief-invariance is socially better than correlated equilibria, and one where all non-belief-invariant equilibria are socially suboptimal. Then, we show that in some cases optimal social welfare is achieved by quantum correlations, which do not need an informed mediator to be implemented. Furthermore, we illustrate potential practical applications: for instance, situations where competing companies can correlate without exposing their trade secrets, or where privacy-preserving advice reduces congestion in a network. Along the way, we highlight open questions on the interplay between quantum information, cryptography, and game theory

Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry

For Hamiltonian systems with spherical symmetry there is a marked difference between zero and non-zero momentum values, and amongst all relative equilibria with zero momentum there is a marked difference between those of zero and those of non-zero angular velocity. We use techniques from singularity theory to study the family of relative equilibria that arise as a symmetric Hamiltonian which has a group orbit of equilibria with zero momentum is perturbed so that the zero-momentum relative equilibrium are no longer equilibria. We also analyze the stability of these perturbed relative equilibria, and consider an application to satellites controlled by means of rotors.Comment: 24 pp; to appear in J. Geometric Mechanic

A finite set of equilibria for the indeterminacy of linear rational expectations models

This paper demonstrates the existence of a finite set of equilibria in the case of the indeterminacy of linear rational expectations models. The number of equilibria corresponds to the number of ways to select n eigenvectors among a larger set of eigenvectors related to stable eigenvalues. A finite set of equilibria is a substitute to continuous (uncountable) sets of sunspots equilibria, when the number of independent eigenvectors for each stable eigenvalue is equal to one

Generalized selfish bin packing

Standard bin packing is the problem of partitioning a set of items with positive sizes no larger than 1 into a minimum number of subsets (called bins) each having a total size of at most 1. In bin packing games, an item has a positive weight, and given a valid packing or partition of the items, each item has a cost or a payoff associated with it. We study a class of bin packing games where the payoff of an item is the ratio between its weight and the total weight of items packed with it, that is, the cost sharing is based linearly on the weights of items. We study several types of pure Nash equilibria: standard Nash equilibria, strong equilibria, strictly Pareto optimal equilibria, and weakly Pareto optimal equilibria. We show that any game of this class admits all these types of equilibria. We study the (asymptotic) prices of anarchy and stability (PoA and PoS) of the problem with respect to these four types of equilibria, for the two cases of general weights and of unit weights. We show that while the case of general weights is strongly related to the well-known First Fit algorithm, and all the four PoA values are equal to 1.7, this is not true for unit weights. In particular, we show that all of them are strictly below 1.7, the strong PoA is equal to approximately 1.691 (another well-known number in bin packing) while the strictly Pareto optimal PoA is much lower. We show that all the PoS values are equal to 1, except for those of strong equilibria, which is equal to 1.7 for general weights, and to approximately 1.611824 for unit weights. This last value is not known to be the (asymptotic) approximation ratio of any well-known algorithm for bin packing. Finally, we study convergence to equilibria

From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem

The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of $N$ player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article we emphasize the role of optimal transport theory in: 1) the passage from Nash to Cournot-Nash equilibria as the number of players tends to infinity, 2) the analysis of Cournot-Nash equilibria