Refinement of solutions to the linear complimentarity problem

Nash equilibrium;game theaory;matrices

Unsupervised Domain Adaptation using Graph Transduction Games

Unsupervised domain adaptation (UDA) amounts to assigning class labels to the unlabeled instances of a dataset from a target domain, using labeled instances of a dataset from a related source domain. In this paper, we propose to cast this problem in a game-theoretic setting as a non-cooperative game and introduce a fully automatized iterative algorithm for UDA based on graph transduction games (GTG). The main advantages of this approach are its principled foundation, guaranteed termination of the iterative algorithms to a Nash equilibrium (which corresponds to a consistent labeling condition) and soft labels quantifying the uncertainty of the label assignment process. We also investigate the beneficial effect of using pseudo-labels from linear classifiers to initialize the iterative process. The performance of the resulting methods is assessed on publicly available object recognition benchmark datasets involving both shallow and deep features. Results of experiments demonstrate the suitability of the proposed game-theoretic approach for solving UDA tasks.Comment: Oral IJCNN 201

Ancient Coin Classification Using Graph Transduction Games

Recognizing the type of an ancient coin requires theoretical expertise and years of experience in the field of numismatics. Our goal in this work is automatizing this time consuming and demanding task by a visual classification framework. Specifically, we propose to model ancient coin image classification using Graph Transduction Games (GTG). GTG casts the classification problem as a non-cooperative game where the players (the coin images) decide their strategies (class labels) according to the choices made by the others, which results with a global consensus at the final labeling. Experiments are conducted on the only publicly available dataset which is composed of 180 images of 60 types of Roman coins. We demonstrate that our approach outperforms the literature work on the same dataset with the classification accuracy of 73.6% and 87.3% when there are one and two images per class in the training set, respectively

Constrained Pure Nash Equilibria in Polymatrix Games

We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs. The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy. These games capture the idea of coordination within a local neighbourhood in the absence of globally common strategies. We study the decision problem of checking whether a given set of strategy choices for a subset of the players is consistent with some pure Nash equilibrium or, alternatively, with all pure Nash equilibria. We identify the most natural tractable cases and show NP or coNP-completness of these problems already for unweighted DAGs.Comment: Extended version of a paper accepted to AAAI1

An Empirical Study on Computation of Exact and Approximate Equilibria

The computation of Nash equilibria is one of the central topics in game theory, which has received much attention from a theoretical point of view. Studies have shown that the problem of finding a Nash equilibrium is PPAD-complete, which implies that we are unlikely to find a polynomial-time algorithm for this problem. Naturally, this has led to a line of work studying the complexity of finding approximate Nash equilibria. This thesis examines the computation of such approximate Nash equilibria within several classes of games from an empirical perspective. In this thesis, we address the computation of approximate Nash equilibria in bimatrix and polymatrix games. For both of these game classes, we provide a library of implementations of algorithms for the computation of exact and approximate Nash equilibria, as well as a suite of game generators which were used as a base for our empirical analysis of the algorithms. We investigate the trade-off between quality of approximation produced by the algorithms and the expected runtime. We provide some insight into the inner workings of the state-of-the-art algorithm for computing ε-Nash equilibria, presenting worst-case examples found for our provided suite of game generators. We then show lower bounds on these algorithms. In the case of polymatrix games, we generate this lower bound from a real-world application of game theory. For bimatrix games, we provide a robust means of generating lower bounds for approximation algorithms with the use of genetic algorithms