Finding all minimal curb sets

Sets closed under rational behavior were introduced by Basu and Weibull (1991) as subsets of the strategy space that contain all best replies to all strategy profiles in the set. We here consider a more restrictive notion of closure under rational behavior: a subset of the strategy space is strongly closed under rational behavior, or sCURB, if it contains all best replies to all probabilistic beliefs over the set. We present an algorithm that computes all minimal sCURB sets in any given finite game. Runtime measurements on two-player games (where the concepts of CURB and sCURB coincide) show that the algorithm is considerably faster than the earlier developed algorithm, that of Benisch et al. (2006)

Iterated Weaker-than-Weak Dominance

We introduce a weakening of standard gametheoretic dominance conditions, called δ-dominance, which enables more aggressive pruning of candidate strategies at the cost of solution accuracy. Equilibria of a game obtained by eliminating a δ-dominated strategy are guaranteed to be approximate equilibria of the original game, with degree of approximation bounded by the dominance parameter, δ. We can apply elimination of δ-dominated strategies iteratively, but the δ for which a strategy may be eliminated depends on prior eliminations. We discuss implications of this order independence, and propose greedy heuristics for determining a sequence of eliminations to reduce the game as far as possible while keeping down costs. A case study analysis of an empirical 2-player game serves to illustrate the technique, and demonstrate the utility of weaker-than-weak dominance pruning.

Fairness with an Honest Minority and a Rational Majority

We provide a simple protocol for secret reconstruction in any threshold secret sharing scheme, and prove that it is fair when executed with many rational parties together with a small minority of honest parties. That is, all parties will learn the secret with high probability when the honest parties follow the protocol and the rational parties act in their own self-interest (as captured by a set-Nash analogue of trembling hand perfect equilibrium). The protocol only requires a standard (synchronous) broadcast channel, tolerates both early stopping and incorrectly computed messages, and only requires 2 rounds of communication. Previous protocols for this problem in the cryptographic or economic models have either required an honest majority, used strong communication channels that enable simultaneous exchange of information, or settled for approximate notions of security/equilibria. They all also required a nonconstant number of rounds of communication.Engineering and Applied Science

Complexity results for some classes of strategic games

Game theory is a branch of applied mathematics studying the interaction of self-interested entities, so-called agents. Its central objects of study are games, mathematical models of real-world interaction, and solution concepts that single out certain outcomes of a game that are meaningful in some way. The solutions thus produced can then be viewed both from a descriptive and from a normative perspective. The rise of the Internet as a computational platform where a substantial part of today's strategic interaction takes place has spurred additional interest in game theory as an analytical tool, and has brought it to the attention of a wider audience in computer science. An important aspect of real-world decision-making, and one that has received only little attention in the early days of game theory, is that agents may be subject to resource constraints. The young field of algorithmic game theory has set out to address this shortcoming using techniques from computer science, and in particular from computational complexity theory. One of the defining problems of algorithmic game theory concerns the computation of solution concepts. Finding a Nash equilibrium, for example, i.e., an outcome where no single agent can gain by changing his strategy, was considered one of the most important problems on the boundary of P, the complexity class commonly associated with efficient computation, until it was recently shown complete for the class PPAD. This rather negative result for general games has not settled the question, however, but immediately raises several new ones: First, can Nash equilibria be approximated, i.e., is it possible to efficiently find a solution such that the potential gain from a unilateral deviation is small? Second, are there interesting classes of games that do allow for an exact solution to be computed efficiently? Third, are there alternative solution concepts that are computationally tractable, and how does the value of solutions selected by these concepts compare to those selected by established solution concepts? The work reported in this thesis is part of the effort to answer the latter two questions. We study the complexity of well-known solution concepts, like Nash equilibrium and iterated dominance, in various classes of games that are both natural and practically relevant: ranking games, where outcomes are rankings of the players; anonymous games, where players do not distinguish between the other players in the game; and graphical games, where the well-being of any particular player depends only on the actions of a small group other players. In ranking games, we further compare the payoffs obtainable in Nash equilibrium outcomes with those of alternative solution concepts that are easy to compute. We finally study, in general games, solution concepts that try to remedy some of the shortcomings associated with Nash equilibrium, like the need for randomization to achieve a stable outcome