Abstract:
Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parametrized by a switching rule, and one of the most natural rules is "all switches" which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game G, a starting strategy s, and an edge e. The problems are: 1. The edge switch problem, namely, is the edge e, ever switched by all-switches strategy improvement when it is started from s on game G? 2. The optimal strategy problem, namely, is the edge e used in the final strategy that is found by strategy improvement when it is started from s on game G? We show PSPACE-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of Voge and Jurdzinski; mean-payoff games with the gain-bias algorithm; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show PSPACE-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for Acyclic Unique Sink Orientations on Cubes.

Abstract:
We study the computational complexity of finding stable outcomes in hedonic games, which are a class of coalition formation games. We restrict our attention to symmetric additively-separable hedonic games, which are a nontrivial subclass of such games that are guaranteed to possess stable outcomes. These games are specified by an undirected edge- weighted graph: nodes are players, an outcome of the game is a partition of the nodes into coalitions, and the utility of a node is the sum of incident edge weights in the same coalition. We consider several stability requirements defined in the literature. These are based on restricting feasible player deviations, for example, by giving existing coalition members veto power. We extend these restrictions by considering more general forms of preference aggregation for coalition members. In particular, we consider voting schemes to decide if coalition members will allow a player to enter or leave their coalition. For all of the stability requirements we consider, the existence of a stable outcome is guaranteed by a potential function argument, and local improvements will converge to a stable outcome. We provide an almost complete characterization of these games in terms of the tractability of computing such stable outcomes. Our findings comprise positive results in the form of polynomial-time algorithms, and negative (PLS-completeness) results. The negative results extend to more general hedonic games.

Abstract:
The simplex method is a well-studied and widely-used pivoting method for solving linear programs. When Dantzig originally formulated the simplex method, he gave a natural pivot rule that pivots into the basis a variable with the most violated reduced cost. In their seminal work, Klee and Minty showed that this pivot rule takes exponential time in the worst case. We prove two main results on the simplex method. Firstly, we show that it is PSPACE-complete to find the solution that is computed by the simplex method using Dantzig's pivot rule. Secondly, we prove that deciding whether Dantzig's rule ever chooses a specific variable to enter the basis is PSPACE-complete. We use the known connection between Markov decision processes (MDPs) and linear programming, and an equivalence between Dantzig's pivot rule and a natural variant of policy iteration for average-reward MDPs. We construct MDPs and show PSPACE-completeness results for single-switch policy iteration, which in turn imply our main results for the simplex method.

Abstract:
We study the deterministic and randomized query complexity of finding approximate equilibria in bimatrix games. We show that the deterministic query complexity of finding an $\epsilon$-Nash equilibrium when $\epsilon < \frac{1}{2}$ is $\Omega(k^2)$, even in zero-one constant-sum games. In combination with previous results \cite{FGGS13}, this provides a complete characterization of the deterministic query complexity of approximate Nash equilibria. We also study randomized querying algorithms. We give a randomized algorithm for finding a $(\frac{3 - \sqrt{5}}{2} + \epsilon)$-Nash equilibrium using $O(\frac{k \cdot \log k}{\epsilon^2})$ payoff queries, which shows that the $\frac{1}{2}$ barrier for deterministic algorithms can be broken by randomization. For well-supported Nash equilibria (WSNE), we first give a randomized algorithm for finding an $\epsilon$-WSNE of a zero-sum bimatrix game using $O(\frac{k \cdot \log k}{\epsilon^4})$ payoff queries, and we then use this to obtain a randomized algorithm for finding a $(\frac{2}{3} + \epsilon)$-WSNE in a general bimatrix game using $O(\frac{k \cdot \log k}{\epsilon^4})$ payoff queries. Finally, we initiate the study of lower bounds against randomized algorithms in the context of bimatrix games, by showing that randomized algorithms require $\Omega(k^2)$ payoff queries in order to find a $\frac{1}{6k}$-Nash equilibrium, even in zero-one constant-sum games. In particular, this rules out query-efficient randomized algorithms for finding exact Nash equilibria.

Abstract:
Softwares like Adobe Photoshop are readily available today which enables us to make desired modifications in any image. But this is limited to only those who have an expertise in using that software. For others, it becomes too difficult. So, selecting a specific domain, the one dealing with images of human faces, we present an algorithm using which anyone can modify various facial features of human face like eyes, nose, lips, ears , hairstyle etc. Moreover, in order to make this procedure user friendly, we implement the algorithm using a GUI in Matlab. By making a GUI, we intend make the task of image morphing simpler and easy to implement

Abstract:
This paper presents the "Game Theory Explorer" software tool to create and analyze games as models of strategic interaction. A game in extensive or strategic form is created and nicely displayed with a graphical user interface in a web browser. State-of-the-art algorithms then compute all Nash equilibria of the game after a mouseclick. In tutorial fashion, we present how the program is used, and the ideas behind its main algorithms. We report on experiences with the architecture of the software and its development as an open-source project.

Abstract:
McLennan and Tourky (2010) showed that "imitation games" provide a new view of the computation of Nash equilibria of bimatrix games with the Lemke-Howson algorithm. In an imitation game, the payoff matrix of one of the players is the identity matrix. We study the more general "unit vector games", which are already known, where the payoff matrix of one player is composed of unit vectors. Our main application is a simplification of the construction by Savani and von Stengel (2006) of bimatrix games where two basic equilibrium-finding algorithms take exponentially many steps: the Lemke-Howson algorithm, and support enumeration.

Abstract:
A class of estimators of the R\'{e}nyi and Tsallis entropies of an unknown distribution $f$ in $\mathbb{R}^m$ is presented. These estimators are based on the $k$th nearest-neighbor distances computed from a sample of $N$ i.i.d. vectors with distribution $f$. We show that entropies of any order $q$, including Shannon's entropy, can be estimated consistently with minimal assumptions on $f$. Moreover, we show that it is straightforward to extend the nearest-neighbor method to estimate the statistical distance between two distributions using one i.i.d. sample from each. (Wit Correction.)