Impact of Risk Aversion in Fuzzy Bimatrix Games

Abstract

In bimatrix games with symmetric triangular fuzzy payoffs, our work defines an (α, β)-risk aversion Nash equilibrium ((α, β)-RANE) and presents its sufficient and necessary condition. Our work also discusses the relationships between the (α, β)-RANE and a mixed-strategy Nash equilibrium (MSNE) in a bimatrix game with a risk-averse player 2 and certain payoffs. Finally, considering 2 × 2 bimatrix games with STFPs, we find the conditions where the increase in player 2’s risk-aversion level hurts or benefits himself/herself.

1. Introduction

As strategic-form games, bimatrix games are used for analyzing the strategic interaction between risk-neutral players. Different types of strategies, including mixed strategies and pure strategies, are often adopted to optimize the payoffs of two risk-neutral players. The payoff matrices in a bimatrix game are determined by the two players’ strategic profiles. Bimatrix games can be easily generalized to n-person non-cooperative games, which means that bimatrix games can exactly model the real decision-making problem. In particular, bimatrix games play key roles in numerous decision-making scenarios when bimatrix games are applied to real-game situations. For each player, it is not easy to exactly know the payoffs of the other player in a bimatrix game. Therefore, it is necessary to explore bimatrix games under uncertain environments.

Classical bimatrix games were investigated based on the expected utility (ET) ([1,2,3,4]). Although the ET dominates the analysis of bimatrix games, a large amount of evidence shows that the ET cannot adequately characterize or predict the behavior of humans ([5,6,7]). Prospect theory (PT), proposed by [8], is regarded as a substitute approach to the ET. PT confirms risk aversion for gains. For instance, 80% of respondents would rather choose a certain outcome of three thousand dollars than opt for an 80% chance of four thousand dollars and a 20% chance of nothing ([6]). This phenomenon is called risk aversion. Nevertheless, scholars have paid little attention to the effects of risk aversion in the game-theoretic field, except for bargaining theory.

In classical game theory, all payoffs in bimatrix games are common knowledge to the two players ([9,10]). In real-game problems, however, players usually lack sufficient information about the payoffs for themselves and other players ([11]) because of the complex relationships between differently strategic selections and their complicated impacts on the payoffs. This results in the phenomenon of uncertainty in the payoffs. The uncertainties in the payoffs are presented as the following two types: random payoffs (RPs) as well as fuzzy payoffs (FPs). Ref. [12] introduced a Bayesian game model (BGM) to address the issues with random payoffs in the game-theoretic field. Ref. [13] investigated zero-sum game problems with RPs by using chance-constrained programming. Ref. [14] examined zero-sum games with RPs by adopting non-convex mathematical programming. Ref. [15] proposed a disordered systems-based model for investigating bimatrix games with random payoffs. Ref. [16] developed statistical mechanics of neural networks for solving zero-sum games with RPs. Ref. [17] applied Cauchy distribution to explore coordination games with RPs. The above works investigated zero-sum games with both players under the hypothesis that plenty of available historical data can be used to adequately characterize RPs. However, BGM fails to adequately model the situation in which few historical data are available in game problems. As a result, the fuzzy game (FG) was introduced by adopting fuzzy set theory developed by [18]. Ref. [19] investigated zero-sum games with FPs. Ref. [20] considered a bimatrix game with FPs by using possibility and necessity measures. Ref. [21] investigated matrix games with FPs and multi-objectives. Ref. [22] adopted minimax equilibria to examine zero-sum games with FPs. Ref. [23] presented an equilibrium to solve a zero-sum game with FPs. Ref. [24] investigated a zero-sum game with FPs in an asymmetric fuzzy environment. Refs. [24,25,26,27,28,29] presented an equilibrium in a zero-sum game with triangular FPs, interval FPs, triangular intuitionistic FPs, and trapezoidal intuitionistic FPs, respectively. Ref. [30] examined the PN equilibrium of matrix games with FPs based on possibility and necessity measures. Differing from the above works that focused on matrix games with risk-neutral players, our work concentrates on bimatrix games with risk-aversion players.

Here, we examine the impact of risk aversion on bimatrix games with symmetric triangular FPs (STFPs). Our work explores the relationship between the (α, β)-risk aversion Nash equilibrium ((α, β)-RANE) and mixed-strategy Nash equilibrium (MSNE) in bimatrix games with certain payoffs. In a bimatrix game with STFPs, let player 2 be the risk-averse one and player 1 be the risk-neutral one. We suppose that there exists the same support for the MSNE in the two cases—one with a risk-neutral player 2 and the other with a risk-averse player 2. This assumption has been used by [9]. As a result, player 2’s equilibrium strategy as well as player 1’s expected payoff cannot be changed. This implies the fact that a player in an MSNE is indifferent between the pure strategies played at a positive probability in a bimatrix game with certain payoffs. Finally, the impacts of risk aversion are explored in a 2 × 2 bimatrix game with STFPs.

The rest of our work is arranged as follows. In Section 2, related definitions as well as notations are reviewed briefly. In Section 3, we introduce the solution concept of the (α, β)-RANE, which is reviewed in bimatrix games with FPs, and explore the relationship between this equilibrium and the MSNE in a bimatrix game with certain payoffs. In Section 4, the impacts of risk aversion are explored in a 2 × 2 bimatrix game with STFPs. Section 5 concludes the work.

2. Preliminaries

For the vectors $e=(e_1,e_2,\dots ,e_n)\in R^n$ and $d=(d_1,d_2,\dots ,d_n)\in R^n$, we have $e\underset{\_}{\underset{\_}{>}}d$ (or $e>d$) if and only if $e_i\underset{\_}{\underset{\_}{>}}{d}_i$ (or $c_i>d_i$), where R^k is the Euclidian k-space and i = 1, 2, …, k.

Definition 1.

Let $\tilde{m}$ denote a fuzzy number and R denote the real number space; then, $\tilde{m}$ is a fuzzy subset F of R with a membership function, i.e., ${\mu }_{\tilde{m}}$: R→[0, 1], where the membership function satisfies the following three conditions:

(i) 

$\tilde{m}$ has a center that is denoted by a real number c, then ${\mu }_{\tilde{m}}(c)=1$;

(ii) 

${\mu }_{\tilde{m}}$ is upper semi-continuous and quasi-concave;

(iii) 

the support supp($\tilde{m}$) of $\tilde{m}$ is compact, where $Supp(\tilde{m})=\{x\in F|{\mu }_{\tilde{m}}(x)=1\}$.

Definition 2

([22]). Given the real number m as well as the positive number h, a fuzzy number is referred to as a symmetric triangular fuzzy number (STFN) if its membership function satisfies the following:

$${\mu }_{\tilde{m}}(x)=\{\begin{array}{ll}1-\frac{|x-m|}{h}& \mathit{for} x\in [m-h,m+h]\\ 0& \mathit{otherwise}\end{array}$$

(1)

where m is the center parameter of $\tilde{m}$ and h is the deviation parameter of $\tilde{m}$. Any STFN $\tilde{m}$ is denoted by $\tilde{m}={(m,h)}^T$.

Definition 3

([22]) . Given a real number and two STFNs, $\tilde{m}={(m,h)}^T$ and $\tilde{n}={(n,t)}^T$, then ${\mu }_{\tilde{m}+\tilde{n}}(x)=\sup \underset{x=u+v}{\min }\{{\mu }_{\tilde{m}}(u),{\mu }_{\tilde{n}}(v)\}$ and ${\mu }_{c\tilde{m}}(x)=\max \{0,\underset{x=cu}{\sup }{\mu }_{\tilde{m}}(u)\}$, where $\sup \{\varnothing \}=-\infty .$

Definition 4

([22]). Given an STFN $\tilde{m}={(m,h)}^T$ and a real number $\alpha \in (0,1]$, then the set ${[\tilde{m}]}^{\alpha }=\{x\in R|{\mu }_{\tilde{m}}(x)\ge \alpha \}$ is referred to as an α-cut of $\tilde{m}$. And ${[\tilde{m}]}^{\alpha }=[m_L^{\alpha },m_R^{\alpha }]$, where $m_L^{\alpha }=\inf {[\tilde{m}]}^{\alpha }$ and $m_R^{\alpha }=\sup {[\tilde{m}]}^{\alpha }$.

For any STFN $\tilde{m}={(m,h)}^T$, by using the operations of intervals that are introduced by [6], then

$$m_L^{\alpha }=m-(1-m)h \mathrm{and} m_R^{\alpha }=m+(1-\alpha )h.$$

(2)

Given the STFNs $\tilde{m}={(m,h)}^T$ and $\tilde{n}={(n,t)}^T$, by adopting the extension principles that are developed by [18,31], then $\tilde{m}+\tilde{n}=(m+n,h+t)$ and $c\tilde{m}=(cm,ch)$ if c > 0.

Definition 5

([32]) . Given the STFNs $\tilde{m}={(m,h)}^T$ and $\tilde{n}={(n,t)}^T$, then $\forall \alpha \in [0,1]$; therefore, there exist the following relationships between $\tilde{m}$ and $\tilde{n}$:

$$\tilde{m}\underset{\_}{\underset{\_}{\succ }}\tilde{n} \mathit{if} \mathit{and} \mathit{only} \mathit{if} {(m_L^{\alpha },m_R^{\alpha })}^T\underset{\_}{\underset{\_}{>}}{(n_L^{\alpha },n_R^{\alpha })}^T$$

(3)

$$\tilde{m}\succ \tilde{n} \mathit{if} \mathit{and} \mathit{only} \mathit{if} {(m_L^{\alpha },m_R^{\alpha })}^T>{(n_L^{\alpha },n_R^{\alpha })}^T.$$

(4)

To describe the inequality relationships between any two STFNs, [33] introduced two indices, as shown in Lemma 1. The reason why our work adopts the indices developed by [33] is that [33] has been cited by a large number of scholars. The number of citations amounts to 53 (see Google Scholar). And [33] has been cited by top journals in the fields of operations research and decision making, such as Fuzzy Sets and Systems and The European Journal of Operational Research. These have proven the effectiveness of the indices developed by [33,34].

Lemma 1

([33,34]). Given the STFNs $\tilde{m}={(m,h)}^T$ and $\tilde{n}={(n,t)}^T$, then

$$\tilde{m}\underset{\_}{\underset{\_}{\succ }}\tilde{n} (\mathrm{or} \tilde{m}\succ \tilde{n}) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} |h-t|\underset{\_}{\underset{\_}{<}}m-n (or |h-t|<m-n).$$

(5)

3. Fuzzy Bimatrix Games

For a bimatrix game with players I and J, the pure strategy set for player I is denoted by L′ = {1,2,…,l} and player J’s is denoted by K = {1, 2,…, k}. Then, player I’s mixed strategy set is as follows:

$$S_I=\{x={(x_1,x_2,\dots ,x_l)}^T\in R^l|x_i\ge 0,{\displaystyle \sum _{i=1}^l{x}_i}=1\}$$

and player I’s mixed strategy set is as follows:

$$S_J=\{y={(y_1,y_2,\dots ,y_k)}^T\in R^k|y_i\ge 0,{\displaystyle \sum _{i=1}^k{y}_i}=1\}$$

Let any two STFNs ${\tilde{m}}_{ij}={(m_{ij},h_{ij})}^T$ and ${\tilde{n}}_{ij}={(n_{ij},t_{ij})}^T$ denote the payoffs of both players when playing the pure strategy profile (i, j), respectively; then, $\tilde{M}={({\tilde{m}}_{ij})}_{l\times k}$ and $\tilde{N}={({\tilde{n}}_{ij})}_{l\times k}$ are the payoff matrices of players I and J, respectively. A bimatrix game with STFPs is denoted by $G_1=\langle \{I,J\},S_I\times S_J,\tilde{M},\tilde{N}\rangle$.

Definition 6

([20]). A strategy profile $(x^{*},y^{*})\in S_I\times S_J$ is referred to as an (α, β)-Nash equilibrium ((α, β)-NE) in G₁ if $(x^{*},y^{*})$ satisfies the following:

$$x^{*T}{M}_R^{\alpha }{y}^{*}=v\underset{\_}{\underset{\_}{>}}{x}^T{M}_R^{\alpha }{y}^{*}\hspace{1em}\forall x\in S_I,$$

(6)

$$x^{*T}{N}_R^{\beta }{y}^{*}=\eta \underset{\_}{\underset{\_}{>}}{x}^{*T}{N}_R^{\beta }y\hspace{1em}\forall y\in S_J,$$

(7)

and a point $(x^{*T}{M}_R^{\alpha }{y}^{*},x^{*T}{N}_R^{\beta }{y}^{*})$ is referred to as an ($\alpha ,\beta$)-equilibrium expected payoff vector.

Let G₃ denote $\langle \{I,J\},S_I\times S_J,M_R^{\alpha },N_R^{\beta }\rangle$.

Lemma 2 shows a sufficient and necessary condition (SNC) that there exists an (α, β)-NE in G₁.

Lemma 2

([20]). For any $\alpha ,\beta \in [0,1]$, in G₁ there exist at least one ($\alpha ,\beta$)-NE. Moreover, in G₁, $(x^{*},y^{*})\in S_I\times S_J$ is an (α, β)-NE if and only if there exists the optimal solution $(x^{*},y^{*},x^{*T}{M}_R^{\alpha }{y}^{*},x^{*T}{N}_R^{\beta }{y}^{*})$ in the programming model, as follows:

$$|\begin{array}{ll}{\max }_{x,y,v,\eta }& x(M_R^{\alpha }+N_R^{\beta })y-v-\eta \\ s.t.& M_R^{\alpha }y\underset{\_}{\underset{\_}{<}}v{1}^l\\ & N_R^{\beta T}x\underset{\_}{\underset{\_}{<}}\eta 1^k\\ & x\in S_I,y\in S_J,v,\eta \in R\end{array}$$

(8)

where $1^l=(\underset{l}{\underbrace{1,1,\cdots ,1}})\in R^l$ and $1^k=(\underset{k}{\underbrace{1,1,\cdots ,1}})\in R^k$. Let $v^{*}=x^{*T}{M}_R^{\alpha }{y}^{*}$ and ${\eta }^{*}=x^{*T}{N}_R^{\beta }{y}^{*}$.

From [35], we derive Lemma 3, which indicates the relationships between the (α, β)-NE in a bimatrix game with STFPs and the MSNE in a bimatrix game with certain payoffs.

Lemma 3.

Given any $\alpha ,\beta \in [0,1]$, $(x^{*},y^{*})\in S_I\times S_J$ is an (α, β)-NE in G₁ if and only if it is an MSNE in the game $G_2=\langle \{I,J\},S_I\times S_J,M_R^{\alpha },N_R^{\beta }\rangle$.

In many situations, however, players often show risk-averse preferences. To characterize such preferences, let φ be a strictly increasing and concave function on [min{$n_{ijR}^{\beta }$}, max{$n_{ijR}^{\beta }$}]. Then, the matrix with element φ ($n_{ijR}^{\beta }$) is denoted by φ ($N_R^{\beta }$). Therefore, there exists a new game $G_3=\langle \{I,J\},S_I\times S_J,M_R^{\alpha },\phi (N_R^{\beta })\rangle$. In G₃, player 2 is more risk averse compared with player 2 in G₂. Let (${\overline{x}}^{*}$, y*) denote the MSNE in G₃. To perform a meaningful comparison, there exists a same support for x* and ${\overline{x}}^{*}$, i.e., $x_1^{*}>0$ ⇔ ${\overline{x}}_1^{*}>0$.

Definition 7.

A strategy profile $({\overline{x}}^{*},y^{*})\in S_I\times S_J$ is referred to as an (α, β)-risk aversion NE ((α, β)-RANE) in G₃ if $({\overline{x}}^{*},y^{*})$ satisfies the following:

$${\overline{x}}^{*T}{M}_R^{\alpha }{y}^{*}=v\underset{\_}{\underset{\_}{>}}{\overline{x}}^T{M}_R^{\alpha }{y}^{*}\hspace{1em}\forall \overline{x}\in S_I,$$

(9)

$${\overline{x}}^{*T}\left(\phi (N_R^{\beta })\right)y^{*}=\eta \underset{\_}{\underset{\_}{>}}{\overline{x}}^{*T}\left(\phi (N_R^{\beta })\right)y\hspace{1em}\forall y\in S_J,$$

(10)

and a point $\left({\overline{x}}^{*T}{M}_R^{\alpha }{y}^{*},{\overline{x}}^{*T}\left(\phi (N_R^{\beta })\right)y^{*}\right)$ is referred to as an ($\alpha ,\beta$)-equilibrium expected payoff vector.

From Lemma 2, we can present an SNC that there exists an (α, β)-RANE in G₃.

Corollary 1.

For any $\alpha ,\beta \in [0,1]$ in G₁ with risk aversion, there exist at least one ($\alpha ,\beta$)-RANE. Moreover, in G₃,  $({\overline{x}}^{*},y^{*})\in S_I\times S_J$ is an (α, β)-RANE if and only if there exists the optimal solution $\left({\overline{x}}^{*},y^{*},{\overline{x}}^{*T}{M}_R^{\alpha }{y}^{*},{\overline{x}}^{*T}\left(\phi (N_R^{\beta })\right)y^{*}\right)$ in the programming model, as follows:

$$|\begin{array}{ll}{\max }_{\overline{x},y,v,\eta }& \overline{x}\left(M_R^{\alpha }+\left(\phi (N_R^{\beta })\right)\right)y-v-\eta \\ s.t.& M_R^{\alpha }y\underset{\_}{\underset{\_}{<}}v{1}^l\\ & N_R^{\beta T}\overline{x}\underset{\_}{\underset{\_}{<}}\eta 1^k\\ & \overline{x}\in S_I,y\in S_J,v,\eta \in R\end{array}$$

(11)

where $1^l=(\underset{l}{\underbrace{1,1,\cdots ,1}})\in R^l$ and $1^k=(\underset{k}{\underbrace{1,1,\cdots ,1}})\in R^k$. Let $v^{*}={\overline{x}}^{*T}{M}_R^{\alpha }{y}^{*}$ and ${\eta }^{*}={\overline{x}}^{*T}\left(\phi (N_R^{\beta })\right)y^{*}$.

Similar to Lemma 3, we need to indicate the relationships between the (α, β)-RANE in a bimatrix game with STFPs and the MSNE in a bimatrix game with risk aversion for player 2 and certain payoffs, as shown in Corollary 2.

Corollary 2.

Given any $\alpha ,\beta \in [0,1]$, $({\overline{x}}^{*},y^{*})\in S_I\times S_J$ is an (α, β)-RANE in G₁ with risk aversion if and only if it is the MSNE in the game $G_3=\langle \{I,J\},S_I\times S_J,M_R^{\alpha },\phi (N_R^{\beta })\rangle$.

4. The (α, β)-NE in 2 × 2 Fuzzy Bimatrix Games with Risk Aversion

When l = k = 2, given a bimatrix game with STFPs, $G_1=\langle \{I,J\},S_I\times S_J,\tilde{M},\tilde{N}\rangle$, where

$$\tilde{M}=\left[\begin{array}{cc}{\tilde{m}}_{11}& {\tilde{m}}_{12}\\ {\tilde{m}}_{21}& {\tilde{m}}_{22}\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\tilde{N}=\left[\begin{array}{cc}{\tilde{n}}_{11}& {\tilde{n}}_{12}\\ {\tilde{n}}_{21}& {\tilde{n}}_{22}\end{array}\right].$$

For i, j = 1, 2, let ${\tilde{m}}_{ij}={(m_{ij},h_{ij})}^T$ and ${\tilde{n}}_{ij}={(n_{ij},t_{ij})}^T$.

Given the real numbers $\alpha ,\beta \in [0,1]$, following Lemma 2, there exists the ($\alpha ,\beta$)-NE in G₁ by deriving the MSNE in the game $G_2=\langle \{I,J\},S_I\times S_J,M_R^{\alpha },N_R^{\beta }\rangle$, where $M_R^{\alpha }={(m_{ijR}^{\alpha })}_{l\times n}$ and ${\tilde{N}}_R^{\beta }={(n_{ijR}^{\beta })}_{l\times k}$. In G₂, there exists a mixed-strategy profile (x, y), where $x=(x_1,1-x_2)$ and $y=(y_1,1-y_2)$. Let (x*, y*) denote the MSNE in G₂, where $x^{*}=(x_1^{*},1-x_1^{*})$ and $y^{*}=(y_1^{*},1-y_1^{*})$.

When player 2 is risk averse, let φ be a strictly increasing and concave function on [min{$n_{ijR}^{\beta }$}, max{$n_{ijR}^{\beta }$}]. Then, the matrix with element φ ($n_{ijR}^{\beta }$) is denoted by φ ($N_R^{\beta }$). Therefore, there exists new game $G_3=\langle \{I,J\},S_I\times S_J,M_R^{\alpha },\phi (N_R^{\beta })\rangle$. Given the real numbers $\alpha ,\beta \in [0,1]$, following Corollary 1, there exists the ($\alpha ,\beta$)-RANE in G₁ by deriving the MSNE in the game $G_3=\langle \{I,J\},S_I\times S_J,M_R^{\alpha },\phi (N_R^{\beta })\rangle$. In G₃, there exists a mixed-strategy profile ($\overline{x}$, y), where $\overline{x}=({\overline{x}}_1,1-{\overline{x}}_1)$ and $y=(y_1,1-y_2)$. Let (${\overline{x}}^{*}$, y*) denote the MSNE in G₃, where ${\overline{x}}^{*}=({\overline{x}}_1^{*},1-{\overline{x}}_1^{*})$ and $y^{*}=(y_1^{*},1-y_1^{*})$.

Player 1 knows that player 2’s utility function depends on a concave and increasing transformation (i.e., player 2’s risk-aversion level). That is, one can regard the payoff matrices $N_R^{\beta }$ and φ ($N_R^{\beta }$) as utilities of monetary outcomes. The monetary outcomes of player 2 are known, but their utilities are not known by player 1.

To explore the impacts of risk aversion in G₁, following [35], for player J, there exist the following assumptions:

$$|t_{11}-t_{12}|\underset{\_}{\underset{\_}{<}}{n}_{11}-n_{12} (|t_{11}-t_{12}|<n_{11}-n_{12}),$$

(12)

$$|t_{22}-t_{21}|\underset{\_}{\underset{\_}{<}}{n}_{22}-n_{21} (|t_{22}-t_{21}|<n_{22}-n_{21})$$

(13)

and

$$|t_{11}-t_{22}|\underset{\_}{\underset{\_}{<}}{n}_{11}-n_{22} (|t_{11}-t_{22}|<n_{11}-n_{22}).$$

(14)

It follows from (19) and Lemma 1 that

$${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{n}_{12} ({\tilde{n}}_{11}\succ {\tilde{n}}_{12}).$$

(15)

From (20) and Lemma 1, we have that

$${\tilde{n}}_{22}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21} ({\tilde{n}}_{22}\succ {\tilde{n}}_{21}).$$

(16)

And from (21) and Lemma 1, we have that

$${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22} ({\tilde{n}}_{11}\succ {\tilde{n}}_{22}).$$

(17)

This leaves the following three exhaustive cases:

Case i.

$|t_{12}-t_{22}|\underset{\_}{\underset{\_}{<}}{n}_{12}-n_{22} (|t_{12}-t_{22}|<n_{12}-n_{22})$;

Case ii.

$|t_{22}-t_{12}|\underset{\_}{\underset{\_}{<}}{n}_{22}-n_{12} (|t_{22}-t_{12}|<n_{22}-n_{12})$ and $|t_{12}-t_{21}|\underset{\_}{\underset{\_}{<}}{n}_{12}-n_{21} (|t_{12}-t_{21}|<n_{12}-n_{21})$;

Case iii.

$|t_{21}-t_{12}|\underset{\_}{\underset{\_}{<}}{n}_{21}-n_{12} (|t_{21}-t_{12}|<n_{21}-n_{12})$.

Let (x*, y*) be an (α, β)-NE in G₁. Then, according to Lemma 2, (x*, y*) is the MSNE in G₂, where

$$x_1^{*}=\frac{n_{22R}^{\beta }-n_{21R}^{\beta }}{n_{11R}^{\beta }+n_{22R}^{\beta }-n_{12R}^{\beta }-n_{21R}^{\beta }}.$$

Theorem 1.

Let (x*, y*) be an (α, β)-NE in G₁ and an MSNE in G₂, and let (${\overline{x}}^{*}$, y*) be an (α, β)-RANE in G₁ with risk aversion and an MSNE in G₃. Then, player 2 benefits from risk aversion at (x*, y*) and (${\overline{x}}^{*}$, y*) if ${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{n}_{12}\underset{\_}{\underset{\_}{\succ }}{n}_{22} \underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21}({\tilde{n}}_{11}\succ {\tilde{n}}_{12}\succ {\tilde{n}}_{22}\succ {\tilde{n}}_{21})$.

Proof. 

For Case i, since $|t_{12}-t_{22}|\underset{\_}{\underset{\_}{<}}{n}_{12}-n_{22} (|t_{12}-t_{22}|<n_{12}-n_{22})$, we obtain the following from Lemma 1:

$${\tilde{n}}_{12}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22} ({\tilde{n}}_{12}\succ {\tilde{n}}_{22})$$

(18)

Combining (16)–(18), we then obtain the following:

$${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22} \underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21}({\tilde{n}}_{11}\succ {\tilde{n}}_{12}\succ {\tilde{n}}_{22}\succ {\tilde{n}}_{21}).$$

(19)

From Definition 5, it follows that

$$\begin{array}{l}{(n_{11L}^{\beta },n_{11R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{12L}^{\beta },n_{12R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{22L}^{\beta },n_{22R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{21L}^{\beta },n_{21R}^{\beta })}^T\\ ({(n_{11L}^{\beta },n_{11R}^{\beta })}^T>{(n_{12L}^{\beta },n_{12R}^{\beta })}^T>{(n_{22L}^{\beta },n_{22R}^{\beta })}^T>{(n_{21L}^{\beta },n_{21R}^{\beta })}^T).\end{array}$$

Thus

$$n_{11R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{12R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{22R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{21R}^{\beta }(n_{11R}^{\beta }>n_{21R}^{\beta }>n_{22R}^{\beta }>n_{12R}^{\beta }).$$

The function φ is normalized such that φ ($n_{11R}^{\beta }$) = $n_{11R}^{\beta }$ and φ ($n_{21R}^{\beta }$) = $n_{21R}^{\beta }$. Since φ is a strictly increasing and concave function, then ($n_{12R}^{\beta }$) ≥ $n_{12R}^{\beta }$ and φ ($n_{22R}^{\beta }$) ≥ $n_{22R}^{\beta }$. Then, for (${\overline{x}}^{*}$, y*) in G₃, we have

$${\overline{x}}_1^{*}=\frac{\phi (n_{22R}^{\beta })-n_{21R}^{\beta }}{n_{11R}^{\beta }+\phi (n_{22R}^{\beta })-\phi (n_{12R}^{\beta })-n_{21R}^{\beta }}\ge x_1^{*}.$$

(20)

Since $n_{11R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{12R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{22R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{21R}^{\beta }(n_{11R}^{\beta }>n_{21R}^{\beta }>n_{22R}^{\beta }>n_{12R}^{\beta })$, then

$$y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }\ge y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta }(y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }>y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta })$$

(21)

Thus, by (20) and (21), we obtain the following:

$$\begin{array}{ll}{\overline{x}}^{*}{N}_R^{\beta }{y}^{*}& ={\overline{x}}_1^{*}[y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta }]\\ & \ge x_1^{*}[y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta }]\\ & =x^{*}{N}_R^{\beta }{y}^{*}.\end{array}$$

(22)

This implies that player 2 benefits from risk aversion in Case i. □

Theorem 2.

Let (x*, y*) be the (α, β)-NE in G₁ and MSNE in G₂, and let (${\overline{x}}^{*}$, y*) be the (α, β)-RANE in G₁ with risk aversion and the MSNE in G₃. Then, player 2 benefits from risk aversion at (x*, y*) and (${\overline{x}}^{*}$, y*) if ${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12} \underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21}({\tilde{n}}_{11}\succ {\tilde{n}}_{22}\succ {\tilde{n}}_{12}\succ {\tilde{n}}_{21})$.

Proof. 

For Case ii, since $|t_{22}-t_{12}|\underset{\_}{\underset{\_}{<}}{n}_{22}-n_{12} (|t_{22}-t_{12}|<n_{22}-n_{12})$ and $|t_{12}-t_{21}|\underset{\_}{\underset{\_}{<}}{n}_{12}-n_{21} (|t_{12}-t_{21}|<n_{12}-n_{21})$, from Lemma 1, we obtain the following:

$${\tilde{n}}_{22}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12} ({\tilde{n}}_{22}\succ {\tilde{n}}_{12}) \mathrm{and} {\tilde{n}}_{12}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21} ({\tilde{n}}_{12}\succ {\tilde{n}}_{21})$$

(23)

Combining (16), (17), and (23), then

$${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12} \underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21}({\tilde{n}}_{11}\succ {\tilde{n}}_{22}\succ {\tilde{n}}_{12}\succ {\tilde{n}}_{21}).$$

(24)

From Definition 5, it follows that

$$\begin{array}{l}{(n_{11L}^{\beta },n_{11R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{22L}^{\beta },n_{22R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{12L}^{\beta },n_{12R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{21L}^{\beta },n_{21R}^{\beta })}^T\\ ({(n_{11L}^{\beta },n_{11R}^{\beta })}^T>{(n_{22L}^{\beta },n_{22R}^{\beta })}^T>{(n_{12L}^{\beta },n_{12R}^{\beta })}^T>{(n_{21L}^{\beta },n_{21R}^{\beta })}^T).\end{array}$$

Thus

$$n_{11R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{22R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{12R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{21R}^{\beta }(n_{11R}^{\beta }>n_{22R}^{\beta }>n_{12R}^{\beta }>n_{21R}^{\beta }).$$

The function φ is normalized such that φ ($n_{11R}^{\beta }$) = $n_{11R}^{\beta }$ and φ ($n_{21R}^{\beta }$) = $n_{21R}^{\beta }$. Since φ is a strictly increasing and concave function, then φ ($n_{12R}^{\beta }$) ≥ $n_{12R}^{\beta }$ and φ ($n_{22R}^{\beta }$) ≥ $n_{22R}^{\beta }$. Then, for (${\overline{x}}^{*}$, y*) in G₃, we have

$${\overline{x}}_1^{*}=\frac{\phi (n_{22R}^{\beta })-n_{21R}^{\beta }}{n_{11R}^{\beta }+\phi (n_{22R}^{\beta })-\phi (n_{12R}^{\beta })-n_{21R}^{\beta }}\ge x_1^{*}.$$

(25)

Since $n_{11R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{22R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{12R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{21R}^{\beta }(n_{11R}^{\beta }>n_{22R}^{\beta }>n_{21R}^{\beta }>n_{12R}^{\beta })$, then

$$y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }\ge y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta }(y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }>y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta })$$

(26)

Thus, from (25) and (26), then

$$\begin{array}{ll}{\overline{x}}^{*}{N}_R^{\beta }{y}^{*}& ={\overline{x}}_1^{*}[y_1^{*}{n}_{11R}^{\alpha }+(1-y_1^{*})n_{12R}^{\alpha }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\alpha }+(1-y_1^{*})n_{22R}^{\alpha }]\\ & \ge x_1^{*}[y_1^{*}{n}_{11R}^{\alpha }+(1-y_1^{*})n_{12R}^{\alpha }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\alpha }+(1-y_1^{*})n_{22R}^{\alpha }]\\ & =x^{*}{N}_R^{\beta }{y}^{*}.\end{array}$$

(27)

This implies that player 2 benefits from risk aversion in Case ii. □

Theorem 3.

Let (x*, y*) be an (α, β)-NE in G₁ and MSNE in G₂, and let (${\overline{x}}^{*}$, y*) be an (α, β)-RANE in G₁ with risk aversion and the MSNE in G₃.

(I) 

For the games G₂ and G₃, where φ (x) = x for all x ≤ $n_{22R}^{\beta }$ and $\phi (n_{11R}^{\beta })=n_{22R}^{\beta }+\frac{1}{2}(n_{11R}^{\beta }-n_{22R}^{\beta })$, then player 2 benefits from risk aversion at (x*, y*) and (${\overline{x}}^{*}$, y*) if ${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12} \underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21}({\tilde{n}}_{11}\succ {\tilde{n}}_{22}\succ {\tilde{n}}_{12}\succ {\tilde{n}}_{21})$;

(II) 

For the games G₂ and G₃, where φ (x) = x for all x ≥ $n_{21R}^{\beta }$ and $\phi (n_{12R}^{\beta })=n_{12R}^{\beta }-\frac{1}{2}(n_{21R}^{\beta }-n_{12R}^{\beta })$, then player 2 is hurt by risk aversion at (x*, y*) and (${\overline{x}}^{*}$, y*) if ${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12} \underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21}({\tilde{n}}_{11}\succ {\tilde{n}}_{22}\succ {\tilde{n}}_{12}\succ {\tilde{n}}_{21})$.

Proof. 

For Case iii, since $|t_{21}-t_{12}|\underset{\_}{\underset{\_}{<}}{n}_{21}-n_{12} (|t_{21}-t_{12}|<n_{21}-n_{12})$, by Lemma 1, we obtain the following:

$${\tilde{n}}_{21}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12} ({\tilde{n}}_{21}\succ {\tilde{n}}_{12})$$

(28)

Combining (16), (17), and (28), then

$${\tilde{n}}_{11}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{22} \underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{21}\underset{\_}{\underset{\_}{\succ }}{\tilde{n}}_{12}({\tilde{n}}_{11}\succ {\tilde{n}}_{22}\succ {\tilde{n}}_{21}\succ {\tilde{n}}_{12}).$$

(29)

From Definition 5, it follows that

$$\begin{array}{l}{(n_{11L}^{\beta },n_{11R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{22L}^{\beta },n_{22R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{21L}^{\beta },n_{21R}^{\beta })}^T\underset{\_}{\underset{\_}{>}}{(n_{12L}^{\beta },n_{12R}^{\beta })}^T\\ ({(n_{11L}^{\beta },n_{11R}^{\beta })}^T>{(n_{22L}^{\beta },n_{22R}^{\beta })}^T>{(n_{21L}^{\beta },n_{21R}^{\beta })}^T>{(n_{12L}^{\beta },n_{12R}^{\beta })}^T).\end{array}$$

Thus

$$n_{11R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{22R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{21R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{12R}^{\beta }(n_{11R}^{\beta }>n_{22R}^{\beta }>n_{21R}^{\beta }>n_{12R}^{\beta }).$$

For the function φ, two cases are shown as follows:

(i)

φ satisfies φ (x) = x for all x ≤ $n_{22R}^{\beta }$ and $\phi (n_{11R}^{\beta })=n_{22R}^{\beta }+\frac{1}{2}(n_{11R}^{\beta }-n_{22R}^{\beta })$;

(ii)

φ satisfies φ (x) = x for all x ≥ $n_{21R}^{\beta }$ and $\phi (n_{12R}^{\beta })=n_{12R}^{\beta }-\frac{1}{2}(n_{21R}^{\beta }-n_{12R}^{\beta })$.

For (i) and for (${\overline{x}}^{*}$, y*) in G₃, then

$${\overline{x}}_1^{*}=\frac{n_{22R}^{\beta }-n_{21R}^{\beta }}{n_{22R}^{\beta }+\frac{1}{2}(n_{11R}^{\beta }-n_{22R}^{\beta })+n_{22R}^{\beta }-n_{12R}^{\beta }-n_{21R}^{\beta }}>\frac{n_{22R}^{\beta }-n_{21R}^{\beta }}{n_{11R}^{\beta }+n_{22R}^{\beta }-n_{12R}^{\beta }-n_{21R}^{\beta }}=x_1^{*}.$$

(30)

For (ii) and for (${\overline{x}}^{*}$, y*) in G₃, then

$${\overline{x}}_1^{*}=\frac{n_{22R}^{\beta }-n_{21R}^{\beta }}{n_{11R}^{\beta }+n_{22R}^{\beta }-(n_{12R}^{\beta }-\frac{1}{2}(n_{21R}^{\beta }-n_{12R}^{\beta }))-n_{21R}^{\beta }}<\frac{n_{22R}^{\beta }-n_{21R}^{\beta }}{n_{11R}^{\beta }+n_{22R}^{\beta }-n_{12R}^{\beta }-n_{21R}^{\beta }}=x_1^{*}.$$

(31)

On the other hand, since $n_{11R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{22R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{21R}^{\beta }\underset{\_}{\underset{\_}{>}}{n}_{12R}^{\beta }(n_{11R}^{\beta }>n_{22R}^{\beta }>n_{21R}^{\beta }>n_{12R}^{\beta })$, the two cases are as follows:

(a)

$y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }\ge y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta }$($y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }>y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta }$);

(b)

$y_1^{*}{n}_{11R}^{\beta }+(1-y_1^{*})n_{12R}^{\beta }\le y_1^{*}{n}_{21R}^{\beta }+(1-y_1^{*})n_{22R}^{\beta }$.

On the other hand, for Case iii, player 2 benefits from risk aversion or is hurt by risk aversion, which depends on y*; $y_1^{*}$ is shown as follows:

$$y_1^{*}=\frac{m_{22R}^{\alpha }-m_{12R}^{\alpha }}{m_{11R}^{\alpha }+m_{22R}^{\alpha }-m_{21R}^{\alpha }-m_{12R}^{\alpha }}.$$

If $\frac{m_{22R}^{\alpha }-m_{12R}^{\alpha }}{m_{11R}^{\alpha }+m_{22R}^{\alpha }-m_{21R}^{\alpha }-m_{12R}^{\alpha }}\ge \frac{n_{22R}^{\beta }-n_{21R}^{\beta }}{n_{11R}^{\beta }+n_{22R}^{\beta }-n_{12R}^{\beta }-n_{21R}^{\beta }}$, then, by (30) and (a), we obtain the following:

$$\begin{array}{ll}{\overline{x}}^{*}{N}_R^{\beta }{y}^{*}& ={\overline{x}}_1^{*}[y_1^{*}{n}_{11R}^{\alpha }+(1-y_1^{*})n_{12R}^{\alpha }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\alpha }+(1-y_1^{*})n_{22R}^{\alpha }]\\ & \ge x_1^{*}[y_1^{*}{n}_{11R}^{\alpha }+(1-y_1^{*})n_{12R}^{\alpha }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\alpha }+(1-y_1^{*})n_{22R}^{\alpha }]\\ & =x^{*}{N}_R^{\beta }{y}^{*}.\end{array}$$

This implies that risk aversion is conducive to player 2.

If $\frac{m_{22R}^{\alpha }-m_{12R}^{\alpha }}{m_{11R}^{\alpha }+m_{22R}^{\alpha }-m_{21R}^{\alpha }-m_{12R}^{\alpha }}\le \frac{n_{22R}^{\beta }-n_{21R}^{\beta }}{n_{11R}^{\beta }+n_{22R}^{\beta }-n_{12R}^{\beta }-n_{21R}^{\beta }}$, then, by (31) and (b), we obtain the following:

$$\begin{array}{ll}{\overline{x}}^{*}{N}_R^{\beta }{y}^{*}& ={\overline{x}}_1^{*}[y_1^{*}{n}_{11R}^{\alpha }+(1-y_1^{*})n_{12R}^{\alpha }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\alpha }+(1-y_1^{*})n_{22R}^{\alpha }]\\ & \le x_1^{*}[y_1^{*}{n}_{11R}^{\alpha }+(1-y_1^{*})n_{12R}^{\alpha }]+(1-{\overline{x}}_1^{*})[y_1^{*}{n}_{21R}^{\alpha }+(1-y_1^{*})n_{22R}^{\alpha }]\\ & =x^{*}{N}_R^{\beta }{y}^{*},\end{array}$$

which means that player 2 is hurt by risk aversion. □

Although the method of reasoning in our work is similar to [35], one of the differences is that our work examines how risk aversion affects player 2 and not player 1, while [35] explores the impacts of loss aversion on both players. The reasons are as follows. To perform a meaningful comparison, we suppose that there exists the same support of the MSNE in the following two cases—one with a risk-neutral player 2 and the other with a risk-averse player 2. As result, player 2’s equilibrium strategy as well as player 1’s expected payoff cannot be changed. This implies the fact that a player in an MSNE is indifferent between the pure strategies played at a positive probability in a bimatrix game with certain payoffs.

5. Conclusions

In real-game problems, due to the complex relationships between differently strategic selections and their complicated impacts on the payoffs, a player tends to have no sufficient information about the payoffs of others, or even his/her own payoffs. To describe such a scenario, STFNs are used to characterize players’ uncertain payoffs. Moreover, given the fact that players represent risk-averse behaviors when facing gains, our work incorporates the risk-averse behaviors of players into bimatrix games with STFPs. An (α, β)-RANE is defined and its SNC is presented. Moreover, our work discusses the relationships between the (α, β)-RANE and MSNE in a bimatrix game with risk aversion for player 2 and certain payoffs. Finally, considering 2 × 2 bimatrix games with STFPs, our work explores how risk aversion affects player 2. We find the conditions that increase player 2’s risk-aversion level which hurts or benefits himself/herself. Future research may explore the impact of risk aversion on n-person strategic-form games with uncertain payoffs, where n > 2.