Two-player Normal Form Games Research Articles

Best-response dynamics in two-person random games with correlated payoffs

We consider finite two-player normal form games with random payoffs. Player A's payoffs are i.i.d. from a uniform distribution. Given p∈[0,1], for any action profile, player B's payoff coincides with player A's payoff with probability p and is i.i.d. from the same uniform distribution with probability 1−p. This model interpolates the model of i.i.d. random payoff used in most of the literature and the model of random potential games. First we study the number of pure Nash equilibria in the above class of games. Then we show that, for any positive p, asymptotically in the number of available actions, best response dynamics reaches a pure Nash equilibrium with high probability.

Identifying Socially Optimal Equilibria Using Combinatorial Properties of Nash Equilibria in Bimatrix Games

Nash equilibrium is arguably the most fundamental concept in game theory, which is used to analyze and predict the behavior of the players. In many games, there exist multiple equilibria, with different expected payoffs for the players, which in turn raises the question of equilibrium selection. In this paper, we study the [Formula: see text]-hard problem of identifying a socially optimal Nash equilibrium in two-player normal-form games (called bimatrix games), which may be represented by a mixed integer linear program (MILP). We characterize the properties of the equilibria and develop several classes of valid inequalities accordingly. We use these theoretical results to provide a decomposition-based reformulation of the MILP, which we solve by a branch-and-cut algorithm. Our extensive computational experiments demonstrate superiority of our approach over solving the MILP formulation through feeding it into a commercial solver or through the “traditional” Benders’ decomposition. Of note, our proposed approach can find provably optimal solutions for many instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms. Funding: This work was supported by the National Institute of Dental and Craniofacial Research [Grant R01DE028283]. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The funding agreements ensured the authors’ independence in designing the study, interpreting the data, writing, and publishing the report. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0072 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0072 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Game-Theoretic Decision-Making and Payoff Design for UAV Collision Avoidance in a Three-Dimensional Airspace

Safety and efficiency are primary goals of air traffic management. With the integration of unmanned aerial vehicles (UAVs) into the airspace, UAV traffic management (UTM) has attracted significant interest in the research community to maintain the capacity of three-dimensional (3D) airspace, provide information, and avoid collisions. We propose a new decision-making architecture for UAVs to avoid collision by formulating the problem into a multi-agent game in a 3D airspace. In the proposed game-theoretic approach, the Ego UAV plays a repeated two-player normal-form game, and the payoff functions are designed to capture both the safety and efficiency of feasible actions. An optimal decision in the form of Nash equilibrium (NE) is obtained. Simulation studies are conducted to demonstrate the performance of the proposed game-theoretic collision avoidance approach in several representative multi-UAV scenarios.

A stochastic stability analysis with observation errors in normal form games

We perform a stochastic stability analysis with observation errors. Players recurrently play a symmetric two-player normal form game with one another and respond to the strategy distribution of other players. In each period, a revising player observes the strategy distribution and chooses a best response to it. Her observation is perturbed with positive probability and she may respond to the misperceived strategy distribution. The robustness of Nash equilibria to such observation errors is examined. We find that only transition probabilities within states where no player plays any strictly dominated strategy matter for stochastic stability. A more precise set of stochastically stable states is characterized for several particular observation error models. For the local interaction model, the set of stochastically stable states is robust to addition of strategies that do not survive iterative elimination of strictly dominated strategies regardless of error types (action/observation) under certain conditions.

When “Better” is better than “Best”

We consider two-player normal form games where each player has the same finite strategy set. The payoffs of each player are assumed to be i.i.d. random variables with a continuous distribution. We show that, with high probability, the better-response dynamics converges to pure Nash equilibrium whenever there is one, whereas best-response dynamics fails to converge, as it is trapped.

A Stochastic Stability Analysis with Observation Errors in Normal Form Games

We perform a stochastic stability analysis with observation errors. Players recurrently play a symmetric two-player normal form game with one another and respond to the strategy distribution of other players. In each period, a revising player observes the strategy distribution and chooses a best response to it. Her observation is perturbed with positive probability and she may respond to the misperceived strategy distribution. The robustness of Nash equilibria to such observation errors is examined. We find that only transition probabilities within states where no player plays any strictly dominated strategy matter for stochastic stability. A more precise set of stochastically stable states is characterized for several particular observation error models. For the local interaction model, the set of stochastically stable states is robust to addition of strategies that do not survive iterative elimination of strictly dominated strategies regardless of error types (action/observation) under certain conditions.

Prospect dynamics and loss dominance

This paper investigates the role of loss-aversion in affecting the long-run equilibria of stochastic evolutionary dynamics. We consider a finite population of loss-averse agents who are repeatedly and randomly matched to play a symmetric two-player normal form game. When an agent revises her strategy, she compares the payoff from each strategy to a reference point. Under the resulting dynamics, called prospect dynamics, risk-dominance is no longer sufficient to guarantee stochastic stability in 2 × 2 coordination games. We propose a stronger concept, loss-dominance: a strategy is loss-dominant if it is risk-dominant and a maximin strategy. In 2 × 2 coordination games, the state where all agents play the loss-dominant strategy is uniquely stochastically stable under prospect dynamics for any degree of loss-aversion and all types of reference points. For symmetric two-player normal form games, a generalized concept, generalized loss-dominance, gives a sufficient condition for stochastic stability under prospect dynamics.

Constant Rank Two-Player Games are PPAD-hard

Finding a Nash equilibrium in a two-player normal form game (2-Nash) is one of the most extensively studied problems within mathematical economics as well as theoretical computer science. Such a game can be represented by two payoff matrices $A$ and $B$, one for each player. $2$-Nash is PPAD-complete in general, while in the case of zero-sum games ($B=-A$) the problem reduces to LP and hence is in P. Extending the notion of zero-sum, in 2005, Kannan and Theobald [Econom. Theory, 42 (2010), pp. 157--174] defined the rank of game $(A, B)$ as $rank(A+B)$, e.g., rank-0 are zero-sum games. They gave an FPTAS for constant rank games and asked if there exists a polynomial time algorithm to compute an exact Nash equilibrium (NE). Adsul et al. [Proceedings of the ACM Symposium on the Theory of Computing, 2011, pp. 195--204] answered this question affirmatively for rank-1 games, leaving rank-$2$ and beyond unresolved. In this paper we show that NE computation in games with rank $\ge3$ is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets and provides a simpler proof of PPAD-hardness for NE computation in two-player games. In addition, we show the following: (i) an equivalence between two-dimensional Linear-FIXP and PPAD, improving on a result of Etessami and Yannakakis [SIAM J. Comput., 39 (2010), pp. 2531--2597] on equivalence between Linear-FIXP and PPAD; (ii) NE computation in a two-player game with convex set of Nash equilibria is as hard as solving a simple stochastic game [A. Condon, Inform. and Comput., 96 (1992), pp. 203--224]; (iii) computing a symmetric NE of a symmetric two-player game with rank $\ge 6$ is PPAD-hard; (iv) computing a ${1}/{poly(n)}$-approximate fixed-point of a piecewise-linear function is PPAD-hard."

Disarmament Games

Much recent work in the AI community concerns algorithms for computing optimal mixed strategies to commit to, as well as the deployment of such algorithms in real security applications. Another possibility is to commit not to play certain actions. If only one player makes such a commitment, then this is generally less powerful than completely committing to a single mixed strategy. However, if players can alternatingly commit not to play certain actions and thereby iteratively reduce their strategy spaces, then desirable outcomes can be obtained that would not have been possible with just a single player committing to a mixed strategy. We refer to such a setting as a disarmament game. In this paper, we study disarmament for two-player normal-form games. We show that deciding whether an outcome can be obtained with disarmament is NP-complete (even for a fixed number of rounds), if only pure strategies can be removed. On the other hand, for the case where mixed strategies can be removed, we provide a folk theorem that shows that all desirable utility profiles can be obtained, and give an efficient algorithm for (approximately) obtaining them.

Two-Player Preplay Negotiation Games with Conditional Offers

We consider an extension of strategic normal form games with a phase before the actual play of the game, where players can make binding offers for transfer of utilities to other players after the play of the game, contingent on the recipient playing the strategy indicated in the offer. Such offers transform the payoff matrix of the original game but preserve its noncooperative nature. The type of offers we focus on here are conditional on a suggested matching offer of the same kind made in return by the receiver. Players can exchange a series of such offers, thus engaging in a bargaining process before a strategic normal form game is played. In this paper, we study and analyze solution concepts for two-player normal form games with such preplay negotiation phase, under several assumptions for the bargaining power of the players, as well as the value of time for the players in such negotiations. We obtain results describing the possible solutions of such bargaining games and analyze the degrees of efficiency and fairness that can be achieved in such negotiation process. We show the similarities and the differences with a variety of frameworks in the literature of bargaining games and games with a preplay phase.

Strategic sophistication and attention in games: An eye-tracking study

We used eye-tracking to measure the dynamic patterns of visual information acquisition in two-player normal-form games. Participants played one-shot games in which either, neither, or only one of the players had a dominant strategy. First, we performed a mixture models cluster analysis to group participants into types according to the pattern of visual information acquisition observed in a single class of games. Then, we predicted agents' choices in different classes of games and observed that patterns of visual information acquisition were game invariant. Our method allowed us to predict whether the decision process would lead to equilibrium choices or not, and to attribute out-of-equilibrium responses to limited cognitive capacities or social motives. Our results suggest the existence of individually heterogeneous-but-stable patterns of visual information acquisition based on subjective levels of strategic sophistication and social preferences.

Circulant games

We study a class of two-player normal-form games with cyclical payoff structures. A game is called circulant if both players’ payoff matrices fulfill a rotational symmetry condition. The class of circulant games contains well-known examples such as Matching Pennies, Rock-Paper-Scissors, as well as subclasses of coordination and common interest games. The best response correspondences in circulant games induce a partition on each player’s set of pure strategies into equivalence classes. In any Nash Equilibrium, all strategies within one class are either played with strictly positive or with zero probability. We further show that, strikingly, a single parameter fully determines the exact number and the structure of all Nash equilibria (pure and mixed) in these games. The parameter itself only depends on the position of the largest payoff in the first row of one of the player’s payoff matrix.

Reinforcement Learning with Restrictions on the Action Set

Consider a two-player normal-form game repeated over time. We introduce an adaptive learning procedure, where the players only observe their own realized payoff at each stage. We assume that agents do not know their own payoff function and have no information on the other player. Furthermore, we assume that they have restrictions on their own actions such that, at each stage, their choice is limited to a subset of their action set. We prove that the empirical distributions of play converge to the set of Nash equilibria for zero-sum and potential games, and games where one player has two actions.

Properties and applications of dual reduction

The dual reduction process, introduced by Myerson, allows a finite game to be reduced to a smaller-dimensional game such that any correlated equilibrium of the reduced game is an equilibrium of the original game. We study the properties and applications of this process. It is shown that generic two-player normal form games have a unique full dual reduction (a known refinement of dual reduction) and all strategies that have probability zero in all correlated equilibria are eliminated in all full dual reductions. Among other applications, we give a linear programming proof of the fact that a unique correlated equilibrium is a Nash equilibrium, and improve on a result due to Nau, Gomez-Canovas and Hansen on the geometry of Nash equilibria and correlated equilibria.

The myth of the Folk Theorem

The Folk Theorem for repeated games suggests that finding Nash equilibria in repeated games should be easier than in one-shot games. In contrast, we show that the problem of finding any Nash equilibrium for a three-player infinitely-repeated game is as hard as it is in two-player one-shot games. More specifically, for any two-player game, we give a simple construction of a three-player game whose Nash equilibria (even under repetition) correspond to those of the one-shot two-player game. Combined with recent computational hardness results for one-shot two-player normal-form games ( Daskalakis et al., 2006; Chen et al., 2006; Chen et al., 2007), this gives our main result: the problem of finding an (epsilon) Nash equilibrium in a given n × n × n game (even when all payoffs are in { − 1 , 0 , 1 } ) is PPAD-hard (under randomized reductions).

Equilibria, fixed points, and complexity classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are, respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in two-player normal form games, and (mixed) Nash equilibria in normal form games with three (or more) players. This paper reviews the underlying computational principles and the corresponding classes.

Evolution in Bayesian games II: Stability of purified equilibria

We study the evolutionary stability of purified equilibria of two-player normal form games, providing simple sufficient conditions for stability and for instability under the Bayesian best response dynamic.

Asymptotic expected number of Nash equilibria of two-player normal form games

The formula given by McLennan [The mean number of real roots of a multihomogeneous system of polynomial equations, Amer. J. Math. 124 (2002) 49–73] is applied to the mean number of Nash equilibria of random two-player normal form games in which the two players have M and N pure strategies respectively. Holding M fixed while N → ∞ , the expected number of Nash equilibria is approximately ( π log N / 2 ) M − 1 / M . Letting M = N → ∞ , the expected number of Nash equilibria is exp ( N M + O ( log N ) ) , where M ≈ 0.281644 is a constant, and almost all equilibria have each player assigning positive probability to approximately 31.5915 percent of her pure strategies.

Equilibrium Departures from Common Knowledge in Games with Non-Additive Expected Utility

Abstract This paper concerns the interpretation of equilibrium in non-additive beliefs in two-player normal form games. We argue that such equilibria involve beliefs and actions which are consistent with a lack of common knowledge of the game. Our argument rests on representation results which show that different notions of equilibrium in games with non-additive beliefs may be reinterpreted as equilibrium in associated games of incomplete information with additive (Bayesian) beliefs where common knowledge of the (original) game does not apply. The representation results show one way of comparing and understanding the various notions of equilibrium, for games with non-additive beliefs, advanced in the literature.

Communication, correlation, and symmetry in bargaining

This paper considers two-player normal form games where each player can send a payoff-irrelevant message prior to play. Let G be the game without communication, and G ∗ ( M) the extended game with message set M. Any convex combination of Nash outcomes in G can be approximated in a subgame perfect equilibrium of G ∗ ( M) for some M. Furthermore, every symmetric game has a symmetric subgame perfect communication equilibrium that is undominated in a limit sense as the message set is enlarged.