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Reducing Stochastic Games to Semidefinite Program Feasibility

Manuel Bodirsky, Georg Loho, Mateusz Skomra

TL;DR

The paper presents a polynomial-time chain of reductions that converts max-plus-average constraint problems into semidefinite program feasibility, thereby subsuming several key game-theoretic problems (mean payoff games, simple stochastic games, parity games) within SDP feasibility. The core innovations are lifting to non-Archimedean SDPs using Puiseux series and then translating to real SDPs through two steps: substituting a large real parameter and encoding large constants with small LMIs via spectrahedral projections. This approach not only links convex optimisation with extensive game-theoretic problems but also suggests that polynomial-time SDP solvers, if available, would yield polynomial-time solutions for the entire referenced class of games. Moreover, the work highlights potential cross-fertilization between convex algebraic geometry and verification, offering new avenues for algorithmic strategies in SDPs inspired by game-theoretic constructions.

Abstract

We present a polynomial-time reduction from max-plus-average constraints to the feasibility problem for semidefinite programs. This shows that Condon's simple stochastic games, stochastic mean payoff games, and in particular mean payoff games and parity games can all be reduced to semidefinite programming.

Reducing Stochastic Games to Semidefinite Program Feasibility

TL;DR

The paper presents a polynomial-time chain of reductions that converts max-plus-average constraint problems into semidefinite program feasibility, thereby subsuming several key game-theoretic problems (mean payoff games, simple stochastic games, parity games) within SDP feasibility. The core innovations are lifting to non-Archimedean SDPs using Puiseux series and then translating to real SDPs through two steps: substituting a large real parameter and encoding large constants with small LMIs via spectrahedral projections. This approach not only links convex optimisation with extensive game-theoretic problems but also suggests that polynomial-time SDP solvers, if available, would yield polynomial-time solutions for the entire referenced class of games. Moreover, the work highlights potential cross-fertilization between convex algebraic geometry and verification, offering new avenues for algorithmic strategies in SDPs inspired by game-theoretic constructions.

Abstract

We present a polynomial-time reduction from max-plus-average constraints to the feasibility problem for semidefinite programs. This shows that Condon's simple stochastic games, stochastic mean payoff games, and in particular mean payoff games and parity games can all be reduced to semidefinite programming.

Paper Structure

This paper contains 8 sections, 10 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Semidefinite Programming
  5. Stochastic Games
  6. From stochastic games to max-average constraints
  7. From max-plus-average constraints to non-Archimedean SDPs
  8. From non-Archimedean to real SDPs

Key Result

Theorem 2

If the primal SDP is strictly feasible and bounded from below, then the dual program attains its supremum and there is no duality gap.

Figures (1)

  • Figure 1: Computational problems that are not known to be in P and not known to be NP-hard, and their relationship. Arcs indicate polynomial-time (many-one) reductions, the dotted arc indicates a polynomial-time Turing reduction.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 2: see Theorem 2.4.1 in ben-tal_nemirovski and the explanations on page 142 and 143
  • Theorem 3: issac2016jsc
  • Definition 4
  • Proposition 5
  • Theorem 6
  • Theorem 7
  • Proposition 8
  • Theorem 9
  • Corollary 10
  • ...and 2 more