Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.07089

Polynomial-Time Computation of Extensive-Form Proper Equilibria in Two-Player Zero-Sum Games

Sam Ganzfried
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a deterministic polynomial-time algorithm for computing an exact extensive-form proper equilibrium in a finite two-player zero-sum extensive-form game with perfect recall and rational data. Properness is understood in the agent normal form: an action at an information set is penalized relative to a better action while all other local behavioral choices, including the player's own future actions, remain fixed. Our algorithm assigns each information set a geometric permutahedron of allowable action probabilities. A scaled doubly stochastic representation embeds these restrictions in a polynomial-size sequence-form polytope; a saddle point then satisfies the local $\varepsilon$-properness inequalities. To obtain an exact equilibrium rather than a solution at an arbitrarily small numerical perturbation, we establish sign stability of a parametric linear system, construct a persistent feasible basis, and extract the limits of behavioral ratios. The resulting probabilities are rational and have polynomial encoding length, including at information sets whose realization probabilities vanish.

Explore related subjects

cs.GTcs.AIcs.MAecon.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sam Ganzfried. 2026-11-04. Polynomial-Time Computation of Extensive-Form Proper Equilibria in Two-Player Zero-Sum Games. https://arxiv.org/abs/2611.07089

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Strongly Polynomial-Time 7/9-Approximation Algorithm for Maximin Share Allocations

We study maximin share (MMS) allocations of indivisible goods under nonnegative additive valuations. We give a strongly polynomial-time algorithm that allocates to every agent a bundle worth at least $7/9$ of her MMS, improving the previous $10/13$ guarantee of Heidari et al. (SODA 2026). Our analysis uses a dynamic witness-allocation framework that tracks the remaining goods from each agent's perspective throughout the allocation process. We obtain the computational result by iteratively refining upper bounds on MMS values, using only polynomially many runs of the allocation algorithm. Finally, we formalize and verify our main results in Lean 4.

cs.GT↗
arXiv2026

Otter: A Provably MEV-Resilient Automated Market Maker via Surplus Redistribution

Miner extractable value (MEV) in automated market makers allows block builders to profit from transaction ordering and injected trades, imposing costs on users and contributing to builder centralization. We introduce Otter (Optimal Truthful Trading with Excess Redistribution), a two-asset batch AMM that achieves provable MEV resilience when the consensus layer provides censorship resilience and block space is uncongested. Otter makes truthful behavior a dominant strategy for both users and builders. Consequently, a builder cannot profit from strategic deviations, including reordering bids or injecting sybil bids, even when the builder is itself a trader with intrinsic value. Moreover, we show that Otter maximizes social welfare, in a strong sense, within a natural class of mechanisms satisfying the desired game-theoretic properties. To achieve these guarantees, we introduce a new paradigm called surplus redistribution, which provably prevents residual surplus from being captured as MEV by redirecting it to the broader community. Specifically, the pool's output tokens need not be distributed entirely among users in the current batch. Instead, residual surplus may be transferred to a smart contract governed by the decentralized community. The accumulated surplus can be used to benefit community members --- for example, by subsidizing transaction fees, rewarding liquidity providers, or returning assets to the pool. Our approach relies on the underlying consensus layer to provide censorship resilience. We motivate its necessity through an impossibility result showing that the desired game-theoretic guarantees become unattainable when the builder is additionally allowed to censor transactions. Thus, our results also provide a mathematically formal demonstration of how consensus-level security guarantees can fundamentally expand what is achievable at the application layer.

cs.GT↗
arXiv2026

Hardness Amplification for Approximate Nash Equilibrium with Constantly Many Players

For every fixed integer $k\ge2$ and fixed parameters $δ,\varepsilon\in(0,1)$ and $η\in[0,1)$, we give a deterministic polynomial-time reduction from finding a $δ$-approximate Nash equilibrium ($δ$-ANE) in a $k$-player game $G$ with payoffs in $[0,1]$ to finding an $(\varepsilon,η)$-weak approximate Nash equilibrium (weak ANE) in a game $H$ with $C$ players, where $C$ depends only on $k,δ,\varepsilon,η$, independently of the action-set sizes. The constructed game is symmetric with binary payoffs. The games and strategies are represented by complete payoff tables and probabilities, with rational entries encoded in binary. Consequently, the existence of deterministic polynomial-time weak-ANE algorithms at one fixed pair of parameters $\varepsilon\in(0,1),η\in[0,1)$ for every fixed number of players would give deterministic polynomial-time $δ$-ANE algorithms at every fixed approximation parameter $δ\in(0,1)$ and number of players. Under deterministic PPAD-ETH, combining our reduction with Rubinstein's hardness result (FOCS 2016) for bimatrix games at some fixed approximation parameter $γ_0\in(0,1)$ rules out a deterministic algorithm for $(\varepsilon,η)$-weak ANE with running time $N^{O((\log N)^{1-ξ})}$ for any fixed $ξ\in(0,1)$. This holds for every fixed $\varepsilon\in(0,1)$ and $η\in[0,1)$ at a suitable constant number $C$ of players, where $N$ is the binary encoding length of the constructed game's complete payoff table.

cs.GT↗