Ryan M. Rogers

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 [Kearns et al. 2014], 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.

Inducing Approximately Optimal Flow Using Truthful Mediators

We revisit a classic coordination problem from the perspective of mechanism design: how can we coordinate a social welfare maximizing flow in a network congestion game with selfish players? The classical approach, which computes tolls as a function of known demands, fails when the demands are unknown to the mechanism designer, and naively eliciting them does not necessarily yield a truthful mechanism. Instead, we introduce a weak mediator that can provide suggested routes to players and set tolls as a function of reported demands. However, players can choose to ignore or misreport their type to this mediator. Using techniques from differential privacy, we show how to design a weak mediator such that it is an asymptotic ex-post Nash equilibrium for all players to truthfully report their types to the mediator and faithfully follow its suggestion, and that when they do, they end up playing a nearly optimal flow. Notably, our solution works in settings of incomplete information even in the absence of a prior distribution on player types. Along the way, we develop new techniques for privately solving convex programs which may be of independent interest.

Max-Information, Differential Privacy, and Post-selection Hypothesis Testing

In this paper, we initiate a principled study of how the generalization properties of approximate differential privacy can be used to perform adaptive hypothesis testing, while giving statistically valid p-value corrections. We do this by observing that the guarantees of algorithms with bounded approximate max-information are sufficient to correct the p-values of adaptively chosen hypotheses, and then by proving that algorithms that satisfy (∈,δ)-differential privacy have bounded approximate max information when their inputs are drawn from a product distribution. This substantially extends the known connection between differential privacy and max-information, which previously was only known to hold for (pure) (∈,0)-differential privacy. It also extends our understanding of max-information as a partially unifying measure controlling the generalization properties of adaptive data analyses. We also show a lower bound, proving that (despite the strong composition properties of max-information), when data is drawn from a product distribution, (∈,δ)-differentially private algorithms can come first in a composition with other algorithms satisfying max-information bounds, but not necessarily second if the composition is required to itself satisfy a nontrivial max-information bound. This, in particular, implies that the connection between (∈,δ)-differential privacy and max-information holds only for inputs drawn from product distributions, unlike the connection between (∈,0)-differential privacy and max-information.