Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead

Stackelberg equilibria have become increasingly important as a solution concept in computational game theory, largely inspired by practical problems such as security settings. In practice, however, there is typically uncertainty regarding the model about the opponent. This paper is, to our knowledge, the first to investigate Stackelberg equilibria under uncertainty in extensive-form games, one of the broadest classes of game. We introduce robust Stackelberg equilibria, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new mixed-integer program for the deterministic limited-lookahead setting. We then extend the program to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a mixed-integer program that is of the same asymptotic size as that for the deterministic setting.Comment: Published at AAAI1

Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead

Stackelberg equilibria have become increasingly important as a solution concept in computational game theory, largely inspired by practical problems such as security settings. In practice, however, there is typically uncertainty regarding the model about the opponent. This paper is, to our knowledge, the first to investigate Stackelberg equilibria under uncertainty in extensive-form games, one of the broadest classes of game. We introduce robust Stackelberg equilibria, where the uncertainty is about the opponent's payoffs, as well as ones where the opponent has limited lookahead and the uncertainty is about the opponent's node evaluation function. We develop a new mixed-integer program for the deterministic limited-lookahead setting. We then extend the program to the robust setting for Stackelberg equilibrium under unlimited and under limited lookahead by the opponent. We show that for the specific case of interval uncertainty about the opponent's payoffs (or about the opponent's node evaluations in the case of limited lookahead), robust Stackelberg equilibria can be computed with a mixed-integer program that is of the same asymptotic size as that for the deterministic setting.Comment: Published at AAAI1

Approximate well-supported Nash equilibria in symmetric bimatrix games

The $\varepsilon$-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than $\varepsilon$ to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant $\varepsilon$ currently known for which there is a polynomial-time algorithm that computes an $\varepsilon$-well-supported Nash equilibrium in bimatrix games is slightly below $2/3$. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a $(1/2+\delta)$-well-supported Nash equilibrium, for an arbitrarily small positive constant $\delta$

Symmetry Breaking for Answer Set Programming

In the context of answer set programming, this work investigates symmetry detection and symmetry breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We contribute a reduction of symmetry detection to a graph automorphism problem which allows to extract symmetries of a logic program from the symmetries of the constructed coloured graph. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process which implicitly represent symmetries and always with exponential compression. These ideas are formulated as preprocessing and implemented in a completely automated flow that first detects symmetries from a given answer set program, adds symmetry-breaking constraints, and can be applied to any existing answer set solver. We demonstrate computational impact on benchmarks versus direct application of the solver. Furthermore, we explore symmetry breaking for answer set programming in two domains: first, constraint answer set programming as a novel approach to represent and solve constraint satisfaction problems, and second, distributed nonmonotonic multi-context systems. In particular, we formulate a translation-based approach to constraint answer set solving which allows for the application of our symmetry detection and symmetry breaking methods. To compare their performance with a-priori symmetry breaking techniques, we also contribute a decomposition of the global value precedence constraint that enforces domain consistency on the original constraint via the unit-propagation of an answer set solver. We evaluate both options in an empirical analysis. In the context of distributed nonmonotonic multi-context system, we develop an algorithm for distributed symmetry detection and also carry over symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201

Separable and Low-Rank Continuous Games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game

Computing Equilibria of Semi-algebraic Economies Using Triangular Decomposition and Real Solution Classification

In this paper, we are concerned with the problem of determining the existence of multiple equilibria in economic models. We propose a general and complete approach for identifying multiplicities of equilibria in semi-algebraic economies, which may be expressed as semi-algebraic systems. The approach is based on triangular decomposition and real solution classification, two powerful tools of algebraic computation. Its effectiveness is illustrated by two examples of application.Comment: 24 pages, 5 figure