We study a setting in which two players play a (possibly approximate) Nash equilibrium of a bimatrix game, while a learner observes only their actions and has no knowledge of the equilibrium or the underlying game. A natural question is whether the learner can rationalize the observed behavior by inferring the players' payoff functions. Rather than producing a single payoff estimate, inverse game theory aims to identify the entire set of payoffs consistent with observed behavior, enabling downstream use in, e.g., counterfactual analysis and mechanism design across applications like auctions, pricing, and security games. We focus on the problem of estimating the set of feasible payoffs with high probability and up to precision $ε$ on the Hausdorff metric. We provide the first minimax-optimal rates for both exact and approximate equilibrium play, in zero-sum as well as general-sum games. Our results provide learning-theoretic foundations for set-valued payoff inference in multi-agent environments.
Caltech（加州理工学院）；University of Maryland, College Park（马里兰大学帕克分校）；MIT（麻省理工学院）
We study two-player zero-sum stochastic games, and propose a form of independent learning dynamics called Doubly Smoothed Best-Response dynamics, which integrates a discrete and doubly smoothed variant of the best-response dynamics into temporal-difference (TD)-learning and minimax value iteration. The resulting dynamics are payoff-based, convergent, rational, and symmetric among players. Our main results provide finite-sample guarantees. In particular, we prove the first-known $\tilde{\mathcal{O}}(1/ε^2)$ sample complexity bound for payoff-based independent learning dynamics, up to a smoothin…
We prove that to find optimal positional strategies for stochastic mean payoff games when the value of every state of the game is known, in general, is as hard as solving such games tout court. This answers a question posed by Daniel Andersson and Peter Bro Miltersen.
We establish that the subgame perfect equilibrium (SPE) threshold problem for mean-payoff games is NP-complete. While the SPE threshold problem was recently shown to be decidable (in doubly exponential time) and NP-hard, its exact worst case complexity was left open.
Quantitative games are two-player zero-sum games played on directed weighted graphs. Total-payoff games (that can be seen as a refinement of the well-studied mean-payoff games) are the variant where the payoff of a play is computed as the sum of the weights. Our aim is to describe the first pseudo-polynomial time algorithm for total-payoff games in the presence of arbitrary weights. It consists of a non-trivial application of the value iteration paradigm. Indeed, it requires to study, as a milestone, a refinement of these games, called min-cost reachability games, where we add a reachability o…
We prove that every repeated game with countably many players, finite action sets, and tail-measurable payoffs admits an $ε$-equilibrium, for every $ε> 0$.
Evolutionarily stable strategy (ESS) is a key concept in evolutionary game theory. ESS provides an evolutionary stability criterion for biological, social and economical behaviors. In this paper, we develop a new approach to evaluate ESS in symmetric two player games with fuzzy payoffs. Particularly, every strategy is assigned a fuzzy membership that describes to what degree it is an ESS in presence of uncertainty. The fuzzy set of ESS characterize the nature of ESS. The proposed approach avoids loss of any information that happens by the defuzzification method in games and handles uncertainty…
In this paper, we study one-player and two-player energy mean-payoff games. Energy mean-payoff games are games of infinite duration played on a finite graph with edges labeled by 2-dimensional weight vectors. The objective of the first player (the protagonist) is to satisfy an energy objective on the first dimension and a mean-payoff objective on the second dimension. We show that optimal strategies for the first player may require infinite memory while optimal strategies for the second player (the antagonist) do not require memory. In the one-player case (where only the first player has choic…
We study the problem of finding robust equilibria in multiplayer concurrent games with mean payoff objectives. A $(k,t)$-robust equilibrium is a strategy profile such that no coalition of size $k$ can improve the payoff of one its member by deviating, and no coalition of size $t$ can decrease the payoff of other players. We are interested in pure equilibria, that is, solutions that can be implemented using non-randomized strategies. We suggest a general transformation from multiplayer games to two-player games such that pure equilibria in the first game correspond to winning strategies in the …
Constructive proof of the existence of Nash Equilibrium in a finite strategic game with sequentially locally non-constant payoff functions by Sperner's lemma
We consider 2-player stochastic games with perfectly observed actions, and study the limit, as the discount factor goes to one, of the equilibrium payoffs set. In the usual setup where current states are observed by the players, we show that the set of stationary equilibrium payoffs always converges, and provide a simple example where the set of equilibrium payoffs has no limit. We then introduce the more general model of hidden stochastic game, where the players publicly receive imperfect signals over current states. In this setup we present an example where not only the limit set of equilibr…
Mean-payoff games are important quantitative models for open reactive systems. They have been widely studied as games of full observation. In this paper we investigate the algorithmic properties of several sub-classes of mean-payoff games where the players have asymmetric information about the state of the game. These games are in general undecidable and not determined according to the classical definition. We show that such games are determined under a more general notion of winning strategy. We also consider mean-payoff games where the winner can be determined by the winner of a finite cycle…
In this paper, we study network games where players are involved in information aggregation processes subject to the differential privacy requirement for players' payoff functions. We propose a Laplace linear-quadratic functional perturbation mechanism, which perturbs players' payoff functions with linear-quadratic functions whose coefficients are produced from truncated Laplace distributions. For monotone games, we show that the LLQFP mechanism maintains the concavity property of the perturbed payoff functions and produces a perturbed NE whose distance from the original NE is bounded and adju…
This paper examines the convergence of no-regret learning in games with continuous action sets. For concreteness, we focus on learning via "dual averaging", a widely used class of no-regret learning schemes where players take small steps along their individual payoff gradients and then "mirror" the output back to their action sets. In terms of feedback, we assume that players can only estimate their payoff gradients up to a zero-mean error with bounded variance. To study the convergence of the induced sequence of play, we introduce the notion of variational stability, and we show that stable e…
Evolutionary game theory classically investigates which behavioral patterns are evolutionarily successful in a single game. More recently, a number of contributions have studied the evolution of preferences instead: which subjective conceptualizations of a game's payoffs give rise to evolutionarily successful behavior in a single game. Here, we want to extend this existing approach even further by asking: which general patterns of subjective conceptualizations of payoff functions are evolutionarily successful across a class of games. In other words, we will look at evolutionary competition of …
We study the problem of finding equilibrium strategies in multi-agent games with incomplete payoff information, where the payoff matrices are only known to the players up to some bounded uncertainty sets. In such games, an ex-post equilibrium characterizes equilibrium strategies that are robust to the payoff uncertainty. When the game is one-shot, we show that in zero-sum polymatrix games, an ex-post equilibrium can be computed efficiently using linear programming. We further extend the notion of ex-post equilibrium to stochastic games, where the game is played repeatedly in a sequence of stag…
In a mean-payoff parity game, one of the two players aims both to achieve a qualitative parity objective and to minimize a quantitative long-term average of payoffs (aka. mean payoff). The game is zero-sum and hence the aim of the other player is to either foil the parity objective or to maximize the mean payoff. Our main technical result is a pseudo-quasi-polynomial algorithm for solving mean-payoff parity games. All algorithms for the problem that have been developed for over a decade have a pseudo-polynomial and an exponential factors in their running times; in the running time of our algor…
Modeling strategic conflict from a game theoretical perspective involves dealing with epistemic uncertainty. Payoff uncertainty models are typically restricted to simple probability models due to computational restrictions. Recent breakthroughs Artificial Intelligence (AI) research applied to Poker have resulted in novel approximation approaches such as counterfactual regret minimization, that can successfully deal with large-scale imperfect games. By drawing from these ideas, this work addresses the problem of arbitrary continuous payoff distributions. We propose a method, Harsanyi-Counterfac…
Two-player games on graphs is central in many problems in formal verification and program analysis such as synthesis and verification of open systems. In this work we consider solving recursive game graphs (or pushdown game graphs) that can model the control flow of sequential programs with recursion. While pushdown games have been studied before with qualitative objectives, such as reachability and $ω$-regular objectives, in this work we study for the first time such games with the most well-studied quantitative objective, namely, mean-payoff objectives. In pushdown games two types of strateg…
We introduce and study incentive equilibria for multi-player meanpayoff games. Incentive equilibria generalise well-studied solution concepts such as Nash equilibria and leader equilibria (also known as Stackelberg equilibria). Recall that a strategy profile is a Nash equilibrium if no player can improve his payoff by changing his strategy unilaterally. In the setting of incentive and leader equilibria, there is a distinguished player called the leader who can assign strategies to all other players, referred to as her followers. A strategy profile is a leader strategy profile if no player, exc…
Recent extensions to dynamic games of the well-known fictitious play learning procedure in static games were proved to globally converge to stationary Nash equilibria in two important classes of dynamic games (zero-sum and identical-interest discounted stochastic games). However, those decentralized algorithms need the players to know exactly the model (the transition probabilities and their payoffs at every stage). To overcome these strong assumptions, our paper introduces regularizations of the systems in (Leslie 2020; Baudin 2022) to construct a family of new decentralized learning algorith…
We consider two-player games played on weighted directed graphs with mean-payoff and total-payoff objectives, two classical quantitative objectives. While for single-dimensional games the complexity and memory bounds for both objectives coincide, we show that in contrast to multi-dimensional mean-payoff games that are known to be coNP-complete, multi-dimensional total-payoff games are undecidable. We introduce conservative approximations of these objectives, where the payoff is considered over a local finite window sliding along a play, instead of the whole play. For single dimension, we show …
In this paper, we provide an effective characterization of all the subgame-perfect equilibria in infinite duration games played on finite graphs with mean-payoff objectives. To this end, we introduce the notion of requirement, and the notion of negotiation function. We establish that the plays that are supported by SPEs are exactly those that are consistent with the least fixed point of the negotiation function. Finally, we show that the negotiation function is piecewise linear, and can be analyzed using the linear algebraic tool box. As a corollary, we prove the decidability of the SPE constr…
We consider concurrent mean-payoff games, a very well-studied class of two-player (player 1 vs player 2) zero-sum games on finite-state graphs where every transition is assigned a reward between 0 and 1, and the payoff function is the long-run average of the rewards. The value is the maximal expected payoff that player 1 can guarantee against all strategies of player 2. We consider the computation of the set of states with value 1 under finite-memory strategies for player 1, and our main results for the problem are as follows: (1) we present a polynomial-time algorithm; (2) we show that whenev…
We consider two-person zero-sum stochastic mean payoff games with perfect information, or BWR-games, given by a digraph $G = (V, E)$, with local rewards $r: E \to \ZZ$, and three types of positions: black $V_B$, white $V_W$, and random $V_R$ forming a partition of $V$. It is a long-standing open question whether a polynomial time algorithm for BWR-games exists, or not, even when $|V_R|=0$. In fact, a pseudo-polynomial algorithm for BWR-games would already imply their polynomial solvability. In this paper, we show that BWR-games with a constant number of random positions can be solved in pseudo…
Mean-payoff games on timed automata are played on the infinite weighted graph of configurations of priced timed automata between two players, Player Min and Player Max, by moving a token along the states of the graph to form an infinite run. The goal of Player Min is to minimize the limit average weight of the run, while the goal of the Player Max is the opposite. Brenguier, Cassez, and Raskin recently studied a variation of these games and showed that mean-payoff games are undecidable for timed automata with five or more clocks. We refine this result by proving the undecidability of mean-payo…
The practical scalability of many optimization algorithms for large extensive-form games is often limited by the games' huge payoff matrices. To ameliorate the issue, Zhang and Sandholm (2020) recently proposed a sparsification technique that factorizes the payoff matrix $\mathbf{A}$ into a sparser object $\mathbf{A} = \hat{\mathbf{A}} + \mathbf{U}\mathbf{V}^\top$, where the total combined number of nonzeros of $\hat{\mathbf{A}}$, $\mathbf{U}$, and $\mathbf{V}$ is significantly smaller. Such a factorization can be used in place of the original payoff matrix in many optimization algorithm, such…
We propose a novel algorithm for the solution of mean-payoff games that merges together two seemingly unrelated concepts introduced in the context of parity games, small progress measures and quasi dominions. We show that the integration of the two notions can be highly beneficial and significantly speeds up convergence to the problem solution. Experiments show that the resulting algorithm performs orders of magnitude better than the asymptotically-best solution algorithm currently known, without sacrificing on the worst-case complexity.
Motivated by the markets operating on fast time scales, we present a framework for online coalitional games with time-varying coalitional values and propose real-time payoff distribution mechanisms. Specifically, we design two online distributed algorithms to track the Shapley value and the core, the two most widely studied payoff distribution criteria in coalitional game theory. We show that the payoff distribution trajectory resulting from our proposed algorithms converges to a neighborhood of the time-varying solutions. We adopt an operator-theoretic perspective to show the convergence of o…
The game-theoretic risk management framework put forth in the precursor work "Towards a Theory of Games with Payoffs that are Probability-Distributions" (arXiv:1506.07368 [q-fin.EC]) is herein extended by algorithmic details on how to compute equilibria in games where the payoffs are probability distributions. Our approach is "data driven" in the sense that we assume empirical data (measurements, simulation, etc.) to be available that can be compiled into distribution models, which are suitable for efficient decisions about preferences, and setting up and solving games using these as payoffs. …
In this work we offer an $O(|V|^2 |E|\, W)$ pseudo-polynomial time deterministic algorithm for solving the Value Problem and Optimal Strategy Synthesis in Mean Payoff Games. This improves by a factor $\log(|V|\, W)$ the best previously known pseudo-polynomial time upper bound due to Brim,~\etal The improvement hinges on a suitable characterization of values, and a description of optimal positional strategies, in terms of reweighted Energy Games and Small Energy-Progress Measures.
