arXiv subjects

Rahul Savani

Publications and source records attributed to Rahul Savani.

62 records · Page 4Linked to original sources

arXiv2014

Game Theory Explorer - Software for the Applied Game Theorist

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.

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arXiv2014

Learning Equilibria of Games via Payoff Queries

A recent body of experimental literature has studied empirical game-theoretical analysis, in which we have partial knowledge of a game, consisting of observations of a subset of the pure-strategy profiles and their associated payoffs to players. The aim is to find an exact or approximate Nash equilibrium of the game, based on these observations. It is usually assumed that the strategy profiles may be chosen in an on-line manner by the algorithm. We study a corresponding computational learning model, and the query complexity of learning equilibria for various classes of games. We give basic results for bimatrix and graphical games. Our focus is on symmetric network congestion games. For directed acyclic networks, we can learn the cost functions (and hence compute an equilibrium) while querying just a small fraction of pure-strategy profiles. For the special case of parallel links, we have the stronger result that an equilibrium can be identified while only learning a small fraction of the cost values.

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arXiv2014

Finding Approximate Nash Equilibria of Bimatrix Games via Payoff Queries

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 $ε$-Nash equilibrium when $ε< \frac{1}{2}$ is $Ω(k^2)$, 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 $(\frac{3 - \sqrt{5}}{2} + ε)$-Nash equilibrium using $O(\frac{k \cdot \log k}{ε^2})$ 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 $ε$-WSNE of a zero-sum bimatrix game using $O(\frac{k \cdot \log k}{ε^4})$ payoff queries, and we then use this to obtain a randomized algorithm for finding a $(\frac{2}{3} + ε)$-WSNE in a general bimatrix game using $O(\frac{k \cdot \log k}{ε^4})$ 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 $Ω(k^2)$ 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.

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arXiv2011

The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement. Extending our result for the Harsanyi-Selten process, we show that several other homotopy-based algorithms for finding equilibria of games are also PSPACE-complete to implement. A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in [Savani and von Stengel]. These results show that our techniques can be widely applied and suggest that the PSPACE-completeness of implementing homotopy methods is a general principle.

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arXiv2011

On the Approximation Performance of Fictitious Play in Finite Games

We study the performance of Fictitious Play, when used as a heuristic for finding an approximate Nash equilibrium of a 2-player game. We exhibit a class of 2-player games having payoffs in the range [0,1] that show that Fictitious Play fails to find a solution having an additive approximation guarantee significantly better than 1/2. Our construction shows that for n times n games, in the worst case both players may perpetually have mixed strategies whose payoffs fall short of the best response by an additive quantity 1/2 - O(1/n^(1-delta)) for arbitrarily small delta. We also show an essentially matching upper bound of 1/2 - O(1/n).

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arXiv2009

Wiretapping a hidden network

We consider the problem of maximizing the probability of hitting a strategically chosen hidden virtual network by placing a wiretap on a single link of a communication network. This can be seen as a two-player win-lose (zero-sum) game that we call the wiretap game. The value of this game is the greatest probability that the wiretapper can secure for hitting the virtual network. The value is shown to equal the reciprocal of the strength of the underlying graph. We efficiently compute a unique partition of the edges of the graph, called the prime-partition, and find the set of pure strategies of the hider that are best responses against every maxmin strategy of the wiretapper. Using these special pure strategies of the hider, which we call omni-connected-spanning-subgraphs, we define a partial order on the elements of the prime-partition. From the partial order, we obtain a linear number of simple two-variable inequalities that define the maxmin-polytope, and a characterization of its extreme points. Our definition of the partial order allows us to find all equilibrium strategies of the wiretapper that minimize the number of pure best responses of the hider. Among these strategies, we efficiently compute the unique strategy that maximizes the least punishment that the hider incurs for playing a pure strategy that is not a best response. Finally, we show that this unique strategy is the nucleolus of the recently studied simple cooperative spanning connectivity game.

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arXiv2009

Linear Complementarity Algorithms for Infinite Games

The performance of two pivoting algorithms, due to Lemke and Cottle and Dantzig, is studied on linear complementarity problems (LCPs) that arise from infinite games, such as parity, average-reward, and discounted games. The algorithms have not been previously studied in the context of infinite games, and they offer alternatives to the classical strategy-improvement algorithms. The two algorithms are described purely in terms of discounted games, thus bypassing the reduction from the games to LCPs, and hence facilitating a better understanding of the algorithms when applied to games. A family of parity games is given, on which both algorithms run in exponential time, indicating that in the worst case they perform no better for parity, average-reward, or discounted games than they do for general P-matrix LCPs.

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arXiv2009

Spanning connectivity games

The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values and Shapley-Shubik indices is #P-complete for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. As a corollary, computing the Shapley value is #P-complete for simple games represented by the set of minimal winning coalitions, Threshold Network Flow Games, Vertex Connectivity Games and Coalitional Skill Games.

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