Existence and computation of monomial families of near-optimal strategies for recursive games

2026-08-08

AI Summary

This work addresses the construction of near-optimal strategies in finite recursive games that are valid for all sufficiently small precision parameters ε. By analyzing the polynomial structure of absorption probabilities through directed forests, the authors express the payoffs of pure strategy responses—under a fixed support—as rational functions sharing a common positive denominator. They compress the asymptotic orders of these expressions into integer weight vectors to construct a unified family of monomial strategies. Employing an elementary approach that avoids semialgebraic selection or Puiseux series, they present the first deterministic polynomial-time algorithm for rational recursive games with a fixed number of states N. The algorithm exactly computes the strategy family and its algebraic coefficients, with both running time and output length bounded by L^{(N+1)^{O(N)}}, where L denotes the input length.

Abstract

In a finite recursive game in the sense of Everett, both players have stationary epsilon-optimal strategies for every epsilon>0. Frederiksen and Miltersen strengthened this result by showing that the strategies for all sufficiently small epsilon can be encoded by finitely many monomials: at every state, all but possibly one of the action probabilities are constants times integer powers of epsilon. The resulting finite symbolic object specifies a strategy for every sufficiently small accuracy. Their proof uses semialgebraic selection and Puiseux series. We give an alternative elementary proof of this regularity theorem for recursive games. We start with stationary strategies that guarantee vectors approaching the value through Everett's one-sided region. After fixing their support, we express, for each pure stationary reply, all absorption probabilities as quotients of directed-forest polynomials with nonnegative coefficients and a common positive denominator. Each payoff is a fixed signed linear combination of these quotients. We then compress the asymptotic orders of the finitely many forest monomials into one integer weight vector. This proof uses neither semialgebraic selection nor Puiseux series. Furthermore, for rational games with a fixed number N of active states, we present a deterministic polynomial-time algorithm that computes a monomial family exactly. It returns all algebraic coefficients in one ordered real univariate representation. The representation length and running time are at most L^{(N+1)^{O(N)}}, where L is the input length.

Problem

• recursive games
• near-optimal strategies
• monomial families
• stationary strategies
• epsilon-optimal

Innovation

• recursive games
• monomial strategies
• absorption probabilities
• deterministic polynomial-time algorithm
• directed-forest polynomials