The Price of Defense

We consider a game on a graph G= ⟨ V, E⟩ with two confronting classes of randomized players: νattackers, who choose vertices and seek to minimize the probability of getting caught, and a single defender, who chooses edges and seeks to maximize the expected number of attackers it catches. In a Nash equilibrium, no player has an incentive to unilaterally deviate from her randomized strategy. The Price of Defense is the worst-case ratio, over all Nash equilibria, of ν over the expected utility of the defender at a Nash equilibrium. We orchestrate a strong interplay of arguments from Game Theory and Graph Theory to obtain both general and specific results in the considered setting: (1) Via a reduction to a Two-Players, Constant-Sum game, we observe that an arbitrary Nash equilibrium is computable in polynomial time. Further, we prove a general lower bound of |V|2 on the Price of Defense. We derive a characterization of graphs with a Nash equilibrium attaining this lower bound, which reveals a promising connection to Fractional Graph Theory; thereby, it implies an efficient recognition algorithm for such Defense-Optimal graphs. (2) We study some specific classes of Nash equilibria, both for their computational complexity and for their incurred Price of Defense. The classes are defined by imposing structure on the players’ randomized strategies: either graph-theoretic structure on the supports, or symmetry and uniformity structure on the probabilities. We develop novel graph-theoretic techniques to derive trade-offs between computational complexity and the Price of Defense for these classes. Some of the techniques touch upon classical milestones of Graph Theory; for example, we derive the first game-theoretic characterization of König-Egerváry graphs as graphs admitting a Matching Nash equilibrium

Multi-Agent Systems for Computational Economics and Finance

In this article we survey the main research topics of our group at the University of Essex. Our research interests lie at the intersection of theoretical computer science, artificial intelligence, and economic theory. In particular, we focus on the design and analysis of mechanisms for systems involving multiple strategic agents, both from a theoretical and an applied perspective. We present an overview of our group’s activities, as well as its members, and then discuss in detail past, present, and future work in multi-agent systems

Role Based Hedonic Games

In the hedonic coalition formation game model Roles Based Hedonic Games (RBHG), agents view teams as compositions of available roles. An agent\u27s utility for a partition is based upon which role she fulfills within the coalition and which additional roles are being fulfilled within the coalition. I consider optimization and stability problems for settings with variable power on the part of the central authority and on the part of the agents. I prove several of these problems to be NP-complete or coNP-complete. I introduce heuristic methods for approximating solutions for a variety of these hard problems. I validate heuristics on real-world data scraped from League of Legends games