Game theory is a foundational framework for analyzing strategic interactions, and its intersection with large language models (LLMs) is a rapidly growing field. However, existing surveys mainly focus narrowly on using game theory to evaluate LLM behavior. This paper provides the first comprehensive survey of the bidirectional relationship between Game Theory and LLMs. We propose a novel taxonomy that categorizes the research in this intersection into four distinct perspectives: (1) evaluating LLMs in game-based scenarios; (2) improving LLMs using game-theoretic concepts for better interpretability and alignment; (3) modeling the competitive landscape of LLM development and its societal impact; and (4) leveraging LLMs to advance game models and to solve corresponding game theory problems. Furthermore, we identify key challenges and outline future research directions. By systematically investigating this interdisciplinary landscape, our survey highlights the mutual influence of game theory and LLMs, fostering progress at the intersection of these fields.
Potential games, originally introduced in the early 1990's by Lloyd Shapley, the 2012 Nobel Laureate in Economics, and his colleague Dov Monderer, are a very important class of models in game theory. They have special properties such as the existence of Nash equilibria in pure strategies. This note introduces graphical versions of potential games. Special cases of graphical potential games have already found applicability in many areas of science and engineering beyond economics, including artificial intelligence, computer vision, and machine learning. They have been effectively applied to the study and solution of important real-world problems such as routing and congestion in networks, distributed resource allocation (e.g., public goods), and relaxation-labeling for image segmentation. Implicit use of graphical potential games goes back at least 40 years. Several classes of games considered standard in the literature, including coordination games, local interaction games, lattice games, congestion games, and party-affiliation games, are instances of graphical potential games. This note provides several characterizations of graphical potential games by leveraging well-known result
A game of information concerns two players transmitting messages that are obscured by noise. A receiver digests the combination of the two information sources and makes an assessment rationally. The aim of the players is to generate opposing assessments for the receiver by choosing signal-to-noise ratios of their information. It is shown that this problem can be reduced into an elementary infinite game on the square, thus admitting a complete equilibrium solution. Three generalisations of the game are proposed.
We consider a sub-class of bi-matrix games which we refer to as two-person (hereafter referred to as two-player) additively-separable sum (TPASS) games, where the sum of the pay-offs of the two players is additively separable. The row player's pay-off at each pair of pure strategies, is the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the column player and the second being independent of the pure strategy chosen by the column player. The column player's pay-off at each pair of pure strategies, is also the sum of two numbers, the first of which may be dependent on the pure strategy chosen by the row player and the second being independent of the pure strategy chosen by the row player. The sum of the inter-dependent components of the pay-offs of the two players is assumed to be zero. We prove the existence of equilibrium for such games and show that the set of equilibria for such games is the projection on the set of strategy pairs of the solutions of a pair of linear programming problems that are dual to each other. This result is a generalization of the corresponding and well-known result for two-person zero-sum games. We also show that a (
Game theoretical techniques have recently become prevalent in many engineering applications, notably in communications. With the emergence of cooperation as a new communication paradigm, and the need for self-organizing, decentralized, and autonomic networks, it has become imperative to seek suitable game theoretical tools that allow to analyze and study the behavior and interactions of the nodes in future communication networks. In this context, this tutorial introduces the concepts of cooperative game theory, namely coalitional games, and their potential applications in communication and wireless networks. For this purpose, we classify coalitional games into three categories: Canonical coalitional games, coalition formation games, and coalitional graph games. This new classification represents an application-oriented approach for understanding and analyzing coalitional games. For each class of coalitional games, we present the fundamental components, introduce the key properties, mathematical techniques, and solution concepts, and describe the methodologies for applying these games in several applications drawn from the state-of-the-art research in communications. In a nutshell,
Eigen mode selection ought to be a practical issue in some real game systems, as it is a practical issue in the dynamics behaviour of a building, bridge, or molecular, because of the mathematical similarity in theory. However, its reality and accuracy have not been known in real games. We design a 5-strategy game which, in the replicator dynamics theory, is predicted to exist two eigen modes. Further, in behaviour game theory, the game is predicted that the mode selection should depends on the game parameter. We conduct human subject game experiments by controlling the parameter. The data confirm that, the predictions on the mode existence as well as the mode selection are significantly supported. This finding suggests that, like the equilibrium selection concept in classical game theory, eigen mode selection is an issue in game dynamics theory.
In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game.
Lipschitz games, in which there is a limit $λ$ (the Lipschitz value of the game) on how much a player's payoffs may change when some other player deviates, were introduced about 10 years ago by Azrieli and Shmaya. They showed via the probabilistic method that $n$-player Lipschitz games with $m$ strategies per player have pure $ε$-approximate Nash equilibria, for $ε\geqλ\sqrt{8n\log(2mn)}$. Here we provide the first hardness result for the corresponding computational problem, showing that even for a simple class of Lipschitz games (Lipschitz polymatrix games), finding pure $ε$-approximate equilibria is PPAD-complete, for suitable pairs of values $(ε(n), λ(n))$. Novel features of this result include both the proof of PPAD hardness (in which we apply a population game reduction from unrestricted polymatrix games) and the proof of containment in PPAD (by derandomizing the selection of a pure equilibrium from a mixed one). In fact, our approach implies containment in PPAD for any class of Lipschitz games where payoffs from mixed-strategy profiles can be deterministically computed.
The rank of a bimatrix game is defined as the rank of the sum of the payoff matrices of the two players. The rank of a game is known to impact both the most suitable computation methods for determining a solution and the expressive power of the game. Under certain conditions on the payoff matrices, we devise a method that reduces the rank of the game without changing the equilibrium of the game. We leverage matrix pencil theory and the Wedderburn rank reduction formula to arrive at our results. We also present a constructive proof of the fact that in a generic square game, the rank of the game can be reduced by 1, and in generic rectangular game, the rank of the game can be reduced by 2 under certain assumptions.
Nearly a decade ago, Azrieli and Shmaya introduced the class of $λ$-Lipschitz games in which every player's payoff function is $λ$-Lipschitz with respect to the actions of the other players. They showed that such games admit $ε$-approximate pure Nash equilibria for certain settings of $ε$ and $λ$. They left open, however, the question of how hard it is to find such an equilibrium. In this work, we develop a query-efficient reduction from more general games to Lipschitz games. We use this reduction to show a query lower bound for any randomized algorithm finding $ε$-approximate pure Nash equilibria of $n$-player, binary-action, $λ$-Lipschitz games that is exponential in $\frac{nλ}ε$. In addition, we introduce ``Multi-Lipschitz games,'' a generalization involving player-specific Lipschitz values, and provide a reduction from finding equilibria of these games to finding equilibria of Lipschitz games, showing that the value of interest is the sum of the individual Lipschitz parameters. Finally, we provide an exponential lower bound on the deterministic query complexity of finding $ε$-approximate correlated equilibria of $n$-player, $m$-action, $λ$-Lipschitz games for strong values of $
Many important real-world settings contain multiple players interacting over an unknown duration with probabilistic state transitions, and are naturally modeled as stochastic games. Prior research on algorithms for stochastic games has focused on two-player zero-sum games, games with perfect information, and games with imperfect-information that is local and does not extend between game states. We present an algorithm for approximating Nash equilibrium in multiplayer general-sum stochastic games with persistent imperfect information that extends throughout game play. We experiment on a 4-player imperfect-information naval strategic planning scenario. Using a new procedure, we are able to demonstrate that our algorithm computes a strategy that closely approximates Nash equilibrium in this game.
We propose a model for games in which the players have shared access to a blockchain that allows them to deploy smart contracts to act on their behalf. This changes fundamental game-theoretic assumptions about rationality since a contract can commit a player to act irrationally in specific subgames, making credible otherwise non-credible threats. This is further complicated by considering the interaction between multiple contracts which can reason about each other. This changes the nature of the game in a nontrivial way as choosing which contract to play can itself be considered a move in the game. Our model generalizes known notions of equilibria, with a single contract being equivalent to a Stackelberg equilibrium, and two contracts being equivalent to a reverse Stackelberg equilibrium. We prove a number of bounds on the complexity of computing SPE in such games with smart contracts. We show that computing an SPE is $\textsf{PSPACE}$-hard in the general case. Specifically, in games with $k$ contracts, we show that computing an SPE is $Σ_k^\textsf{P}$-hard for games of imperfect information. We show that computing an SPE remains $\textsf{PSPACE}$-hard in games of perfect informati
We present a fully polynomial-time approximation scheme (FPTAS) for computing equilibria in congestion games, under smoothed running-time analysis. More precisely, we prove that if the resource costs of a congestion game are randomly perturbed by independent noises, whose density is at most $φ$, then any sequence of $(1+\varepsilon)$-improving dynamics will reach an $(1+\varepsilon)$-approximate pure Nash equilibrium (PNE) after an expected number of steps which is strongly polynomial in $\frac{1}{\varepsilon}$, $φ$, and the size of the game's description. Our results establish a sharp contrast to the traditional worst-case analysis setting, where it is known that better-response dynamics take exponentially long to converge to $α$-approximate PNE, for any constant factor $α\geq 1$. As a matter of fact, computing $α$-approximate PNE in congestion games is PLS-hard. We demonstrate how our analysis can be applied to various different models of congestion games including general, step-function, and polynomial cost, as well as fair cost-sharing games (where the resource costs are decreasing). It is important to note that our bounds do not depend explicitly on the cardinality of the play
The recent rise of renewable energy produced by many decentralized sources yields interesting market design challenges for electrical grids. Balancing supply and demand in such networks is both a temporal and spatial challenge due to capacity constraints. The recent surge in the number of household-owned batteries, especially in regions with rooftop solar adoption, offers mitigation potential but often acts misaligned with grid-level objectives. In fact, the decision to charge or discharge a household-owned battery is a strategic choice by each battery owner governed by selfish incentives. This calls for an analysis from a game-theoretic point of view. We initiate this timely research direction by considering a game-theoretic setting where selfish agents strategically charge or discharge their batteries to increase their profit. In particular, we study a Stackelberg-like market model where a third party introduces price incentives, aiming to optimize renewable energy utilization while preserving grid feasibility. For this, we study the existence and the quality of equilibria under various pricing strategies. We find that the existence of equilibria crucially depends on the chosen p
Game theory is a powerful framework for reasoning about strategic interactions, with applications in domains ranging from day-to-day life to international politics. However, applying formal reasoning tools in such contexts is challenging, as these scenarios are often expressed in natural language. To address this, we introduce a framework for the autoformalization of game-theoretic scenarios, which translates natural language descriptions into formal logic representations suitable for formal solvers. Our approach utilizes one-shot prompting and a solver that provides feedback on syntactic correctness to allow LLMs to refine the code. We evaluate the framework using GPT-4o and a dataset of natural language problem descriptions, achieving 98% syntactic correctness and 88% semantic correctness. These results show the potential of LLMs to bridge the gap between real-life strategic interactions and formal reasoning.
Strategic interactions between competitive entities are generally considered from the perspective of complete revelation of benefits achieved from those interactions, in the form of public payoff functions and/or beliefs, in the announced games. However, there exist strategic interplays between competitors where the players have a choice to strategise under the availability of private payoffs, in similar competitive settings. In this contribution, we propose a formal framework for a competitive ecosystem where each player is permitted to defect from publicly optimal strategies under certain private payoffs greater than announced payoffs, given that these defections have certain acceptable bounds in the long run as agreed by all players. We call this game theoretic construction an Intention Game. We formally define an Intention Game, and notions of participational equilibria that exist in such interactions that permit public defections. We compare Intention Games with conventional strategic form games, and demonstrate a type-theoretic construction of Intention Games. In a partially honest setting, we give Intention Game instances of a Cournot competition, secure interactions between
In a two-person Rock-Paper-Scissors (RPS) game, if we set a loss worth nothing and a tie worth 1, and the payoff of winning (the incentive a) as a variable, this game is called as generalized RPS game. The generalized RPS game is a representative mathematical model to illustrate the game dynamics, appearing widely in textbook. However, how actual motions in these games depend on the incentive has never been reported quantitatively. Using the data from 7 games with different incentives, including 84 groups of 6 subjects playing the game in 300-round, with random-pair tournaments and local information recorded, we find that, both on social and individual level, the actual motions are changing continuously with the incentive. More expressively, some representative findings are, (1) in social collective strategy transit views, the forward transition vector field is more and more centripetal as the stability of the system increasing; (2) In the individual behavior of strategy transit view, there exists a phase transformation as the stability of the systems increasing, and the phase transformation point being near the standard RPS; (3) Conditional response behaviors are structurally chan
In this study, we formulate positive and negative externalities caused by changes in the supply of shared vehicles as ride sharing games. The study aims to understand the price of anarchy (PoA) and its improvement via a coordination technique in ride sharing games. A critical question is whether ride sharing games exhibit a pure Nash equilibrium (pNE) since the PoA bound assumes it. Our result shows a sufficient condition for a ride sharing game to have a finite improvement property and a pNE similar to potential games. This is the first step to analyze PoA bound and its improvement by coordination in ride sharing games. We also show an example of coordinating players in ride sharing games using signaling and evaluate the improvement in the PoA.
A network of cognitive transmitters is considered. Each transmitter has to decide his power control policy in order to maximize energy-efficiency of his transmission. For this, a transmitter has two actions to take. He has to decide whether to sense the power levels of the others or not (which corresponds to a finite sensing game), and to choose his transmit power level for each block (which corresponds to a compact power control game). The sensing game is shown to be a weighted potential game and its set of correlated equilibria is studied. Interestingly, it is shown that the general hybrid game where each transmitter can jointly choose the hybrid pair of actions (to sense or not to sense, transmit power level) leads to an outcome which is worse than the one obtained by playing the sensing game first, and then playing the power control game. This is an interesting Braess-type paradox to be aware of for energy-efficient power control in cognitive networks.
In this work, we provide a structural characterization of the possible Nash equilibria in the well-studied class of security games with additive utility. Our analysis yields a classification of possible equilibria into seven types and we provide closed-form feasibility conditions for each type as well as closed-form expressions for the expected outcomes to the players at equilibrium. We provide uniqueness and multiplicity results for each type and utilize our structural approach to propose a novel algorithm to compute equilibria of each type when they exist. We then consider the special cases of security games with fully protective resources and zero-sum games. Under the assumption that the defender can perturb the payoffs to the attacker, we study the problem of optimizing the defender expected outcome at equilibrium. We show that this problem is weakly NP- hard in the case of Stackelberg equilibria and multiple attacker resources and present a pseudopolynomial time procedure to solve this problem for the case of Nash equilibria under mild assumptions. Finally, to address non-additive security games, we propose a notion of nearest additive game and demonstrate the existence and un