Fast Payoff Matrix Sparsification Techniques for Structured Extensive-Form Games

The practical scalability of many optimization algorithms for large extensive-form games is often limited by the games' huge payoff matrices. To ameliorate issue, Zhang and Sandholm recently proposed a sparsification technique that factorizes matrix A into sparser object = Â + UVᵀ, where total combined number nonzeros Â, U, V, significantly smaller. Such factorization can be used in place original algorithm, such as interior-point second-order methods, thus increasing size handled. Their sparsifies poker (end)games, standard benchmarks computational game theory, AI, more broadly. We show existence extremely sparse factorizations tied to their particular Kronecker-product structure. clarify how structure arises introduce connection between sparsification. By leveraging structure, we give two ways computing strong sparsifications (as well any other with similar structure) are i) orders magnitude faster compute, ii) numerically stable, iii) produce dramatically smaller than prior technique. Our techniques enable—for first time—effective computation high-precision Nash equilibria strategies subject constraints on amount allowed randomization. Furthermore, they speed up parallel first-order game-solving algorithms; state-of-the-art GPU.