Random Payoff Research Articles

Impact of environmental stochastic fluctuations on the evolutionary stability of imitation dynamics.

To show the impact of environmental noise on imitation dynamics, the stochastic stability and stochastic evolutionary stability of a discrete-time imitation dynamics with random payoffs are studied in this paper. Based on the stochastic local stability of fixation states and constant interior equilibria in a two-phenotype model, we extend the concept of stochastic evolutionary stability to the stochastic imitation dynamics, which is defined as a strategy such that, if all the members of the population adopt it, then the probability for any mutant strategy to invade the population successfully under the influence of natural selection is arbitrarily low. Our main results show clearly that the stochastic evolutionary stability of the system depends only on the properties of the mean matrix of the random payoff matrix and is independent of the randomness of the random payoff matrix. Moreover, as two examples, we show also that under the framework of stochastic imitation dynamics, the noise intensity affects the evolution of cooperative behavior in a stochastic prisoner's dilemma game and the system's nonlinear dynamic behavior.

Distributionally Robust Chance-Constrained Markov Decision Processes with Random Payoff

Distributionally Robust Chance-Constrained Markov Decision Processes with Random Payoff

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.

The emergence of cooperative behavior based on random payoff and heterogeneity of concerning social image

The heterogeneity among individuals in a group usually leads to the volatility of the individual’s income in the game and the difference in the attention paid to the individual’s social image. This paper proposes a spatial prisoner’s dilemma model with random fluctuations of temptation returns and the social images of individuals considered. The payoff of a defector in the game with a cooperator is set as a random variable. And the social image value of an individual who holds the cooperative strategy is set to a positive value, while the social image value of an individual who holds defective strategy is set to zero. Some individuals in the group care about the social images, while others do not. Social image values and benefits are combined into the fitness of individual. The learning strategy adopted is the combination of optimal fitness and Fermi rule based on return difference.Numerical experiments indicate that the volatility of the temptation value is beneficial for cooperation under low temptation, but the volatility of the defector’s income is unfavorable for cooperation under high temptation. And the volatility increases the complexity of system evolution, thereby prolonging the evolution time required to reach steady state. The presence of individuals in the system who care about social image can promote cooperation under low temptation, but the proportion of individuals concerned with social image has little effect on the cooperation behavior. When the social image value of individuals with cooperative strategy increases, the cooperative behavior characteristics of the system do not change qualitatively.

Picking winners: Diversification through portfolio optimization

We develop a general framework for selecting a small pool of candidate solutions to maximize the chances that one will be optimal for a combinatorial optimization problem, under a linear and additive random payoff function. We formulate this problem using a two‐stage distributionally robust model, with a mixed 0–1 semidefinite program. This approach allows us to exploit the “diversification” effect inherent in the problem to address how different candidate solutions can be selected to improve the chances that one will attain a high ex post payoff. More interestingly, using this distributionally robust optimization approach, our model recovers the “evil twin” strategy, well known in the field of football pool betting, under appropriate settings. We also address the computational challenges of scaling up our approach to construct a moderate number of candidate solutions to increase the chances of finding one that performs well. To this end, we develop a sequential optimization approach based on a compact semidefinite programming reformulation of the problem. Extensive numerical results show the superiority of our approach over existing methods.

Complementarity formulation of games with random payoffs

We consider an n-player non-cooperative game where the payoff function of each player follows a multivariate distribution. This formulation is adopted to model a zonal electricity market in which generators operate by running conventional and renewable-based plants. The players in the market compete as in a Cournot model. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. A full empirical analysis has been conducted on the Italian electricity market to test the impact of renewable generators in the light of decarbonization of the market and the impact of the volatility of the cost of conventional plants, mainly related to the volatility of gas prices. We finally test the robustness of the chance constraint formulation with an out of sample analysis.

Random Games Under Elliptically Distributed Dependent Joint Chance Constraints

We study an n-player game with random payoffs and continuous strategy sets. The payoff function of each player is defined by its expected value and the strategy set of each player is defined by a joint chance constraint. The random constraint vectors defining the joint chance constraint are dependent and follow elliptically symmetric distributions. The Archimedean copula is used to capture the dependence among random constraint vectors. We propose a reformulation of the joint chance constraint of each player. Under mild assumptions on the players’ payoff functions and 1-dimensional spherical distribution functions, we show that there exists a Nash equilibrium of the game.

Average abundancy of cooperation in multi-player games with random payoffs

We consider interactions between players in groups of size [Formula: see text] with payoffs that not only depend on the strategies used in the group but also fluctuate at random over time. An individual can adopt either cooperation or defection as strategy and the population is updated from one time step to the next by a birth-death event according to a Moran model. Assuming recurrent symmetric mutation and payoffs to cooperators and defectors according to the composition of the group whose expected values, variances, and covariances are of the same small order, we derive a first-order approximation for the average abundance of cooperation in the selection-mutation equilibrium. In general, we show that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation. As for the effect of the covariance between any payoff for cooperation and any payoff for defection, we show that it depends on the number of cooperators in the group associated with these payoffs. We study in particular the public goods game, the stag hunt game, and the snowdrift game, all social dilemmas based on random benefit b and random cost c for cooperation, which lead to correlated payoffs to cooperators and defectors within groups. We show that a decrease in the scaled variance of b or c, or an increase in their scaled covariance, makes it easier for weak selection to favor the abundance of cooperation in the stag hunt game and the snowdrift game. The same conclusion holds for the public goods game except that the variance of b has no effect on the average abundance of C. Moreover, while the mutation rate has little effect on which strategy is more abundant at equilibrium, the group size may change it at least in the stag hunt game with a larger group size making it more difficult for cooperation to be more abundant than defection under weak selection.

Z-equilibrium in random bi-matrix games: Definition and computation

This paper deals with bi-matrix games with random payoffs. Using probability tools, we propose a solution based on the concept of Z-equilibrium. Then, we give sufficient conditions of its existence. Further, the problem of computation of this solution is transformed into the determination of Pareto optimal solutions of a deterministic bi-criteria minimization problem. Finally, we provide illustrative numerical examples.

Stochastic replicator dynamics and evolutionary stability.

To develop the concept of evolutionary stability in a stochastic environment, we investigate the continuous-time dynamics of a two-phenotype linear evolutionary game with generally correlated random payoffs in pairwise interactions. By using the Gram-Schmidt orthogonalization procedure and Itô's formula, we deduce a stochastic differential equationfor the phenotype frequencies that extends the replicator equation, called the stochastic replicator equation. We give conditions for stochastic stability of a fixation state or a constant interior equilibrium point with respect to the stochastic dynamics of the two phenotypes. We show that, if a fixation state is stochastically stable, then the pure strategy corresponding to this fixation state must be stochastically evolutionarily stable with respect to mixed strategies. However, this is not the case for a mixed strategy that corresponds to a stochastically stable constant interior equilibrium point with respect to the two phenotypes.

Stochastic evolutionary stability in matrix games with random payoffs.

Evolutionary game theory and the concept of an evolutionarily stable strategy have been not only extensively developed and successfully applied to explain the evolution of animal behavior, but also widely used in economics and social sciences. Recently, in order to reveal the stochastic dynamical properties of evolutionary games in randomly fluctuating environments, the concept of stochastic evolutionary stability based on conditions for stochastic local stability for a fixation state was developed in the context of a symmetric matrix game with two phenotypes and random payoffs in pairwise interactions [Zheng et al., Phys. Rev. E 96, 032414 (2017)2470-004510.1103/PhysRevE.96.032414]. In this paper, we extend this study to more general situations, namely, multiphenotype symmetric as well as asymmetric matrix games with random payoffs. Conditions for stochastic local stability and stochastic evolutionary stability are established. Conditions for a fixation state to be stochastically unstable and almost everywhere stochastically unstable are distinguished in a multiphenotype setting according to the initial population state. Our results provide some alternative perspective and a more general theoretical framework for a better understanding of the evolution of animal behavior in a stochastic environment.

Evolution of cooperation with respect to fixation probabilities in multi-player games with random payoffs

We study the effect of variability in payoffs on the evolution of cooperation (C) against defection (D) in multi-player games in a finite well-mixed population. We show that an increase in the covariance between any two payoffs to D, or a decrease in the covariance between any two payoffs to C, increases the probability of ultimate fixation of C when represented once, and decreases the corresponding fixation probability for D. This is also the case with an increase in the covariance between any payoff to C and any payoff to D if and only if the sum of the numbers of C-players in the group associated with these payoffs is large enough compared to the group size. In classical social dilemmas with random cost and benefit for cooperation, the evolution of C is more likely to occur if the variances of the cost and benefit, as well as the group size, are small, while the covariance between cost and benefit is large.

Risk-Averse Bargaining in a Stochastic Optimization Context

Problem definition: Bargaining situations are ubiquitous in economics and management. We consider the problem of bargaining for a fair ex ante distribution of random profits arising from a cooperative effort of a fixed set of risk-averse agents. Our approach integrates optimal managerial decision making into bargaining situations with random outcomes and explicitly models the impact of risk aversion. The proposed solution rests on a firm axiomatic foundation and yet allows to compute concrete bargaining solutions for a wide range of practically relevant problems. Methodology/results: We model risk preferences using coherent acceptability functionals and base our bargaining solution on a set of axioms that can be considered a natural extension of Nash bargaining to our setting. We show that the proposed axioms fully characterize a bargaining solution, which can be efficiently computed by solving a stochastic optimization problem. We characterize special cases where random payoffs of players are simple functions of overall project profit. In particular, we show that, for players with distortion risk functionals, the optimal bargaining solution can be represented by an exchange of standard options contracts with the project profit as the underlying asset. We illustrate the concepts in the paper with a detailed example of risk-averse households that jointly invest into a solar plant. Managerial implications: We demonstrate that there is no conflict of interest between players about management decisions and that risk aversion facilitates cooperation. Furthermore, our results on the structure of optimal contracts as a basket of option contracts provides valuable guidance for negotiators. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.1076 .

Noise-Induced Quasi-Heteroclinic Cycle in a Rock–Paper–Scissors Game with Random Payoffs

Noise-Induced Quasi-Heteroclinic Cycle in a Rock–Paper–Scissors Game with Random Payoffs

Fuzzy Random Bimatrix Games Based on Possibility and Necessity Measures

In this study, we formulate bimatrix games with fuzzy random payoffs, and introduce equilibrium solution concepts based on possibility and necessity measures. It is assumed that each player has linear fuzzy goals for his/her payoff. To obtain equilibrium solutions based on the possibility and necessity measures, we propose two algorithms in which quadratic programming problems are solved repeatedly until equilibrium conditions are satisfied.

Chance-constrained games with mixture distributions

In this paper, we consider an n-player non-cooperative game where the random payoff function of each player is defined by its expected value and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We consider the case when the probability distribution of each random constraint vector belongs to a subset of elliptical distributions as well as the case when it is a finite mixture of the probability distributions from the subset. We propose a convex reformulation of the joint chance constraint of each player and derive the bounds for players’ confidence levels and the weights used in the mixture distributions. Under mild conditions on the players’ payoff functions, we show that there exists a Nash equilibrium of the game when the players’ confidence levels and the weights used in the mixture distributions are within the derived bounds. As an application of these games, we consider the competition between two investment firms on the same set of portfolios. We use a best response algorithm to compute the Nash equilibria of the randomly generated games of different sizes.

Stochastic Superiority Equilibrium in Game Theory

The definition of best response for a player in the Nash equilibrium is based on maximizing the expected utility given the strategy of the rest of the players in a game. In this work, we consider stochastic games, that is, games with random payoffs, in which a finite number of players engage only once or at most a limited number of times. In such games, players may choose to deviate from maximizing their expected utility. This is because maximizing expected utility strategy does not address the uncertainty in payoffs. We instead define a new notion of a stochastic superiority best response. This notion of best response results in a stochastic superiority equilibrium in which players choose to play the strategy that maximizes the probability of them being rewarded the most in a single round of the game rather than maximizing the expected received reward, subject to the actions of other players. We prove the stochastic superiority equilibrium to exist in all finite games, that is, games with a finite number of players and actions, and numerically compare its performance to Nash equilibrium in finite-time stochastic games. In certain cases, we show the payoff under the stochastic superiority equilibrium is 70% likely to be higher than the payoff under Nash equilibrium.

Dynamic survival bias in optimal stopping problems

This paper studies the optimal inference from observing an ongoing experiment. An experimenter sequentially chooses whether to continue with costly trials that yield random payoffs. The experimenter sees the full history of the trial results, while an outside observer sees only the recent trial results, not the earlier prehistory. I contrast the optimal sophisticated posterior of the observer based on a full Bayesian inference that accounts for the prehistory and the naive posterior based solely on the observed history. The resulting dynamic bias grows with longer prehistory if we see enough early successes. Observing more failures may increase the sophisticated posterior if they come early. Revealing a success (failure) in the prehistory always increases (lowers) the sophisticated posterior. Uncovering a more recent signal leads to a larger change than an older one. Seeing a future failure may increase the sophisticated posterior.

The equal-surplus Shapley value for chance-constrained games on finite sample spaces

Many interactions from linear production problems, financial markets, or sequencing problems are modeled by cooperative games where payoffs to a coalition of players is a random variable. For this class of cooperative games, we introduce a two-stage value as an ex-ante agreement among players. Players are first promised their prior Shapley shares which are exactly their respective shares by the Shapley value of the expectation game. The final payoff vector is obtained by equally re-allocating the surplus when a realization of the random payoff of the grand coalition is observed. In support of the tractability of the newly introduced value called equal-surplus Shapley value, we provide a simple and compact formula. Depending on which probability distributions over the sample spaces are admissible, we present several characterization results of the equal-surplus Shapley value. This is achieved by using some classical axioms together with some other appealing axioms such as the independence of local duplication which simply requires that individual shares in a game remain unchanged when only certain events are duplicated in the sample space of a coalition without altering the probability of observing the others.

Uncertainty quantification in game theory

Abstract This work examines the applicability of Uncertainty Quantification (UQ) in Game Theory. We consider the classical games “matching pennies” and “Hawk an Dove” in situations involving uncertainty. The first game examined is “matching pennies”: a first situation concerns the game where the probabilities of choice between Heads and Tails are unknown and must be determined from observations. A second situation concerns fluctuations in the implementation of the Nash equilibrium. Instability is evidentiated and a strategy based on statistical estimation is introduced. A third situation considers random payoffs having an unknown distribution: observations are used to generate an UQ representation of the real distribution of the payoffs, without any supplementary assumption on the nature of the distribution. Finally, we analyze the effects of uncertainties on the associated replicator dynamics: UQ is applied to generate mean trajectories and mean orbits - in this step, we need to manipulate statistics of curves, which are objects defined by functions, belonging to infinitely dimensional vector spaces. The second game is “Hawk and Dove”. We examine the situation where the reward and the cost of an injury are both uncertain and only a small sample of values is available. The methods of UQ are applied to determine the mean evolution of the system and confidence intervals for the evolution of the fractions of Hawks and Doves. The UQ methods involved are described and simple examples are given to facilitate understanding and application to other situations.