Simple Stochastic Stopping Games: A Generator and Benchmark Library

Simple Stochastic Games (SSGs) were introduced by Anne Condon in 1990, as the simplest version of Stochastic Games for which there is no known polynomial-time algorithm. Condon showed that Stochastic Games are polynomial-time reducible to SSGs, which in turn are polynomial-time reducible to Stopping Games. SSGs are games where all decisions are binary and every move has a random outcome with a known probability distribution. Stopping Games are SSGs that are guaranteed to terminate. There are many algorithms for SSGs, most of which are fast in practice, but they all lack theoretical guarantees for polynomial-time convergence. The pursuit of a polynomial-time algorithm for SSGs is an active area of research. This paper is intended to support such research by making it easier to study the graphical structure of SSGs. Our contributions are: (1) a generating algorithm for Stopping Games, (2) a proof that the algorithm can generate any game, (3) a list of additional polynomial-time reductions that can be made to Stopping Games, (4) an open source generator for generating fully reduced instances of Stopping Games that comes with instructions and is fully documented, (5) a benchmark set of such instances, (6) and an analysis of how two main algorithm types perform on our benchmark set.Comment: 18 pages, 1 figure, 4 table

Solving Simple Stochastic Games with Few Random Nodes Faster Using Bland\u27s Rule

The best algorithm so far for solving Simple Stochastic Games is Ludwig\u27s randomized algorithm [Ludwig, 1995] which works in expected 2^{O(sqrt{n})} time. We first give a simpler iterative variant of this algorithm, using Bland\u27s rule from the simplex algorithm, which uses exponentially less random bits than Ludwig\u27s version. Then, we show how to adapt this method to the algorithm of Gimbert and Horn [Gimbert and Horn, 2008] whose worst case complexity is O(k!), where k is the number of random nodes. Our algorithm has an expected running time of 2^{O(k)}, and works for general random nodes with arbitrary outdegree and probability distribution on outgoing arcs

An Empirical Analysis of Algorithms for Simple Stochastic Games

This thesis presents the findings of a computational study on algorithms for Simple Stochastic Games (SSG). Simple Stochastic Games are a restriction of the Shapley stochastic model motivated by their applications in AI planning, logic synthesis, and theoretical computer science. This thesis seeks to empirically assess the performance of these algorithms to compensate for their lack of strong complexity results. Where applicable, we include both variations of algorithms where stable strategies are computed by a linear-programming and naive approach. These algorithms are evaluated on random inputs, in addition to specific difficult cases that were identified experimentally. We are interested in identifying difficult cases in particular because the worst case for the best performing algorithm, the Hoffman-Karp algorithm, is not known. The Hoffman-Karp algorithm for SSG is generally thought to take exponential iterations in the worst case, but this has not been proven. Despite both exhaustive and randomized searches of possible inputs, a case taking more than a linear number of iterations was not found. In this linear case, the otherwise fastest algorithms become inefficient and the naive algorithms surpass their linear-programming counterparts