Designing cost-sharing methods for Bayesian games

We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players

Asymptotically Truthful Equilibrium Selection in Large Congestion Games

Studying games in the complete information model makes them analytically tractable. However, large $n$ player interactions are more realistically modeled as games of incomplete information, where players may know little to nothing about the types of other players. Unfortunately, games in incomplete information settings lose many of the nice properties of complete information games: the quality of equilibria can become worse, the equilibria lose their ex-post properties, and coordinating on an equilibrium becomes even more difficult. Because of these problems, we would like to study games of incomplete information, but still implement equilibria of the complete information game induced by the (unknown) realized player types. This problem was recently studied by Kearns et al. and solved in large games by means of introducing a weak mediator: their mediator took as input reported types of players, and output suggested actions which formed a correlated equilibrium of the underlying game. Players had the option to play independently of the mediator, or ignore its suggestions, but crucially, if they decided to opt-in to the mediator, they did not have the power to lie about their type. In this paper, we rectify this deficiency in the setting of large congestion games. We give, in a sense, the weakest possible mediator: it cannot enforce participation, verify types, or enforce its suggestions. Moreover, our mediator implements a Nash equilibrium of the complete information game. We show that it is an (asymptotic) ex-post equilibrium of the incomplete information game for all players to use the mediator honestly, and that when they do so, they end up playing an approximate Nash equilibrium of the induced complete information game. In particular, truthful use of the mediator is a Bayes-Nash equilibrium in any Bayesian game for any prior.Comment: The conference version of this paper appeared in EC 2014. This manuscript has been merged and subsumed by the preprint "Robust Mediators in Large Games": http://arxiv.org/abs/1512.0269

Privacy and Truthful Equilibrium Selection for Aggregative Games

We study a very general class of games --- multi-dimensional aggregative games --- which in particular generalize both anonymous games and weighted congestion games. For any such game that is also large, we solve the equilibrium selection problem in a strong sense. In particular, we give an efficient weak mediator: a mechanism which has only the power to listen to reported types and provide non-binding suggested actions, such that (a) it is an asymptotic Nash equilibrium for every player to truthfully report their type to the mediator, and then follow its suggested action; and (b) that when players do so, they end up coordinating on a particular asymptotic pure strategy Nash equilibrium of the induced complete information game. In fact, truthful reporting is an ex-post Nash equilibrium of the mediated game, so our solution applies even in settings of incomplete information, and even when player types are arbitrary or worst-case (i.e. not drawn from a common prior). We achieve this by giving an efficient differentially private algorithm for computing a Nash equilibrium in such games. The rates of convergence to equilibrium in all of our results are inverse polynomial in the number of players $n$. We also apply our main results to a multi-dimensional market game. Our results can be viewed as giving, for a rich class of games, a more robust version of the Revelation Principle, in that we work with weaker informational assumptions (no common prior), yet provide a stronger solution concept (ex-post Nash versus Bayes Nash equilibrium). In comparison to previous work, our main conceptual contribution is showing that weak mediators are a game theoretic object that exist in a wide variety of games -- previously, they were only known to exist in traffic routing games

Signaling and forward induction in a market entry context

Game Theory;Market Structure

Designing Cost-Sharing Methods for Bayesian Games

We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE). We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players

Robust Predictions in Games with Incomplete Information

We analyze games of incomplete information and offer equilibrium predictions which are valid for all possible private information structures that the agents may have. Our characterization of these robust predictions relies on an epistemic result which establishes a relationship between the set of Bayes Nash equilibria and the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior information of the analyst refines the set of equilibrium distribution. As an application, we obtain new results regarding the optimal information sharing policy of firms under demand uncertainty. Finally, we reverse the perspective and investigate the identification problem under concerns for robustness to private information. We show how the presence of private information leads to partial rather than complete identification of the structural parameters of the game. As a prominent example we analyze the canonical problem of demand and supply identification.Incomplete information, Correlated equilibrium, Robustness to private information, Moments restrictions, Identification, Information bounds