Normal Form Games Research Articles

On the $$\gamma $$-core of asymmetric aggregative games

This paper analyzes the core of cooperative games generated by asymmetric aggregative normal-form games, i.e., games where the payoff of each player depends on his strategy and the sum of the strategies of all players. We assume that each coalition calculates its worth presuming that the outside players stand alone and select individually best strategies (Hart and Kurz Econometrica 51:1047–1064, 1983; Chander and Tulkens Int J Game Theory 26:379–401, 1997). We show that under some mild monotonicity assumptions on payoffs, the resulting cooperative game is balanced and has a non-empty core (which is the $$\gamma $$-core). Our paper thus offers an existence result for a core notion which is frequently encountered in the theory and applications of cooperative games with externalities.

Quasi-Transfer Continuity and Nash Equilibrium

We introduce a new notion of continuity, called quasi-transfer continuity, and show that it is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. This holds true in a large class of discontinuous games. We show that our result strictly generalizes the pure strategy existence theorem of Carmona [Carmona, G. [2009] An existence result for discontinuous games, J. Econ. Theory 144, 1333–1340]. We also show that our result is neither implied by nor does it imply the existence theorems of Reny [Reny, J. P. [1999] On the existence of pure and mixed strategy Nash equilibria in discontinuous games, Econometrica 67, 1029–1056] and Baye et al. [Baye, M. R., Tian, G. and Zhou, J. [1993] Characterizations of the existence of equilibria in games with discontinuous and nonquasiconcave payoffs, Rev. Econ. Studies 60, 935–948].

Non-altruistic Equilibria

Which choice will a player make if he can make one of two choices in which his own payoffs are equal, but his rival’s payoffs are not equal, that is, one with a large payoff for his rival and the other with a small payoff for his rival? This paper introduces non-altruistic equilibria for normal-form games and extensive-form non-altruistic equilibria for extensive-form games as equilibrium concepts of non-cooperative games by discussing such a problem and examines the connections between their equilibrium concepts and Nash and subgame perfect equilibria that are important and frequently encountered equilibrium concepts.

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

The search problem of computing a Stackelberg (or leader-follower)equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers’ game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader’s commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless textsf {P}=textsf {NP}. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless textsf {P}=textsf {NP}. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader’s strategy where such value is attained, also an alpha -approximate strategy where alpha > 0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances.

Rough games modeled via L-fuzzy ambiguous representations of semilattices

Monotonic predicates and L-fuzzy ambiguous representations of continuous semilattices with bottom elements are applied to describe rough normal-form games and rough extensive-form two-player zero-sum games with imperfect information. Payoff predicates are introduced to find best guaranteed gains of the players. These predicates are calculated recursively via backward payoff predicate transformers, which are also studied. It is shown how payoff predicates determine almost optimal strategies. An approximate minimax theorem is also stated.

Translucent players: Explaining cooperative behavior in social dilemmas

In the past few decades, numerous experiments have shown that humans do not always behave so as to maximize their material payoff. Cooperative behavior when noncooperation is a dominant strategy (with respect to the material payoffs) is particularly puzzling. Here we propose a novel approach to explain cooperation, assuming what Halpern and Pass call translucent players. Typically, players are assumed to be opaque, in the sense that a deviation by one player in a normal-form game does not affect the strategies used by other players. However, a player may believe that if he switches from one strategy to another, the fact that he chooses to switch may be visible to the other players. For example, if he chooses to defect in Prisoner’s Dilemma, the other player may sense his guilt. We show that by assuming translucent players, we can recover many of the regularities observed in human behavior in well-studied games such as Prisoner’s Dilemma, Traveler’s Dilemma, Bertrand Competition, and the Public Goods game. The approach can also be extended to take into account a player’s concerns that his social group (or God) may observe his actions. This extension helps explain prosocial behavior in situations in which previous models of social behavior fail to make correct predictions (e.g. conflict situations and situations where there is a trade-off between equity and efficiency).

Refinements of Nash equilibrium in a Pentagonal Fuzzy Bimatrix Game

In this paper, we introduce the concept of symmetric pentagonal fuzzy bimatrix and constant pentagonal fuzzy matrix for which we define the existence of Nash equilibrium in both pure and mixed strategies along with some numerical illustrations. By applying Ranking to the pay-offs, we convert the fuzzy valued game problem to crisp valued game problem. Therefore, Solution concept of perfect equilibrium for normal form game is reviewed and a new concept of proper equilibrium of pentagonal fuzzy bimatrix game is discussed.

Algorithm Selection in Adversarial Settings: From Experiments to Tournaments in StarCraft

Algorithm selection—the mapping of problem instances to algorithms—has been successfully applied to a variety of complex theoretical and practical problems, including computer games. In this paper, we extend the traditional framework, which considers a single decision maker, to adversarial settings, by modeling algorithm selection as a normal-form game. In this “game of algorithm selection,” agents select algorithms to play a computer game on their behalf. The game's payoff matrix stores the relative performance among algorithms. We also consider nonstationary scenarios, where algorithms can learn from previous matches. We apply this approach to real-time strategy game StarCraft , using bots developed for the game as our algorithms. Our experiments suggest that minimax-Q is a suitable method for algorithm selection in both stationary and nonstationary conditions. We proceed by implementing our approach in MegaBot, a fully capable StarCraft bot. MegaBot showed robustness to nonstationarity, competing in the difficult scenario of StarCraft AI tournaments. MegaBot successfully learns how to select algorithms, exhibiting increasing win rates with tournament progress. In 2016, its debut year, it left 60% of opponents behind in two out of three tournaments. In 2017, however, MegaBot faced difficulties as its algorithm portfolio got outdated compared to newer entries.

A differentiable homotopy method to compute perfect equilibria

The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large.

On the uniqueness of Nash equilibrium in discontinuous ordinal and normal form games

Our aim is to investigate the existence and uniqueness of Nash equilibrium in the general setting of ordinal games. Using a new property, called strict condition, we identify a class of discontinuous games where there exists a unique Nash equilibrium if and only if the single deviation property is satisfied.

Analysis of Hannan consistent selection for Monte Carlo tree search in simultaneous move games

Hannan consistency, or no external regret, is a key concept for learning in games. An action selection algorithm is Hannan consistent (HC) if its performance is eventually as good as selecting the best fixed action in hindsight. If both players in a zero-sum normal form game use a Hannan consistent algorithm, their average behavior converges to a Nash equilibrium of the game. A similar result is known about extensive form games, but the played strategies need to be Hannan consistent with respect to the counterfactual values, which are often difficult to obtain. We study zero-sum extensive form games with simultaneous moves, but otherwise perfect information. These games generalize normal form games and they are a special case of extensive form games. We study whether applying HC algorithms in each decision point of these games directly to the observed payoffs leads to convergence to a Nash equilibrium. This learning process corresponds to a class of Monte Carlo Tree Search algorithms, which are popular for playing simultaneous-move games but do not have any known performance guarantees. We show that using HC algorithms directly on the observed payoffs is not sufficient to guarantee the convergence. With an additional averaging over joint actions, the convergence is guaranteed, but empirically slower. We further define an additional property of HC algorithms, which is sufficient to guarantee the convergence without the averaging and we empirically show that commonly used HC algorithms have this property.

Reasoning about Social Choice and Games in Monadic Fixed-Point Logic

Whether it be in normal form games, or in fair allocations, or in voter preferences in voting systems, a certain pattern of reasoning is common. From a particular profile, an agent or a group of agents may have an incentive to shift to a new one. This induces a natural graph structure that we call the improvement graph on the strategy space of these systems. We suggest that the monadic fixed-point logic with counting, an extension of monadic first-order logic on graphs with fixed-point and counting quantifiers, is a natural specification language on improvement graphs, and thus for a class of properties that can be interpreted across these domains. The logic has an efficient model checking algorithm (in the size of the improvement graph).

Learning Deviation Payoffs in Simulation-Based Games

We present a novel approach for identifying approximate role-symmetric Nash equilibria in large simulation-based games. Our method uses neural networks to learn a mapping from mixed-strategy profiles to deviation payoffs—the expected values of playing pure-strategy deviations from those profiles. This learning can generalize from data about a tiny fraction of a game’s outcomes, permitting tractable analysis of exponentially large normal-form games. We give a procedure for iteratively refining the learned model with new data produced by sampling in the neighborhood of each candidate Nash equilibrium. Relative to the existing state of the art, deviation payoff learning dramatically simplifies the task of computing equilibria and more effectively addresses player asymmetries. We demonstrate empirically that deviation payoff learning identifies better approximate equilibria than previous methods and can handle more difficult settings, including games with many more players, strategies, and roles.

Descriptive complexity and revealed preference theory

This paper formalizes revealed preference theory using the notion of Ramsey eliminability in logic, and shows how the language required to state a revealed preference axiom for some choice theory is closely connected with the computational tractability of testing that theory. The connection is made through results in descriptive complexity theory, a relatively new field in finite model theory. It is shown that checking whether observed choices of players in normal form games are Nash rationalizable is an NP-complete problem. This also means that there does not exist an analogue (in a precise sense) of the Strong Axiom of Revealed Preference for Nash equilibrium.

Quantifying Commitment in Nash Equilibria

To quantify a player’s commitment in a given Nash equilibrium of a finite dynamic game, we map the corresponding normal-form game to a “canonical extension,” which allows each player to adjust his or her move with a certain probability. The commitment measure relates to the average overall adjustment probabilities for which the given Nash equilibrium can be implemented as a subgame-perfect equilibrium in the canonical extension.

On seeking efficient Pareto optimal points in multi-player minimum cost flow problems with application to transportation systems

In this paper, we propose a multi-player extension of the minimum cost flow problem inspired by a transportation problem that arises in modern transportation industry. We associate one player with each arc of a directed network, each trying to minimize its cost function subject to the network flow constraints. In our model, the cost function can be any general nonlinear function, and the flow through each arc is an integer. We present algorithms to compute efficient Pareto optimal point(s), where the maximum possible number of players (but not all) minimize their cost functions simultaneously. The computed Pareto optimal points are Nash equilibriums if the problem is transformed into a finite static game in normal form.

N-Non-Cooperative Games Under a Dynamic Multidimensional Graphical Prism

This paper explores the concept of multidimensional Nash equilibrium as a prospective solution in a non-cooperative multi-player normal form game. The paper suggests a paradigm shift: a new multidimensional geometric approach to capture all symmetrical and asymmetrical strategic graphical movements in real space and time. The Multidimensional Non-Cooperative Games Graphical Framework (MNCGG-Framework) extends the traditional two-dimensional normal form game into a multidimensional graphical framework that synchronizes the optimal agent interactions with corresponding payoffs in real-time. The paper applies the proposed graphical framework on John Forbes Nash’s six examples on non-cooperative games taken from his doctoral dissertation.

Measuring Competitiveness and Cooperativeness

We develop an index of competitiveness and cooperativeness which is based on the primitives of a normal-form game, i.e., players, strategies and payoffs. The index relies on a unique decomposition of a given game into a zero-sum game and a common-interest game. The index decreases in the distance to its zero-sum part and it increases in the distance to its common-interest part. Alternatively, the index increases if the share of variation in payoffs captured by the zero-sum part increases. We compute our index for well-known classes of games such as Prisoner's Dilemma, games with Strategic Complements and Substitutes, All-pay auctions, Tullock contests, and Public Goods games. The comparative statics of our index coincide with economic intuition. The index does well in explaining experimental findings in the sense that more cooperative and less competitive behavior correlates with lower values of the index.

On the Solutions of Games in Normal Forms: Particular Models based on Nash Equilibrium Theory

Abstract The main objective of this paper is to present in a deductive way, solutions for general games played under normal conditions following competitive paths, applying core principles of Nash equilibrium. Here the normal approach implies strategic choices available for each player, formulated and implemented without any information concerning specific choices to be made by others players. It is convenient to keep in mind that John von Neumann and Oskar Morgenstern outlined a set of conditions for Nash equilibrium for a game in normal form, proposed as the basic framework to analyze the conditions and requirements for a particular Nash equilibrium to be the solution of the game. Theorems that exhibit imbedding relations among the Nash equilibriums of the game are given to examine the role of pre-play communication and the imbedding order in equilibrium selection. A core argument to claim here is that a generic case of Nash equilibriums that are strategically unstable relative to maxi-min strategies is given to emphasize the role of moves of the third kind and pre-play communication in correlated and coordinated solutions and the need to account for cases where Nash equilibriums are not plausible or even desirable as solutions for a game in normal form.

Best-reply sets

We provide results concerning, in a given normal-form game, which sets of actions are best replies to some belief. Proposition 1 states that for any set S of actions, there is a belief under which all actions in S are simultaneously best replies if and only if no mixture of actions in S is strictly dominated. Similarly, Proposition 2 states that for any set S of actions, there is a full-support belief under which all actions in S are best replies if and only if no mixture of actions in S is weakly dominated. One important consequence is Corollary 1: a two-player game has a totally mixed Nash equilibrium if and only if neither player has a pair of mixed strategies such that one weakly dominates the other.