Normal Form Games Research Articles

Joint Coverage, Connectivity, and Charging Strategies for Distributed UAV Networks

This paper proposes deployment strategies for consumer unmanned aerial vehicles (UAVs) to maximize the stationary coverage of a target area and to guarantee the continuity of the service through energy replenishment operations at ground charging stations. The three main contributions of our work are as follows. 1) A centralized optimal solution is proposed for the joint problem of UAV positioning for a target coverage ratio and scheduling the charging operations of the UAVs that involves travel to the ground station. 2) A distributed game-theory-based scheduling strategy is proposed using normal-form games with rigorous analysis on performance bounds. Furthermore, a bio-inspired scheme using attractive/repulsive spring actions are used for distributed positioning of the UAVs. 3) The cost-benefit tradeoffs of different levels of cooperation among the UAVs for the distributed charging operations is analyzed. This paper demonstrates that the distributed deployment using only 1-hop messaging achieves approximation of the centrally computed optimum, in terms of coverage and lifetime.

Strongly Subgame-Consistent Core in Stochastic Games

This paper investigates stochastic games on finite tree graphs. A given n-player normal-form game is defined at each node of a tree. Transition to a next node of the tree is random and depends on the strategy profile realized in a current game. We construct a cooperative solution of the game by maximizing the total expected payoff of the players. The core is used as the solution concept of the cooperative game. We introduce the definition of a strongly subgame-consistent (strongly time-consistent) core. Finally, we suggest a method for designing a cooperative distribution procedure of an imputation from the core that guarantees its strong subgame consistency.

Beyond Coincidence: The Reasoning Process Underlying Utility Proportional Beliefs Process

Abstract We provide insights into (Bach, Christian W., and Andrés Perea’s. 2014. “Utility Proportional Beliefs.” International Journal of Game Theory: 1–22) concept of utility proportional beliefs and characterize its underlying reasoning process in normal-form games. Our analysis suggests that assumptions on the sensitivity to utilities influence players’ reasoning process and therefore the computations necessary to obtain a belief about the opponent. Under the assumption that more complex computations take longer to complete, we develop additional hypotheses about the players’ reaction times. These additional hypotheses allow for more rigorous testing of the concept than pure accordance with predictions. Using (Nauerz, Christian T., Marion Collewet, and Frauke Meyer. 2015. “Explaining Beliefs in Strictly Competitive One-shot Games.” Working Paper) data set we confirm our hypotheses about players’ reaction times, strengthening our trust in the concept.

The reinforcement heuristic in normal form games

We analyze simple reinforcement-based behavioral rules in 3 × 3 games through choice data and response times. We argue that there is a large overlap between reinforcement-based heuristics (win-stay, lose-shift) and the more “rational” behavioral rule of myopic best reply. However, evidence from response times shows that choices in agreement with the common prescription of those rules are comparatively fast, and choices of the form “lose-shift” occur more frequently for larger differences with bygone payoffs. Both observations speak in favor of reinforcement processes as a cognitive shortcut for apparent myopic best reply, and advise caution when interpreting behavioral results in favor of optimizing behavior.

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.

Riemannian game dynamics

We study a class of evolutionary game dynamics defined by balancing a gain determined by the game's payoffs against a cost of motion that captures the difficulty with which the population moves between states. Costs of motion are represented by a Riemannian metric, i.e., a state-dependent inner product on the set of population states. The replicator dynamics and the (Euclidean) projection dynamics are the archetypal examples of the class we study. Like these representative dynamics, all Riemannian game dynamics satisfy certain basic desiderata, including positive correlation, local stability of interior ESSs, and global convergence in potential games. When the underlying Riemannian metric satisfies a Hessian integrability condition, the resulting dynamics preserve many further properties of the replicator and projection dynamics. We examine the close connections between Hessian game dynamics and reinforcement learning in normal form games, extending and elucidating a well-known link between the replicator dynamics and exponential reinforcement learning.

Partial cooperation in strategic multi-sided decision situations

We consider a normal-form game in which there is a single exogenously given coalition of cooperating players that can write a binding agreement on pre-selected actions. The actions representing other dimensions of the strategy space remain under the sovereign, individual control of the players. We consider a standard extension of the Nash equilibrium concept denoted as a partial cooperative equilibrium as well as an equilibrium concept in which the coalition of cooperators has a leadership position. Existence results are stated and we identify conditions under which the various equilibrium concepts are equivalent. We apply this framework to existing models of multi-market oligopolies and international pollution abatement. In a multi-market oligopoly, typically, a merger paradox emerges in the partial cooperative equilibrium. The paradox vanishes if the cartel attains a leadership position. For international pollution abatement treaties, cooperation by a sufficiently large group of countries results in a Pareto improvement over the standard tragedy of the commons outcome described by the Nash equilibrium.

The conditions in the existence results for discontinuous games by Reny and by Simon and Zame are incomparable

A natural way to relate the existence theorems of Reny (1999) and Simon and Zame (1990) would be to show the following: Each game with an endogenous sharing rule satisfying the assumptions of Simon and Zame (1990) is such that the payoff correspondence has a measurable selection inducing a normal-form game whose mixed extension satisfies the assumptions in Reny (1999). We present a result showing that this is not so in general, even when the assumptions in Reny (1999) are weakened to those in Barelli and Meneghel (2013).

Evolutionary games and matching rules

This study considers evolutionary games with non-uniformly random matching when interaction occurs in groups of nge 2 individuals using pure strategies from a finite strategy set. In such models, groups with different compositions of individuals generally co-exist and the reproductive success (fitness) of a specific strategy varies with the frequencies of different group types. These frequencies crucially depend on the matching process. For arbitrary matching processes (called matching rules), we study Nash equilibrium and ESS in the associated population game and show that several results that are known to hold for population games under uniform random matching carry through to our setting. In our most novel contribution, we derive results on the efficiency of the Nash equilibria of population games and show that for any (fixed) payoff structure, there always exists some matching rule leading to average fitness maximization. Finally, we provide a series of applications to commonly studied normal-form games.

Interactive Information Design

We study the interaction between multiple information designers who try to influence the behavior of a set of agents. When the set of messages available to each designer is finite, such games always admit subgame perfect equilibria. When designers produce public information about independent pieces of information, every equilibrium of the direct game (in which the set of messages coincides with the set of states) is an equilibrium with larger (possibly infinite) message sets. The converse is true for a class of Markovian equilibria only. When designers produce information for their own corporation of agents, pure strategy equilibria exist and are characterized via an auxiliary normal form game. In an infinite-horizon multi-period extension of information design games, a feasible outcome which Pareto dominates a more informative equilibrium of the one-period game is supported by an equilibrium of the multi-period game.

Stabilizing Cooperative Outcomes in Two-Person Games: Theory and Cases

We show that a cooperative outcome—one that is at least next-best for the players—is not a Nash equilibrium (NE) in 19 of the 57 2 x 2 strict ordinal conflict games (33%), including Prisoners’ Dilemma and Chicken. Auspiciously, in 16 of these games (84%), cooperative outcomes are nonmyopic equilibria (NMEs) when the players make farsighted calculations, based on backward induction; in the other three games, credible threats induce cooperation. More generally, in all finite normal-form games, if players’ preferences are strict, farsighted calculations stabilize at least one Pareto-optimal NME. We illustrate the choice of NMEs that are not NEs by two cases in international relations: (i) no first use of nuclear weapons, chosen by the protagonists in the 1962 Cuban missile crisis and since adopted by some nuclear powers; and (ii) the 2015 agreement between Iran, and a coalition of the United States and other countries, that has been abrogated by the United States but has forestalled Iran’s possible development of nuclear weapons.

A Regression Approach for Modeling Games With Many Symmetric Players

We exploit player symmetry to formulate the representation of large normal-form games as a regression task. This formulation allows arbitrary regression methods to be employed in in estimating utility functions from a small subset of the game's outcomes. We demonstrate the applicability both neural networks and Gaussian process regression, but focus on the latter. Once utility functions are learned, computing Nash equilibria requires estimating expected payoffs of pure-strategy deviations from mixed-strategy profiles. Computing these expectations exactly requires an infeasible sum over the full payoff matrix, so we propose and test several approximation methods. Three of these are simple and generic, applicable to any regression method and games with any number of player roles. However, the best performance is achieved by a continuous integral that approximates the summation, which we formulate for the specific case of fully-symmetric games learned by Gaussian process regression with a radial basis function kernel. We demonstrate experimentally that the combination of learned utility functions and expected payoff estimation allows us to efficiently identify approximate equilibria of large games using sparse payoff data.

The modified stochastic game

We present a new tool for the study of multiplayer stochastic games, namely the modified game, which is a normal-form game that depends on the discount factor, the initial state, and for every player a partition of the set of states and a vector that assigns a real number to each element of the partition. We study properties of the modified game, like its equilibria, min–max value, and max–min value. We then show how this tool can be used to prove the existence of a uniform equilibrium in a certain class of multiplayer stochastic games.

A theory of focal points in 2 × 2 games

Since the classic work of Schelling, the notion of a focal point has been widely applied to explain coordinated behavior. However, focal points remain largely outside the formal apparatus of game theory. This paper develops a model of play in 2 × 2 games where payoff differences determine what strategy players will perceive as “salient” and choose to play. The model uniquely predicts which outcome will emerge for virtually the entire class of 2 × 2 normal form games. For the subset of such games involving coordination and asymmetric payoffs, payoff differences identify focal outcomes and strategies in the same way shared social knowledge produces coordination in Schelling’s symmetric games. The model characterizes situations when Nash equilibria are likely to be played, even in a one-shot interaction, and predicts which equilibrium will obtain in games containing more than one. It identifies other circumstances in which a non-equilibrium outcome will predominate. Finally, the theory specifies when a player is likely to select a strategy based solely on consideration of her own payoffs, and when and why the same player will be prompted to act strategically. Experimental results are presented that test the predictions of the model.

Weight of fitness deviation governs strict physical chaos in replicator dynamics.

Replicator equation-a paradigm equation in evolutionary game dynamics-mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here, we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions-fixed points, periodic orbits, and chaotic trajectories-are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate on which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states, useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.

Game-theoretical model for marketing cooperative in fisheries

The classical game-theoretical models described the conflict in fisheries arising from harvesting a ‘common pool resource’ which without an efficient regulation leads to an overexploitation of a renewable but not unlimited resource, known as the ‘tragedy of the commons’. Unlike these studies, the present paper deals with a marketing cooperative of micro or small enterprises in fishing industry, formed to negotiate a contracted price with large buyers, sharing risk among members of the cooperative. In the paper a game-theoretical model for the behaviour in this cooperative is set up. By the time of the actual commercialization of the product, the market price may be higher than what the cooperative can guarantee for members, negotiated on beforehand. Therefore some “unfaithful” members may be interested in selling at least a part of their product on the free market, the cooperative, however, can punish them for this. This conflict is modelled with a multi-person normal form game. An evolutionary dynamics is proposed for the continuous change of the applied strategies, which in the long term leads to a particular Nash equilibrium, considered the solution of the game. This strategy dynamics is continuously influenced by an “exosystem” describing the dynamics of fishing, based on a classical fishing effort model. This approach focuses only on the conflict within the marketing cooperative, since it is supposed that the single enterprises fish from independent resources.

Essential equilibrium in normal-form games with perturbed actions and payoffs

A Nash equilibrium of a normal-form game G is essential if it is robust to perturbations of G. A game is essential if all of its Nash equilibria are essential. This paper provides conditions on the primitives of a (possibly) discontinuous game that guarantee the generic existence of essential games. Unlike the extant literature, the present analysis allows for perturbations of the players’ action spaces, in addition to the standard payoff perturbations.

Noisy Belief Equilibrium

Quantal response equilibrium (QRE) of McKelvey and Palfrey (1995) relaxes the rationality requirement of Nash equilibrium by allowing for probabilistic mistakes or “noise in actions” while maintaining that beliefs are correct. QRE is well-studied, and much is known about the falsifiable restrictions imposed by different variants of the model. By contrast, little is understood of the empirical content of equilibrium models which relax the other condition of Nash equilibrium by allowing for “noise in beliefs” while maintaining best response. To fill this gap, we introduce noisy belief equilibrium (NBE) for normal form games in which axioms restrict belief distributions to be unbiased with respect to and responsive to changes in the opponents’ behavior. The axioms impose testable restrictions both within and across games, and we compare these restrictions to those of regular QRE (Goeree et al. 2005) in which axioms are placed on the quantal response function as the primitive. We show that NBE generates similar predictions as QRE such as the “own payoff effect”, and yet is more consistent with the empirically documented effects of changes in payoff magnitude. Unlike QRE, NBE is a refinement of rationalizability and invariant to affine transformations of payoffs.

Incentive Magnitude Effects in Experimental Games: Bigger is not Necessarily Better

In experimental games, task-related incentives are payments to experimental subjects that vary according to their strategy choices and the consequent outcomes of the games. Limited evidence exists regarding incentive magnitude effects in experimental games. We examined one-off strategy choices and self-reported reasons for choices in eight 3 × 3 and four 4 × 4 normal-form games under task-related incentives of conventional magnitude and compared them with choices and reasons in the same games under incentives five times as large. Both strategy choices and self-reported reasons for choices were almost indistinguishable between the two conditions. These results are in line with earlier findings on individual decision making and with a parametric model, in which the incentive elasticity of effort is very small when compared with other factors, such as the complexity of the decision problem.

An Alternative Interpretation of Mixed Strategies in n-Person Normal Form Games via Resource Allocation

In this paper, we give an interpretation of mixed strategies in normal form games via resource allocation games, where all players utilize the same resource. We define a game in normal form such that each player allocates to each of his pure strategies a fraction of the maximum resource he has available. However, he does not necessarily allocate all of the resource at his disposal. The payoff functions in the resource allocation games vary with how each player allocates his resource. We prove that a Nash equilibrium always exists in mixed strategies for n-person resource allocation games. On the other hand, we show that a mixed Berge equilibrium may not exist in such games.