Search arXiv⌕ Search

arXiv subjects

Sam Ganzfried

Publications and source records attributed to Sam Ganzfried.

At least 19 recordsLinked to original sources
arXiv2026

Variable Bound Tightening for Nash Equilibrium Computation in Multiplayer Imperfect-Information Games

There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer strategic-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. Recently, an approach has been presented for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically constrained program based on a nonlinear complementarity problem formulation derived from the sequence-form game representation. This formulation was solved using Gurobi's nonconvex quadratic solver, which employs spatial branch-and-bound to iteratively refine variable bounds by solving convex relaxations of bilinear terms. During presolve, Gurobi introduces auxiliary variables and, in some cases, binary variables, leading to an internal mixed-integer quadratically constrained program. This approach was demonstrated to outperform prior algorithms from the Gambit software suite and quickly solve three-player Kuhn poker after removal of dominated actions; however, the algorithm was not able to solve the full version of the game within 24 hours. In this paper, we derive finite bounds on slack and multiplier variables in the nonlinear complementarity formulation. These bounds can strengthen the convex relaxations used within spatial branch-and-bound. We compare uniform corollary bounds with finer proposition-based bounds on full three-player Kuhn poker. All eight bounded configurations return numerically verified equilibria in under 17 seconds, while the unbounded formulation finds no feasible solution within 300 seconds. The finest bounds do not consistently yield the fastest computation.

cs.GT↗
arXiv2026

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

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.

cs.GT↗
arXiv2026

Safe Equilibrium

The standard game-theoretic solution concept, Nash equilibrium, assumes that all players behave rationally. If we follow a Nash equilibrium and opponents are irrational (or follow strategies from a different Nash equilibrium), then we may obtain an extremely low payoff. On the other hand, a maximin strategy assumes that all opposing agents are playing to minimize our payoff (even if it is not in their best interest), and ensures the maximal possible worst-case payoff, but results in exceedingly conservative play. We propose a new solution concept called safe equilibrium that models opponents as behaving rationally with a specified probability and behaving potentially arbitrarily with the remaining probability. We prove that a safe equilibrium exists in all strategic-form games (for all possible values of the rationality parameters), and prove that its computation is PPAD-hard. We present exact algorithms for computing a safe equilibrium in both 2 and $n$-player games, as well as scalable approximation algorithms.

cs.GT↗
arXiv2026

Observable Perfect Equilibrium

While Nash equilibrium has emerged as the central game-theoretic solution concept, many important games contain several Nash equilibria and we must determine how to select between them in order to create real strategic agents. Several Nash equilibrium refinement concepts have been proposed and studied for sequential imperfect-information games, the most prominent being trembling-hand perfect equilibrium, quasi-perfect equilibrium, proper equilibrium, and recently one-sided quasi-perfect equilibrium. These concepts are robust to certain arbitrarily small mistakes, and are guaranteed to always exist; however, we argue that none of these is the correct concept for developing strong agents in sequential games of imperfect information. We define a new equilibrium refinement concept for extensive-form games called observable perfect equilibrium in which the solution is robust over trembles in publicly-observable action probabilities (not necessarily over all action probabilities that may not be observable by opposing players). Observable perfect equilibrium correctly captures the assumption that the opponent is playing as rationally as possible given mistakes that have been observed (while previous solution concepts do not). We prove that observable perfect equilibrium is always guaranteed to exist, and demonstrate that it selects a unique equilibrium in a simplified no-limit poker game where the prior extensive-form refinements retain a continuum of equilibria. We expect observable perfect equilibrium to be a useful equilibrium refinement concept for modeling many important imperfect-information games of interest in artificial intelligence.

cs.GT↗
arXiv2026

Consistent Opponent Modeling in Imperfect-Information Games

The goal of agents in multi-agent environments is to maximize total reward against the opposing agents that are encountered. Following a game-theoretic solution concept, such as Nash equilibrium, may obtain a strong performance in some settings; however, such approaches fail to capitalize on historical and observed data from repeated interactions against our opponents. Opponent modeling algorithms integrate machine learning techniques to exploit suboptimal opponents utilizing available data; however, the effectiveness of such approaches in imperfect-information games to date is quite limited. We show that existing opponent modeling approaches fail to satisfy a simple desirable property even against static opponents drawn from a known prior distribution; namely, they do not guarantee that the model approaches the opponent's true strategy even in the limit as the number of iterations approaches infinity. We develop a new algorithm that achieves this property given sufficiently informative observations and runs efficiently by solving a convex minimization problem based on the sequence-form game representation.

cs.GT↗
arXiv2026

Computing Evolutionarily Stable Strategies in Imperfect-Information Games

We present an algorithm for computing evolutionarily stable strategies (ESSs) in symmetric two-player perfect-recall extensive-form games of imperfect information. The algorithm operates directly in sequence form, avoiding explicit enumeration of normal-form supports. In a biologically motivated cancer signaling game, the algorithm recovers all seven symmetric Nash equilibria and identifies all three ESSs, each with a clear biological interpretation. Experiments on random imperfect-information games evaluate scalability as the numbers of private signals and available actions increase.

cs.GT↗
arXiv2026

Evolutionarily Stable Stackelberg Equilibrium

We present a new solution concept called evolutionarily stable Stackelberg equilibrium (SESS). We study the Stackelberg evolutionary game setting in which there is a single leading player and a symmetric population of followers. The leader selects an optimal mixed strategy, anticipating that the follower population plays an evolutionarily stable strategy (ESS) in the induced subgame and may satisfy additional ecological conditions. We consider both leader-optimal and leader-pessimal selection among ESSs, which arise as special cases of our framework. Prior approaches to Stackelberg evolutionary games either define the follower response via evolutionary dynamics or assume rational best-response behavior, without explicitly enforcing stability against invasion by mutations. We present algorithms for computing SESS in discrete and continuous games, and validate the latter empirically. Our model applies naturally to biological settings; for example, in cancer treatment the leader represents the physician and the followers correspond to competing cancer cell phenotypes.

cs.GT↗
arXiv2026

Dominated Actions in Imperfect-Information Games

Dominance is a fundamental concept in game theory. In normal-form games dominated strategies can be identified in polynomial time. As a consequence, iterative removal of dominated strategies can be performed efficiently as a preprocessing step for reducing the size of a game before computing a Nash equilibrium. For imperfect-information games in extensive form, we could convert the game to normal form and then iteratively remove dominated strategies in the same way; however, this conversion may cause an exponential blowup in game size. In this paper we define and study the concept of dominated actions in imperfect-information games. Our main result is a polynomial-time algorithm for determining whether an action is dominated (strictly or weakly) by any mixed strategy in two-player perfect-recall games with publicly observable actions, which can be extended to iteratively remove dominated actions. This allows us to efficiently reduce the size of the game tree as a preprocessing step for Nash equilibrium computation. We explore the role of dominated actions empirically in "All In or Fold" No-Limit Texas Hold'em poker.

cs.GT↗
arXiv2026

Quadratic Programming Approach for Nash Equilibrium Computation in Multiplayer Imperfect-Information Games

There has been significant recent progress in algorithms for approximation of Nash equilibrium in large two-player zero-sum imperfect-information games and exact computation of Nash equilibrium in multiplayer strategic-form games. While counterfactual regret minimization and fictitious play are scalable to large games and have convergence guarantees in two-player zero-sum games, they do not guarantee convergence to Nash equilibrium in multiplayer games. We present an approach for exact computation of Nash equilibrium in multiplayer imperfect-information games that solves a quadratically-constrained program based on a nonlinear complementarity problem formulation from the sequence-form game representation. This approach capitalizes on recent advances for solving nonconvex quadratic programs. Our algorithm is able to quickly solve three-player Kuhn poker after removal of dominated actions. Of the available algorithms in the Gambit software suite, only the logit quantal response approach is successfully able to solve the game; however, the approach takes longer than our algorithm and also involves a degree of approximation. Our formulation also leads to a new approach for computing Nash equilibrium in multiplayer strategic-form games which we demonstrate to outperform a previous quadratically-constrained program formulation.

cs.GT↗
arXiv2026

Projected Exploitability Descent for Nash Equilibrium Computation in Multiplayer Imperfect-Information Games

Many important games have more than two players and imperfect information. Existing approaches for computing Nash equilibrium, the central game-theoretic solution concept, in such games either lack scalability or obtain poor performance. In this paper we introduce a new algorithm called projected exploitability descent (PED) for approximating Nash equilibria in multiplayer games of imperfect information. The algorithm works by running projected subgradient descent minimizing a proxy for the multiplayer generalized exploitability function. The objective is nonconvex and nonsmooth, but can be represented as the sum of the maxima of linear functions, for which a subgradient can easily be computed and projected to the polytope of feasible sequence-form strategies. We explore performance of PED on a generalized version of the well-studied benchmark game three-player Kuhn poker. No prior exact algorithms scale to the version of the game with deck size larger than 4, and we compare performance to the popular algorithms of fictitious play (FP) and counterfactual regret minimization (CFR). We find that PED obtains a consistent near-monotonic improvement throughout all runs, though both FP and CFR perform significantly better in the initial iterations. This inspires a hybrid algorithm FP-PED that runs FP for an initial burn-in period before switching to PED for stable long-run refinement. We can alternatively view this as a multi-step algorithm that runs FP as a pre-processing step to obtain a strong initialization for PED.

cs.GT↗
arXiv2026

Human strategic decision making in parametrized games

Many real-world games contain parameters which can affect payoffs, action spaces, and information states. For fixed values of the parameters, the game can be solved using standard algorithms. However, in many settings agents must act without knowing the values of the parameters that will be encountered in advance. Often the decisions must be made by a human under time and resource constraints, and it is unrealistic to assume that a human can solve the game in real time. We present a new framework that enables human decision makers to make fast decisions without the aid of real-time solvers. We demonstrate applicability to a variety of situations including settings with multiple players and imperfect information.

cs.GT↗
arXiv2024

Nonparametric Strategy Test

We present a nonparametric statistical test for determining whether an agent is following a given mixed strategy in a repeated strategic-form game given samples of the agent's play. This involves two components: determining whether the agent's frequencies of pure strategies are sufficiently close to the target frequencies, and determining whether the pure strategies selected are independent between different game iterations. Our integrated test involves applying a chi-squared goodness of fit test for the first component and a generalized Wald-Wolfowitz runs test for the second component. The results from both tests are combined using Bonferroni correction to produce a complete test for a given significance level $α.$ We applied the test to publicly available data of human rock-paper-scissors play. The data consists of 50 iterations of play for 500 human players. We test with a null hypothesis that the players are following a uniform random strategy independently at each game iteration. Using a significance level of $α= 0.05$, we conclude that 305 (61%) of the subjects are following the target strategy.

stat.ME↗
arXiv2024

Opponent Modeling in Multiplayer Imperfect-Information Games

In many real-world settings agents engage in strategic interactions with multiple opposing agents who can employ a wide variety of strategies. The standard approach for designing agents for such settings is to compute or approximate a relevant game-theoretic solution concept such as Nash equilibrium and then follow the prescribed strategy. However, such a strategy ignores any observations of opponents' play, which may indicate shortcomings that can be exploited. We present an approach for opponent modeling in multiplayer imperfect-information games where we collect observations of opponents' play through repeated interactions. We run experiments against a wide variety of real opponents and exact Nash equilibrium strategies in three-player Kuhn poker and show that our algorithm significantly outperforms all of the agents, including the exact Nash equilibrium strategies.

cs.GT↗
arXiv2024

Empirical Analysis of Fictitious Play for Nash Equilibrium Computation in Multiplayer Games

While fictitious play is guaranteed to converge to Nash equilibrium in certain game classes, such as two-player zero-sum games, it is not guaranteed to converge in non-zero-sum and multiplayer games. We show that fictitious play in fact leads to improved Nash equilibrium approximation over a variety of game classes and sizes than (counterfactual) regret minimization, which has recently produced superhuman play for multiplayer poker. We also show that when fictitious play is run several times using random initializations it is able to solve several known challenge problems in which the standard version is known to not converge, including Shapley's classic counterexample. These provide some of the first positive results for fictitious play in these settings, despite the fact that worst-case theoretical results are negative.

cs.GT↗
arXiv2023

Random Initialization Solves Shapley's Fictitious Play Counterexample

In 1964 Shapley devised a family of games for which fictitious play fails to converge to Nash equilibrium. The games are two-player non-zero-sum with 3 pure strategies per player. Shapley assumed that each player played a specific pure strategy in the first round. We show that if we use random (mixed) strategy profile initializations we are able to converge to Nash equilibrium approximately 1/3 of the time for a representative game in this class.

cs.GT↗
arXiv2023

Stable Relationships

We study a dynamic model of the relationship between two people where the states depend on the "power" in the relationship. We perform a comprehensive analysis of stability of the system, and determine a set of conditions under which stable relationships are possible. In particular, stable relationships can occur if both people are dominant, but the sum of dominances is below a bound determined by the model's parameters. Stable relationships can also occur if one person is dominant and the other is submissive, provided the level of dominance exceeds the level of submissiveness but not beyond a threshold. We also conclude that a stable relationship is not possible if both people are submissive. While our model is motivated by a social or romantic relationship, it can also be applied to professional or business relationships as well as diplomatic relationships between nations.

cs.MA↗
arXiv2023

Fast Complete Algorithm for Multiplayer Nash Equilibrium

We describe a new complete algorithm for computing Nash equilibrium in multiplayer general-sum games, based on a quadratically-constrained feasibility program formulation. We demonstrate that the algorithm runs significantly faster than the prior fastest complete algorithm on several game classes previously studied and that its runtimes even outperform the best incomplete algorithms.

cs.GT↗