Abstract

Inspired by Shalev’s model of loss aversion, we investigate the effect of loss aversion on a bimatrix game where the payoffs in the bimatrix game are characterized by triangular fuzzy variables. First, we define three solution concepts of credibilistic loss aversion Nash equilibria, and their existence theorems are presented. Then, three sufficient and necessary conditions are given to find the credibilistic loss aversion Nash equilibria. Moreover, the relationship among the three credibilistic loss aversion Nash equilibria is discussed in detail. Finally, for bimatix game with triangular fuzzy payoffs, we investigate the effect of loss aversion coefficients and confidence levels on the three credibilistic loss aversion Nash equilibria. It is found that an increase of loss aversion levels of a player leads to a decrease of his/her own payoff. We also find that the equilibrium utilities of players are decreasing (increasing) as their own confidence levels when players employ the optimistic (pessimistic) value criterion.

1. Introduction

Since established by von Neumann and Morgenstern (1944), game theory has been applied to modeling strategic interactions for decision problem in many fields, such as economics, political science, and psychology. A bimatrix game is a finite noncooperative game in normal (or strategic) form, which is often used to analyze interaction between two rational players who behave strategically. In this game, each player desires achieving as much payoffs as possible by choosing one from her/his pure strategy set. A bimatrix game is modeled by two payoff matrices determined by the two players’ strategic profiles. A mixed strategy describes a situation that each player would rather randomly choose a pure strategy with a given probability distribution than choose a particular pure strategy. Nash [1] proved that there exists at least one mixed strategy Nash equilibrium such that no player can improve her/his gains by unilaterally deviating from such strategy profiles.

As a matter of fact, uncertainty for parameters plays an important role in bimatrix games, which is often called incomplete information. The most striking style of the uncertainty for parameters is the uncertainty for the payoffs, which includes random payoffs and fuzzy payoffs. As pointed out by Harsanyi [2], each player often lacks the information about her/his opponents’ or even her/his own payoffs. In order to deal with games with incomplete information, Harsanyi [2] proposed Bayesian game model that is often used to deal with games with random payoffs. Since then the game with random payoffs has been investigated by many researchers [38]. When it comes to games with fuzzy payoffs, the Bayesian game model fails to be applied to such games. Fuzzy set theory provides an efficient alternative to model the uncertainty about payoffs that are modeled by fuzzy variables. Campos [9] firstly investigated two-person zero-sum games with fuzzy payoffs. Since then two-person games with fuzzy payoffs, including two-person zero-sum games and bimatrix games, have been investigated by many researchers [1022]. Moreover, some credibilistic games are proposed to deal with the games with fuzzy payoffs, including credibilistic strategic game [2326] and credibilistic coalitional game [27].

It is worthwhile noting that the existing literature on games with fuzzy payoffs ignores the effect of decision-maker’s behavior characteristics on equilibrium strategies. In real game problems, players often show bounded rational behavior characteristics. A number of experimental works in the psychological and the economic literature suggest that decision-makers are motivated to minimize losses (relative to a reference point) much more than they are motivated to maximize gains [2832]. In other words, decision-makers are loss-averse. Players’ loss aversion plays an important role in real game problems. For example, Langsha Group, the largest socks manufacturer of China, decided to terminate cooperation with Wal-Mart in 2007, since Langsha Group could no longer benefit from cooperation with Wal-Mart. That is, Wal-Mart grabbed too much profit to make Langsha Group incur losses. Thus, to incorporate loss aversion into games, Shalev [33] proposed an elegant and simple model of loss aversion to measure loss aversion: the utility of an outcome below the reference point is obtained from the basic utility by subtracting a disutility that is equal to the size of the loss multiplied by the loss aversion coefficient. Since then Shalev’s model of loss aversion has received attention in the field of game theory based on the assumption that the payoffs are common knowledge to two players [3438]. However, it is very likely that this assumption is violated, since the decision environment is often characterized by lots of strategies, complicated relations between strategic choices, and their intricate influences on payoffs, which leads to the uncertainty of the payoffs. For example, Langsha Group develops its direct selling business in 2017; it cannot be exactly predicted whether Langsha Group can benefit from the direct selling because of the uncertainty of the consumer demand and the investment cost in direct selling business. Thus, to deal with such situation faced by Langsha Group, this paper investigates the impact of loss aversion on equilibrium strategies in bimatrix games with fuzzy payoffs.

In this paper, we extend the analysis of bimatrix games with fuzzy payoffs to incorporate loss aversion. Inspired by Shalev’s model of loss aversion, we first give a formula that relates outcomes and reference points to utility. This formula, where the parameters except loss aversion coefficient are modeled as triangular fuzzy variables, is called loss aversion utility function with triangular fuzzy parameters. Then, given an extended bimatrix game with triangular fuzzy payoffs, including a bimatrix game with triangular fuzzy payoffs and loss aversion coefficients, three ranking methods are used to model different situations. And we define the solution concepts of three credibilistic loss aversion equilibria and discuss the relationship among the three credibilistic loss aversion equilibria. Finally, by means of 2×2 bimatrix game with triangular fuzzy payoffs, we investigate the effect loss aversion coefficients and confidence levels on the three credibilistic loss aversion Nash equilibria.

The rest of this paper is organized as follows. In Section 2, credibility theory and loss aversion are briefly reviewed. In Section 3, the concepts of three credibilistic loss aversion Nash equilibria are proposed, and their existence theorems and sufficient and necessary conditions are given. Moreover, the relationship of the three credibilistic loss aversion Nash equilibria is discussed. In Section 4, we investigate the effect of loss aversion coefficients and confidence levels on the three credibilistic loss aversion Nash equilibria. In Section 5, conclusions are shown.

2. Preliminaries

2.1. Credibility Theory

As a branch of mathematics, credibility theory is often used to model the behavior of fuzzy phenomena. Since proposed by Liu [39], credibility theory has been used extensively in many fields, such as portfolio selection [40, 41] and transportation planning [42, 43].

Definition 1 (see [44]). Let be a nonempty set and be the power set of ; a set function is called a credibility measure if it satisfies the following four axioms.
Axiom 1 (normality). .
Axiom 2 (monotonicity). , where .
Axiom 3 (self-duality). for any .
Axiom 4 (maximality). for any with .

Definition 2 (see [44]). Let be a nonempty set, be the power set of , and Cr be a credibility measure; then the triplet is called a credibility space.

Definition 3 (see [44]). A fuzzy variable is a measurable function from a credibility space to the set of real numbers.

Lemma 4 (see [44]). Let be a fuzzy variable with membership function ; then for any set B of real numbers, we haveAs a special fuzzy variable with the triangular fuzzy membership function , a triangular fuzzy variable is denoted by [45], where is the mean of and and are the low bound and the bound of , respectively. Its membership function is given byObviously, if , then the triangular fuzzy variable is reduced to a real number. For a triangular fuzzy number , if and , we call nonnegative triangular fuzzy number. Let and be two triangular fuzzy numbers; then according to extension principle proposed by Zadeh [46] and Negoita et al. [47], we have the following arithmetical operations:(i);(ii);(iii) if  .It follows from extension principle that . However, in optimization decision problems, it can be desirable to have crisp values for . In order to deal with such situations, Gani and Assarudeen [48] developed a new operation for subtraction on triangular fuzzy numbers.

Lemma 5 (see [48]). Let and be two triangular fuzzy numbers; then we have

Definition 6 (see [44]). The fuzzy variables are said to be independent iffor any sets of .

Definition 7 (see [49]). Let be a fuzzy variable; then the expected value of is defined byprovided that at least one of the two integrals is finite.

Lemma 8 (see [50]). Let and be independent fuzzy variables with finite expected values; then for any numbers a and b, we have

Definition 9 (see [50]). Let be a fuzzy variable and be a confidence level; then, for a real number r,is called α-optimistic value to , andis called α-pessimistic value to .
This means that the fuzzy variable will reach upwards of the α-optimistic value with credibility α and will be below the α-pessimistic value with credibility α. In other words, the α-optimistic value is the supremum value that achieves with credibility α, and the α-pessimistic value is the infimum value that achieves with credibility α.

Lemma 10 (see [50]). Let and be independent fuzzy variables; then for any and any nonnegative numbers a and b, we have

Lemma 11 (Liu and Liu, 2007). Let be a fuzzy variable; then for any , we have

In order to rank fuzzy variables, the following methods are often used.

Definition 12 (see [51]). Let and be independent fuzzy variables; then we have
expected value criterion: if and only if ;
optimistic value criterion: if and only if for some predetermined confidence level ;
pessimistic value criterion: if and only if for some predetermined confidence level .
The expected value criterion is used to deal with the situation where a player wants to optimize the expected value of her/his payoff. The optimistic value criterion is applied to coping with the situation in which a player strives to optimize the optimistic value of her/his payoff at given a confidence level α. The pessimistic value criterion is used to model the situation that a player strives to optimize the pessimistic value of her/his payoff at given a confidence level α.
Let be independent fuzzy variables; then is called the maximum of , and is called the minimum of .

2.2. Loss Aversion

Kahneman and Tversky [29] first proposed loss aversion; its central assumption is that gains are smaller than losses. For instance, the decrease in utility of loss 10 dollars if one has 100 dollars is larger than the increase in utility of gain 10 dollars if one has 90 dollars. Shalev [33] proposed an elegant and simple model of loss aversion to measure this. The utility function is given by the following transformationor equivalentlywhere λ is nonnegative and w0 is a reference point. Shalev’s model of loss aversion is similar to the value function given by Tversky and Kahneman [32]. The loss aversion aspect of the utility function is retained, which is that the marginal utility in losses is larger than in gains. Moreover, in order to reflect heterogeneity of loss aversion, the loss aversion coefficients are different for different players.

3. Credibilistic Loss Aversion Nash Equilibria

In this section, according to three decision criteria from credibility theory, i.e., expected value criterion, optimistic value criterion, and pessimistic value criterion, three credibilistic loss aversion Nash equilibria are proposed.

3.1. Bimatrix Game with Triangular Fuzzy Payoffs and Loss Aversion

Let I and J be two players and and be the pure strategy sets of players I and J, respectively. The mixed strategy set of player is denoted byand the mixed strategy set of player J is denoted by

A bimatrix game is denoted by , where and are payoff matrices of players I and J, whose elements are and . For any mixed strategy profile , it determines the outcomes of the bimatrix game (), where and are expected payoffs of players I and J, respectively.

Lemma 13 (see [1]). There exists at least one mixed strategy Nash equilibrium in a finite bimatrix game.

Since the decision environment is often characterized by lots of strategies, intricate relations between strategic choices and their influences on players’ payoffs, it is impossible that players make accurate or probabilistic estimation of the payoff matrices. Thus, we consider a bimatrix game with fuzzy payoffs. Here, players’ payoffs are modeled by triangular fuzzy variables. Triangular fuzzy numbers and are denoted by the payoffs that players I and J obtain when the pure strategy profile (i, j) is played. This bimatrix game with triangular fuzzy payoffs is denoted by , where and are m×n payoff matrices of players I and J, whose i, j elements are and . Furthermore, we assume that and are independent triangular fuzzy variables for all pure strategy profiles (i, j).

In many situations, decision-makers are motivated to minimize losses (relative to a reference point) much more than they are motivated to maximize gains. We extend the analysis of the bimatrix game with fuzzy payoffs to incorporate loss aversion. An extended bimatrix game with triangular fuzzy payoffs has two additional elements—the loss aversion coefficients of players I and J. Let and specify the levels of loss aversion of players I and J, respectively. The higher the values of and are, the higher the degrees of loss aversion of players I and J are. Players I and J seek maximum of gains if and , which means that the utility functions of players I and J are not reference dependent. Inspired by (13), let be a reference point of player I and and be independent triangular fuzzy variables. Given a reference point and a basic utility value , then for player I, the final loss aversion utility is given bySimilarly, let be a reference point of player J and and be independent triangular fuzzy variables. Given a reference point and a basic utility value , then for player J, the final loss aversion utility is given by

For exogenously given and , an extended bimatrix game with triangular fuzzy payoffs can be transformed into a standard game with triangular fuzzy payoffs by evaluating the utility of each payoff according to (16) and (17). Given an extended bimatrix game G3 and reference points and , a transformationis defined, where the utility of each payoff for each player is transformed by using the appropriate reference points and loss aversion coefficients according to (16) or (17).

For an extended bimatrix game with triangular fuzzy payoffs G3, we definewhere and are the minimum and the maximum in matrices and .

3.2. Definitions of Credibilistic Loss Aversion Nash Equilibrium Strategies

Suppose that the expected value criterion is employed. Given an extended bimatrix game G3 and reference points and , the loss aversion utilities of player I and J are given, respectively:

For a strategy profile () and a reference point pair (), players I and J have expected final loss aversion utilities and , respectively. Note that the expected value criterion is used to deal with the situation that players want to maximize the expected values of their own expected final loss aversion utility. Thus, for player I, we have and . Since is a continuous function of , then there exists a such that for . Furthermore, is unique since is nonincreasing on . Similarly, for player J, there exists a such that is unique and satisfies for . Thus, the best responses of player I to a strategy are the optimal solutions of the fuzzy expected value modeland the best responses of player J to a strategy are the optimal solutions of the fuzzy expected value model

Definition 14. A strategy profile () is an expected loss aversion Nash equilibrium (ELANE) of an extended bimatrix game G3 if () satisfies is called an expected value of the game G3.
Suppose that the optimistic value criterion is employed. Given an extended bimatrix game G3 and reference points and , we havewhere , .
For player I, it follows from Lemma 4 and Definition 9 thatandThus, we haveSimilarly, for player J, we haveFor a strategy profile () and a reference point pair (), players I and J want to maximize the optimistic value of their expected final loss aversion utility at their own confidence level. Thus, for player I, we have and . Since is a continuous function of , then there exists a such that for . Furthermore, is unique since is nonincreasing on . Similarly, for player J, there exists a such that is unique and satisfies for . Thus, the best responses of player I to a strategy are the optimal solutions of the following chance-constrained programming model:where α is a predetermined confidence level of player I and uI is a real number.
And the best responses of player J to a strategy are the optimal solutions of the following chance-constrained programming modelwhere β is a predetermined confidence level of player J and is a real number.

Definition 15. A strategy profile () is ()-optimistic loss aversion Nash equilibrium (()-OLANE) of an extended bimatrix game G3 if () satisfiesand is called an optimistic value of the game G3.
Suppose that the pessimistic value criterion is employed. Given an extended bimatrix game G3 and reference points and , we havewhere , .
For player I, it follows from Lemma 4 and Definition 9 thatandThus, we haveSimilarly, for player J, we haveFor a strategy profile () and a reference point pair (), player I wants to maximize the pessimistic value of her/his expected final loss aversion utility at a confidence level α and player J wants to maximize the pessimistic value of her/his expected final loss aversion utility at a confidence level β. Thus, for player I, we have and . Since is a continuous function of , then there exists a such that for . Furthermore, is unique since is nonincreasing on . Similarly, for player J, there exists a such that is unique and satisfies for . Thus, the best responses of player I to a strategy are the optimal solutions of the following chance-constrained programming model:where α is a predetermined confidence level of player I and is a real number.
The best responses of player J to a strategy are the optimal solutions of the following chance-constrained programming model:where β is a predetermined confidence level of player J and is a real number.

Definition 16. A strategy profile () is ()-pessimistic loss aversion Nash equilibrium (()-PLANE) of an extended bimatrix game G3 if () satisfiesand is called a pessimistic value of the game G3.

3.3. Existence Theorems and Sufficient and Necessary Conditions of Credibilistic Loss Aversion Equilibria

Now, we present the existence theorems of the three credibilistic loss aversion equilibrium strategies.

Theorem 17. For any extended bimatrix game G3, then
there exists at least one ELANE (), and the expected value of the game G3 is , where , , , ;
there exists at least one ()-OLANE (), and the ()-optimistic loss aversion value of the game is , where , , , ;
there exists at least one ()-PLANE (), and the ()-pessimistic loss aversion value of the game is , where , , , .

Proof. The proofs of the three parts are similar. Here, for simplicity, we only prove part .
Given an extended game , for any and , we have and . Thus, part can be proven by proving that there exists Nash equilibrium () in the game such that , For the game , we define the correspondence f from to itself as follows. if () is a best response to () in the game for player I (J), where is the payoff that player I obtains from () and is the payoff that player J obtains from () in the game .
SI, SJ, and are nonempty, compact, and convex, so it is their product. The correspondence is nonempty for every . For player I (J), there exists a best response () to () in the game . And let be the payoff that player I obtains from () and be the payoff that player J obtains from () in the game . Then we have an element in . If there exist at least two best responses for a player, then the payoffs that this player obtains from these best responses are the same, and so does any convex combination of these best responses. Thus, the correspondence is convex. Since functions of and as payoffs of players I and J are continuous in and , and the best response function has a closed graph, then the correspondence has a closed graph. By Kakutani’s fixed point theorem, there exist () and such that . Given payoffs of and for players I and J, it follows from the definition of f that () is a Nash equilibrium of the game . Thus, there exists at least one ()-OLANE in the game G3.
These complete the proof of part . Theorem 17 is proven.

To find the three credibilistic loss aversion Nash equilibrium strategies of the game G3, we give the sufficient and necessary conditions.

Theorem 18. For any extended bimatrix game G3,
a strategy profile is an ELANE if and only if the point is an optimal solution to the following quadratic programming model:where , , , , is a vector, and is a vector.
a strategy profile is a ()-OLANE if and only if the point is an optimal solution to the following quadratic programming model:where , , , , is a vector, and is a vector.
a strategy profile is a ()-PLANE if and only if the point is an optimal solution to the following quadratic programming model:where , , , ,