Solving parity games through fictitious play

摘要

The thesis aims to find an efficient algorithm for solving parity games. Parity games are graph-based, 0-sum, 2-person games with infinite plays. It is known that these games are determined: all nodes in these games are won by exactly one player. Solving parity games is equivalent to the model checking problem of modal mu-calculus; an efficient solution has important implications to program verification and controller synthesis. Although the decision problem of which player wins a given node is generally believed to be in PTIME, all known algorithms so far have been shown to run in (sub)exponential time. The design of existing algorithms either derives from the determinacy proof of parity games or from a purely graph theoretical perspective, using certain rank functions to iteratively search for an optimal solution. Since parity games are 2-person, 0-sum games, in this thesis I borrow ideas of game theory and investigate the viability of using fictitious play to solve them. Fictitious play is a method where two players choose strategies in strict alternation, and where these choices are “best responses” against the last k (so called bounded recall length) or against all strategies (unbounded recall length) of the other player chosen so far.udI use this method to design an algorithm that can solve partity games andudstudy its theoretical and experimental properties. For example, I prove that the basic algorithm solves fully connected games in polynomial time through a number of iterations that is bounded by a small constant. Although the proof is not extended to the general cases in the thesis, the basic algorithm performs demonstrably well against existing solvers in experiments over a large number and variety of games. In particular, the empirically obtained number of iterations that our basic algorithm requires appears to increase polynomially against the game sizes for all the games tested. Furthermore, the algorithm is conjectured to have a run time complexity bounded by O(n4 log2(n)) and I provide a discussion of strategy graphs and their emperically observed properties that motivates this conjecture.udOne caveat of fictitious play with bounded recall length is that the algorithm may fail to converge to the optimal solution due to the presence of nonoptimal strategy cycles of length greater than 2. In this thesis, I observe that in practice such cases account for less than 0.01% of the games tested. Different cycle resolution methods are explored in the thesis to address this. One particular method combines our basic algorithm and the discrete strategy solver together such that the resulting algorithm is guaranteed to terminate with the optimal solution. Also, this combined solver shares the runtime performance of fictitious play.