Players with Bounded Randomness Capabilities

Abstract

In this thesis I study the effect of bounded randomness capabilities on the outcomes of games, and their payoffs to the players. I study this subject from two perspectives. The first perspective is the ability to share randomness across team members playing against an opposing team. The second perspective is the capability to store the underlying distribution of the mixed strategy a player intends to play.

The first perspective is the ability to share randomness across team members playing against an opposing team. I consider team zero-sum network congestion games played between a team of n agents and a team of k interceptors over a graph G. The agents aim to minimize their collective cost of sending traffic over paths, while the interceptors aim to maximize the collective cost by adding tolls or congestion to road segments. I consider two cases, the correlated case where agents have access to a shared source of randomness, and the uncorrelated case, where each agent has access to only its own source of randomness. I show that the additional cost that the agents have to incur due to being unable to share random bits is bounded by O(min(m_c(G),n)), where m_c(G) is the mincut size of G.

The second perspective is the capability to store the underlying distribution of the mixed strategy a player intends to play. I define a measure of the complexity of finite probability distributions and study the complexity required to play Nash equilibria in finite two-player n times n games with rational payoffs. My central results show that there exist games in which there is an exponential vs. linear gap in the complexity of the mixed distributions that the two players play in the (in these games unique) Nash equilibrium of these games. This gap induces asymmetries in the amounts of space required by players to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. I also establish exponential upper and lower bounds on the complexity of Nash equilibria in normal-form games.