Utility Functions Of Players Research Articles

On nash equilibria in normal-form games with vectorial payoffs

We provide an in-depth study of Nash equilibria in multi-objective normal-form games (MONFGs), i.e., normal-form games with vectorial payoffs. Taking a utility-based approach, we assume that each player’s utility can be modelled with a utility function that maps a vector to a scalar utility. In the case of a mixed strategy, it is meaningful to apply such a scalarisation both before calculating the expectation of the payoff vector as well as after. This distinction leads to two optimisation criteria. With the first criterion, players aim to optimise the expected value of their utility function applied to the payoff vectors obtained in the game. With the second criterion, players aim to optimise the utility of expected payoff vectors given a joint strategy. Under this latter criterion, it was shown that Nash equilibria need not exist. Our first contribution is to provide a sufficient condition under which Nash equilibria are guaranteed to exist. Secondly, we show that when Nash equilibria do exist under both criteria, no equilibrium needs to be shared between the two criteria, and even the number of equilibria can differ. Thirdly, we contribute a study of pure strategy Nash equilibria under both criteria. We show that when assuming quasiconvex utility functions for players, the sets of pure strategy Nash equilibria under both optimisation criteria are equivalent. This result is further extended to games in which players adhere to different optimisation criteria. Finally, given these theoretical results, we construct an algorithm to compute all pure strategy Nash equilibria in MONFGs where players have a quasiconvex utility function.

Caching resource management of mobile edge network based on Stackelberg game

Mobile edge caching technology is gaining more and more attention because it can effectively improve the Quality of Experience (QoE) of users and reduce backhaul burden. This paper aims to improve the utility of mobile edge caching technology from the perspectie of caching resource management by examining a network composed of one operator, multiple users and Content Providers (CPs). The caching resource management model is constructed on the premise of fully considering the QoE of users and the servicing capability of the Base Station (BS). In order to create the best caching resource allocation scheme, the original problem is transformed into a multi-leader multi-follower Stackelberg game model through the analysis of the system model. The strategy combinations and the utility functions of players are analyzed. The existence and uniqueness of the Nash Equilibrium (NE) solution are also analyzed and proved. The optimal strategy combinations and the best responses are deduced in detail. Simulation results and analysis show that the proposed model and algorithm can achieve the optimal allocation of caching resource and improve the QoE of users.

The economics of attitudes: A different approach to utility functions of players in tourism marketing coalitional networks

The foundation of destination collaboration is based on the interdependency of the organizations involved in producing destination products. The high rate of destination collaboration failure underscores the need for conflict studies. Unlike previous studies, which depend solely on the collaboration monetary values, this study proposes a new approach to define its utility functions based on the attitudinal and motivational values. We employ the network theory to define the utility function of four major players and the game theory to examine three distribution solutions of coalitional activities' values. The results support the notion of “free riders” mentioned in collaboration studies and explains why free riding is a natural phenomenon in tourism destinations’ marketing activities. The findings suggest that individual entities and hospitality are the two players with the highest admission fee and the least contribution. We suggest the concepts of fairness and stability to be considered in incentive policies to encourage collaboration among higher admission players.

Nonlinear dynamics from discrete time two-player status-seeking games

We study the dynamics of two-player status-seeking games where moves are made simultaneously in discrete time. For such games, each player's utility function will depend on both non-positional goods and positional goods (the latter entering into status). In order to understand the dynamics of such games over time, we sample a variety of different general utility functions, such as CES, composite log-Cobb-Douglas, and King-Plosser-Rebelo utility functions (and their various simplifications). For the various cases considered, we determine asymptotic dynamics of the two-player game, demonstrating the existence of stable equilibria, periodic orbits, or chaos, and we show that the emergent dynamics will depend strongly on the utility functions employed. For periodic orbits, we provide bifurcation diagrams to show the existence or non-existence of period doubling or chaos resulting from bifurcations due to parameter shifts. In cases where multiple feasible solution branches exist at each iteration, we consider both cases where deterministic or random selection criteria are employed to select the branch used, the latter resulting in a type of stochastic game.

Łukasiewicz Games

Boolean games provide a simple, compact, and theoretically attractive abstract model for studying multiagent interactions in settings where players will act strategically in an attempt to achieve individual goals. A standard critique of Boolean games, however, is that the strictly dichotomous nature of the preference relations induced by Boolean goals inevitably trivialises the nature of such strategic interactions: a player is assumed to be indifferent between all outcomes that satisfy her goal, and indifferent between all outcomes that do not satisfy her goal. While various proposals have been made to overcome this limitation, many of these proposals require the inclusion of nonlogical structures into games to capture nondichotomous preferences. In this article, we introduce Łukasiewicz games, which overcome this limitation by allowing goals to be specified using Łukasiewicz logics . By expressing goals as formulae of Łukasiewicz logics, we can express a much richer class of utility functions for players than is possible using classical Boolean logic: we can express every continuous piecewise linear polynomial function with rational coefficients over [0, 1] n as well as their finite-valued restrictions over {0, 1/ k , …, ( k − 1)/ k , 1} n . We thus obtain a representation of nondichotomous preference structures within a purely logical framework. After introducing the formal framework of Łukasiewicz games, we present a number of detailed worked examples to illustrate the framework, and then investigate some of their theoretical properties. In particular, we present a logical characterisation of the existence of Nash equilibria in finite and infinite Łukasiewicz games. We conclude by briefly discussing issues of computational complexity.

Data-driven estimation in equilibrium using inverse optimization

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and nonparametric estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.

Money-egalitarian-equivalent and gain-maximin allocations of indivisible items with monetary compensation

Suppose that a certain quantity M of money and a finite number of indivisible items are to be distributed among n people, all of whom have equal claims on the whole. Different allocations are presented using various criteria of fairness in the special case where each player's utility function is additively separable. An allocation is “money-egalitarian-equivalent” (MEE) if each player's monetary valuation of his or her bundle is a fixed constant. We show that there is an essentially unique allocation that is MEE and Pareto-optimal; it is also envy-free. Alternatively, the “gain” of a player may be defined as the difference between how the player evaluates his bundle and an exact nth part of the whole according to his numerical evaluation of the whole. A “gain-maximin” criterion would maximize the minimum gain obtained by any player. We show that Knaster's procedure finds an allocation which is optimal under the gain-maximin criterion. That allocation is not necessarily envy-free, so we also find the envy-free allocation that is optimal under the gain-maximin criterion among all envy-free allocations. It turns out that, even though there exist allocations that are simultaneously envy-free and Pareto-optimal, this optimal allocation may fail to be Pareto-optimal, and it may also violate monotonicity criteria.

Nonconvexity in noncooperative game theory

Recent modeling efforts which attempted to apply the classical Nash equilibrium theory were found unsatisfactory because one or more players in the game had nonconvex loss functions (or nonconcave utility functions). In order to handle these difficult and important models, a generalization of the equilibrium is proposed which is denoted the pre-equilibrium set. This set contains all solutions to the first-order necessary optimality conditions for each player's utility function with respect to every opponent's necessary optimality conditions solutions. It contains all the Nash equilibria, as well as other points which may be of interest in their own right.

The Role of Attitude Toward Risk in Strictly Competitive Decision-Making Situations

The impact of decision makers' attitudes toward risk has been studied extensively in the context of noncompetitive decision making but not in the context of competitive decision making. In this paper we study the effects of the utility functions of the players in a strictly competitive decision-making situation on (a) their strategies and (b) their expected monetary payoffs. The setting is that of a two-by-two, two-person, zero-sum (in money) game with exponential utility functions for the competitors. The monetary payoffs and utility functions are assumed to be public knowledge. Combinations of utility functions that preserve the zero-sum condition as well as combinations that do not preserve this condition are considered. As might be expected, results from noncompetitive decision making do not necessarily carry over into competitive situations. For example, an increase in the risk aversion of a player may actually increase that player's expected monetary payoff because of altered strategies on the part of the competitor. In general, the effects of nonlinear utility functions on a player's strategy and expected monetary payoff depend on the player's utility function, the opponent's utility function, and the relationship among the entries in the payoff table. Furthermore, it does not appear possible to express this dependence in the form of simple rules of thumb, although the results presented here indicate the direction of movement of strategies and expected payoffs as the inputs to the problem are varied in certain systematic ways.