Exact stationary equilibria with three or more discounted players

GT-035 | Stochastic games and computational complexity

P | Corrected unresolved multi-player complexity target; two-player classification known

Definitions and mathematical statement

A discounted perfect-information stochastic game has finite states $S$, a controlling player $c(s)\in\{1,\ldots,n\}$ at each state, finite actions $A(s)$ for its controller, rational payoffs $r_i(s,a)$ and transition probabilities $P(s'\mid s,a)$, and rational $\gamma\in(0,1)$. All data are explicitly listed in binary. A stationary profile is $x(\cdot\mid s)\in\Delta(A(s))$. Its normalized discounted payoff from $s$ is \[ V_i^x(s)=(1-\gamma)\mathbb E_{s,x}\sum_{t\geq0}\gamma^t r_i(s_t,a_t). \] The exact search relation asks for $x$ that is a Nash equilibrium from every initial state against arbitrary behavioral unilateral deviations. Equivalently, with $r_i^x(s)=\sum_a x(a\mid s)r_i(s,a)$ and $P^x(s'\mid s)=\sum_a x(a\mid s)P(s'\mid s,a)$, it asks for real $x,V$ satisfying the simplex constraints and \begin{align*} V_i(s)&=(1-\gamma)r_i^x(s)+\gamma\sum_{s'}P^x(s'\mid s)V_i(s'),\\ (1-\gamma)r_{c(s)}(s,a)+\gamma\sum_{s'}P(s'\mid s,a)V_{c(s)}(s')&\leq V_{c(s)}(s) \quad(s\in S,\ a\in A(s)). \end{align*} Classify this exact search problem for $n\geq3$, with fixed-$n$ subcases distinguished; in particular, determine whether the source's FIXP upper bound can be matched by FIXP-hardness under the corresponding exact fixed-point reductions.

Short English statement

What is the exact complexity of stationary equilibrium computation once the discounted game has at least three players?

Variants and scope boundaries

Exact solutions can be irrational, so an ordinary finite rational-output requirement is not adequate. FIXP treats exact real fixed points of rational algebraic circuits as solutions; bit approximation and approximate equilibrium are different search relations. Every-state optimality is part of the source convention. The number of players, discount encoding, and action counts cannot be hidden when stating a reduction.

Source relationship and status

[S1] already proves PPAD-completeness for the two-player case (Theorem 1). It leaves the precise complexity for three or more players open, with irrational examples and a four-player SqrtSum-hardness result (Theorem 3). Section 6 identifies possible FIXP-hardness as a remaining question. The original unrestricted "complexity unresolved" wording is therefore corrected.

References

[S1] Kristoffer Arnsfelt Hansen and Xinhao Nie, On the Complexity of Stationary Nash Equilibria in Discounted Perfect Information Stochastic Games, arXiv:2510.11550v1 (2025). Locators: abstract; Section 2.1 (game and equilibrium convention); Theorems 1 and 3; Section 6 (remaining complexity and FIXP-hardness questions). https://arxiv.org/html/2510.11550v1

Common conventions: Source mathematical conventions

Finite games. For a finite set $A$, $\Delta(A)$ is its probability simplex. Players choose independent mixed strategies in Nash equilibrium; correlation is allowed only when explicitly part of the solution concept. In a game with payoff functions $u_i$ and strategy profile $x$, an additive $\varepsilon$ Nash equilibrium satisfies \[ u_i(x)\geq u_i(x_i',x_{-i})-\varepsilon \quad\text{for every player $i$ and every admissible deviation $x_i'$}. \] For numerical approximation, payoffs are normalized as locally specified. An $\varepsilon$ payoff residual is distinct from distance at most $\varepsilon$ to the set of exact equilibria. Point convergence, convergence of empirical play, and convergence in distance to an equilibrium set are also distinct.

Computational representations. Unless stated otherwise, a complexity question takes rational data with binary encoding; polynomial time is measured in total bit length. A fixed-accuracy polynomial algorithm is not necessarily polynomial in $1/\varepsilon$, and neither is necessarily polynomial in $\log(1/\varepsilon)$. Succinct games and value, demand, or sampling oracles must declare their interfaces. Exact real solutions require an explicit representation; an unspecified list of arbitrary real numbers is not a finite computational output. A claimed algorithmic barrier is meaningful only for its specified model and reductions.

Dynamic games. Histories, information, observation, and allowable randomization are part of the model. A stationary strategy depends only on the current state; a positional strategy in a deterministic graph game chooses one action at each controlled vertex. Nash equilibrium at an initial state and equilibrium after every history are different requirements. An expectation of a pathwise $\liminf$ payoff, a $\liminf$ of expected averages, a limiting expected average, and a uniform finite-horizon equilibrium are different criteria. None is silently substituted for another.

Allocation and mechanisms. A complete allocation assigns every item exactly once unless otherwise stated. Zero-valued goods, negative values, capacity constraints, feasibility, lotteries, and copies of goods affect fairness definitions and are specified locally. Dominant-strategy incentive compatibility, Bayesian incentive compatibility, universal truthfulness, and truthfulness in expectation impose different inequalities. Whenever an objective ratio has zero denominator, its convention is stated or those instances are excluded.

Infinite-dimensional models. Expectations, integrals, stochastic equations, and optimization objectives require their local regularity, integrability, filtrations, and admissible domains. A backward--forward mean-field system is a boundary-value problem; it is not an initial-value semiflow without an additional construction. Long-time, large-population, and vanishing-noise limits require an order of limits and a convergence topology. If a minimum need not be attained, the objective is its infimum or supremum and the approximation requirement is stated separately.