Semi-algebraic Tools in Stochastic Games

In this thesis we consider two-person zero-sum stochastic games with a special focus on how tools from the mathematical field of semi-algebraic geometry have been applied to these games. In the first of two parts of the thesis we introduce stochastic games and prove a complexity result about computing the value of a type of stochastic games called concurrent reachability games. We show that the value of a finite-state concurrent reachability game can be approximated to arbitrary precision in TFNP[NP], that is, in the polynomial time hierarchy. Previously, no better bound than PSPACE was known for this problem. In the second part we go into the parts of semi-algebraic geometry theory that have been applied to stochastic games, and we provide a survey of the results in stochastic games that have been proved using these tools. We then go on to use semi-algebraic geometry to show that for limiting average games, the most general class of stochastic games we consider, there is a particularly nice class of stationary strategies that we will call monomial strategies which are -optimal among stationary strategies. This implies that all concurrent reachability games have a monomial family of -optimal strategies. We follow this up by proving numerically precise versions of some sampling theorems from semi-algebraic geometry. We apply these to the results of [40] to get precise bounds for the patience of concurrent reachability games, and obtain bounds for the monomial strategies that we have proven to exist.