Games over Probability Distributions Revisited: New Equilibrium Models and Refinements (original) (raw)
Related papers
arXiv (Cornell University), 2015
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. While preferences among distributions turn out to be quite simple if nonparametric methods (kernel density estimates) are used, computing Nash-equilibria in games using such models is discovered as inefficient (if not impossible). In fact, we give a counterexample in which fictitious play fails to converge for the (specifically unfortunate) choice of payoff distributions in the game, and introduce a suitable tail approximation of the payoff densities to tackle the issue. The overall procedure is essentially a modified version of fictitious play, and is herein described for standard and multicriteria games, to iteratively deliver an (approximate) Nash-equilibrium. An exact method using linear programming is also given.
arXiv: Economics, 2015
Optimal behavior in (competitive) situation is traditionally determined with the help of utility functions that measure the payoff of different actions. Given an ordering on the space of revenues (payoffs), the classical axiomatic approach of von Neumann and Morgenstern establishes the existence of suitable utility functions, and yields to game-theory as the most prominent materialization of a theory to determine optimal behavior. Although this appears to be a most natural approach to risk management too, applications in critical infrastructures often violate the implicit assumption of actions leading to deterministic consequences. In that sense, the gameplay in a critical infrastructure risk control competition is intrinsically random in the sense of actions having uncertain consequences. Mathematically, this takes us to utility functions that are probability-distribution-valued, in which case we loose the canonic (in fact every possible) ordering on the space of payoffs, and the o...
The Mathematical Underpinnings of Utility Functions and their Broader Implications on Game Theory
Academia.edu, 2020
The predominating objective of this short paper is to lay forth a series of results derived from the optimization of canonical utility and indifference functions (insofar as they are used conventionally), and then analogizing them in the context of payoffs. It concerns the categorizations of utility functions, indifference curves and other analytical constraints. To commence, it draws a generalization as to how utilities change with regards to marginal rates of substitution. Following this, it invokes a series of arguments as to how they can be maximized in both univariate and multivariate states.
Potential games are necessary to ensure pure Nash equilibria in cost sharing games
Mathematics of Operations Research
We consider the problem of designing distribution rules to share 'welfare' (cost or revenue) among individually strategic agents. There are many known distribution rules that guarantee the existence of a (pure) Nash equilibrium in this setting, e.g., the Shapley value and its weighted variants; however, a characterization of the space of distribution rules that guarantee the existence of a Nash equilibrium is unknown. Our work provides an exact characterization of this space for a specific class of scalable and separable games, which includes a variety of applications such as facility location, routing, network formation, and coverage games. Given arbitrary local welfare functions W, we prove that a distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is equivalent to a generalized weighted Shapley value on some 'ground' welfare functions W', which can be distinct from W. However, if budget-balance is required in addition to the existence of a Nash equilibrium, then W' must be the same as W. We also provide an alternate characterization of this space in terms of 'generalized' marginal contributions, which is more appealing from the point of view of computational tractability. A possibly surprising consequence of our result is that, in order to guarantee equilibrium existence in all games with any fixed local welfare functions, it is necessary to work within the class of potential games.
Distributionally robust chance-constrained games: existence and characterization of Nash equilibrium
Optimization Letters, 2016
We consider an n-player finite strategic game. The payoff vector of each player is a random vector whose distribution is not completely known. We assume that the distribution of a random payoff vector of each player belongs to a distributional uncertainty set. We define a distributionally robust chanceconstrained game using worst-case chance constraint. We consider two types of distributional uncertainty sets. We show the existence of a mixed strategy Nash equilibrium of a distributionally robust chance-constrained game corresponding to both types of distributional uncertainty sets. For each case, we show a one-to-one correspondence between a Nash equilibrium of a game and a global maximum of a certain mathematical program.
Characterizing distribution rules for cost sharing games
We consider the problem of designing the distribution rule used to share "welfare" (cost or revenue) among individually strategic agents. There are many distribution rules known to guarantee the existence of a (pure Nash) equilibrium in this setting, e.g., the Shapley value and its weighted variants; however a characterization of the space of distribution rules that yield the existence of a Nash equilibrium is unknown. Our work provides a step towards such a characterization. We prove that when the welfare function is strictly submodular, a budgetbalanced distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is a weighted Shapley value.
An Introduction to Applicable Game Theory
Journal of Economic Perspectives, 1997
This paper offers an introduction to game theory for applied economists. The author gives simple definitions and intuitive examples of four kinds of games and their corresponding solution concepts: Nash equilibrium in static games of complete information; subgame-perfect Nash equilibrium in dynamic games of complete information; Bayesian Nash equilibrium in static games with incomplete (or 'private') information; and perfect Bayesian (or sequential) equilibrium in dynamic games with incomplete information. The main theme of the paper is that there are important differences among the games but important similarities among the solution concepts.
International Journal of Game Theory
We present a general existence result for a type of equilibrium in normal-form games, which extends the concept of Nash equilibrium. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a legitimate equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a legitimate equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptu...
Risk-Averse Equilibrium for Games
ArXiv, 2020
The term rational has become synonymous with maximizing expected payoff in the definition of the best response in Nash setting. In this work, we consider stochastic games in which players engage only once, or at most a limited number of times. In such games, it may not be rational for players to maximize their expected payoff as they cannot wait for the Law of Large Numbers to take effect. We instead define a new notion of a risk-averse best response, that results in a risk-averse equilibrium (RAE) 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 risk-averse equilibrium to exist in all finite games and numerically compare its performance to Nash equilibrium in finite-time stochastic games.
Modeling Attacker-Defender Games with Risk Preference
This project will study how the incorporation of more realistic decision making frameworks namely expected utility and cumulative prospect theories (EUT and CPT), affects the equilibrium behavior of players in attacker-defender games.