# Probabilistic Linguistic Matrix Game Based on Fuzzy Envelope and Prospect Theory with Its Application

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review and Research Gaps

#### 2.1. Fuzzy Matrix Games

#### 2.2. PLTSs

#### 2.3. Prospect Theory

#### 2.4. Research Gaps

- PLTS permits players to select multiple linguistic terms from a linguistic term set (LTS) and assign them with probabilities, which can describe players’ judgments more exactly [23]. A PLTS combines fuzziness, hesitancy and accurate information in a comprehensive form. Although PLTSs have been widely used by investigators [24,25,26,27,28] since they were proposed, probabilistic linguistic information is rarely used to depict the payoffs in matrix games. This is the first research gap intended to be narrowed.
- There are many defuzzification techniques [5,12,13,14,15,16,18,29] to address uncertain information. For matrix games with linguistic information, the semantics of linguistic terms may be lost by substituting symbolic computation for the operation of membership functions during computation [5]. The use of membership function is very important in the defuzzification of probabilistic linguistic information [6], but these methods [18,29] integrated the linguistic terms in the game without introducing the membership function. However, to the best of our knowledge, there is no research on the trapezoidal membership function of probabilistic linguistic information. This is the second research gap intended to be filled.
- Although PT has been extended into many fuzzy environments [6,31,32,33,34], most of them [31,32,33,34] are applied to solve MADM problems. The PT applied to solve a matrix game problem was first studied in [6]. However, as far as we know, there is no research on introducing PT under a probabilistic linguistic environment to solve a matrix game problem. This is the third research gap intended to be filled.

## 3. Methodology

#### 3.1. Preliminaries

#### 3.1.1. Related Concepts for TrFNs, HFLTSs and PLTSs

**Definition**

**1**

**.**A TrFN can be represented by a four-tuple $\tilde{e}=({e}_{1},{e}_{2},{e}_{3},{e}_{4})$, where ${e}_{i}$ $(i=1,2,3,4)$ are real numbers and meet ${e}_{1}\le {e}_{2}\le {e}_{3}\le {e}_{4}$. According to cut set theory, a TrFN $\tilde{e}=({e}_{1},{e}_{2},{e}_{3},{e}_{4})$ is equivalent to the following interval:

**Definition**

**2**

**.**Let $S=\{{s}_{0},{s}_{1},\cdots ,{s}_{\tau}\}$ be a linguistic term set. A HFLTS (denoted by ${H}_{S}$) is an ordered subset of the consecutive linguistic terms in $S$.

**Example**

**1.**

_{0}: extremely bad, s

_{1}: very bad, s

_{2}: bad, s

_{3}: medium, s

_{4}: good, s

_{5}: very good, s

_{6}: extremely good} be a linguistic term set. Then, ${H}_{S}^{1}=\left\{{s}_{2}\right\}$, ${H}_{S}^{2}=\{{s}_{0},{s}_{1},{s}_{2}\}$, ${H}_{S}^{3}=\{{s}_{3},{s}_{4},{s}_{5}\}$ and ${H}_{S}^{4}=\{{s}_{4},{s}_{5},{s}_{6}\}$ are four HFLTSs.

- ${T}_{{G}_{H}}({s}_{i})=\left\{{s}_{i}\right|{s}_{i}\in S\}$;
- ${T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{most}\text{}{s}_{i})=\left\{{s}_{j}\right|{s}_{j}\in S\text{}\mathrm{and}\text{}{s}_{j}\le {s}_{i}\}=\{{s}_{0},{s}_{1},\cdots ,{s}_{i-1},{s}_{i}\}$;
- ${T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{least}\text{}{s}_{i})=\left\{{s}_{j}\right|{s}_{j}\in S\text{}\mathrm{and}\text{}{s}_{j}\ge {s}_{i}\}=\{{s}_{i},{s}_{i+1},\cdots ,{s}_{\tau}\}$;
- ${T}_{{G}_{H}}(\mathrm{between}\text{}{s}_{i}\text{}\mathrm{and}\text{}{s}_{j})=\left\{{s}_{k}\right|{s}_{k}\in S\text{}\mathrm{and}\text{}{s}_{i}\le {s}_{k}\le {s}_{j}\}=\{{s}_{i},{s}_{i+1},\cdots ,{s}_{j-1},{s}_{j}\}$.

**Definition**

**3**

**.**Let $S=\{{s}_{0},{s}_{1},\cdots ,{s}_{\tau}\}$ be a linguistic term set. A PLTS is defined as

**Definition**

**4.**

- $P{T}_{{G}_{H}}({s}_{i})=\{{s}_{i}({p}_{i})|{s}_{i}\in S,0\le {p}_{i}\le 1\}$;
- $\begin{array}{ll}P{T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{most}\text{}{s}_{i})& =\{{s}_{j}({p}_{j})|{s}_{j}\in S,\text{}{s}_{j}\le \text{}{s}_{i},0\le {p}_{j}\le 1,{\displaystyle {\sum}_{j=0}^{i}{p}_{j}\le 1}\}\\ & =\{{s}_{0}({p}_{0}),\cdots ,{s}_{i-1}({p}_{i-1}),\text{}{s}_{i}({p}_{i})\}\end{array}$;
- $\begin{array}{ll}P{T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{least}\text{}{s}_{i})& =\{{s}_{j}({p}_{j})|{s}_{j}\in S,{s}_{j}\ge \text{}{s}_{i},0\le {p}_{j}\le 1,{\displaystyle {\sum}_{j=i}^{\tau}{p}_{j}\le 1}\}\\ & =\{{s}_{i}({p}_{i}),{s}_{i+1}\text{}({p}_{i+1})\text{},\cdots ,{s}_{\tau}({p}_{\tau})\}\end{array}$;
- $\begin{array}{ll}P{T}_{{G}_{H}}(\mathrm{between}\text{}{s}_{i}\text{}\mathrm{and}\text{}{s}_{j})& =\{{s}_{k}({p}_{k})|{s}_{k}\in S,\text{}{s}_{i}\le \text{}{s}_{k}\le \text{}{s}_{j},0\le {p}_{k}\le 1,{\displaystyle {\sum}_{k=i}^{j}{p}_{k}\le 1}\}\\ & =\{{s}_{i}({p}_{i}),\cdots ,{s}_{j}({p}_{j})\}\end{array}$.

#### 3.1.2. Ordered Weighted Averaging Operator

**Definition**

**5**

**.**Let $U=\{{u}_{1},{u}_{2},\cdots ,{u}_{n}\}$ be a set of arguments and ${a}_{k}$ be the k-th largest argument among set $U$. An ordered weighted averaging (OWA) operator can be defined as

**Definition**

**6**

**.**Let $\lambda \in [0,1]$ be a parameter. The first and second kinds of OWA weight vectors ${W}^{o}={({w}_{1}^{o},{w}_{2}^{o},\cdots ,{w}_{n}^{o})}^{T}$ and ${W}^{p}={({w}_{1}^{p},{w}_{2}^{p},\cdots ,{w}_{n}^{p})}^{T}$ are respectively defined as ${w}_{1}^{o}=\lambda ,\text{}{w}_{2}^{o}=\lambda (1-\lambda ),\cdots ,{w}_{k}^{o}=\lambda {(1-\lambda )}^{k-1},\cdots ,{w}_{n}^{o}={(1-\lambda )}^{n-1}$ and ${w}_{1}^{p}={\lambda}^{n-1},{w}_{2}^{p}=(1-\lambda ){\lambda}^{n-2},\cdots ,{w}_{k}^{p}=(1-\lambda ){\lambda}^{n-k},\cdots ,{w}_{n}^{p}=1-\lambda $.

#### 3.1.3. Prospect Theory

#### 3.2. Research Flow

## 4. Fuzzy Envelope and Cosine Similarity Measure for PLTSs

#### 4.1. An Improved Probabilistic Ordered Weighted Averaging Operator

**Definition**

**7.**

**Remark**

**1.**

**Remark**

**2.**

#### 4.2. A New Fuzzy Envelope for PLTSs

- The different probabilities of linguistic terms imply the different importance of such terms.
- Trapezoidal fuzzy membership function has strong ability to portray the fuzziness of the comparative linguistic terms.
- The parameters of the trapezoidal fuzzy membership function are calculated by an aggregation operator, which can embody the different importance of the linguistic terms in PLTS.

_{0}: extremely bad, s

_{1}: very bad, s

_{2}: bad, s

_{3}: medium, s

_{4}: good, s

_{5}: very good, s

_{6}: extremely good} be a LTS (i.e., $\tau =6$). Afterwards, S with its semantics depicted by triangular fuzzy membership function can be visually displayed in Figure 2 [43].

**Definition**

**8.**

- Fuzzy envelope for $P{T}_{{G}_{H}}({s}_{i})$. The parameters ${b}_{1}$, ${b}_{2}$, ${b}_{3}$ and ${b}_{4}$ are determined as $({b}_{1},{b}_{2},{b}_{3},{b}_{4})=({a}_{i}^{l},{a}_{i}^{m},{a}_{i}^{m},{a}_{i}^{u})$;
- Fuzzy envelope for $P{T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{most}\text{}{s}_{i})$. ${T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{most}\text{}{s}_{i})$ can be transformed into $\tilde{A}=\{{a}_{0}^{l},{a}_{0}^{m},{a}_{1}^{m},\cdots ,{a}_{i}^{m},{a}_{i}^{u}\}$. The parameters ${b}_{1}$, ${b}_{2}$, ${b}_{4}$ and ${b}_{3}$ are determined as ${b}_{1}={b}_{2}=\mathrm{min}\{{a}_{0}^{l},{a}_{0}^{m},{a}_{1}^{m},\cdots ,{a}_{i}^{m},{a}_{i}^{u}\}={a}_{0}^{l}$, ${b}_{4}=\mathrm{max}\{{a}_{0}^{l},{a}_{0}^{m},{a}_{1}^{m},\cdots ,{a}_{i}^{m},{a}_{i}^{u}\}={a}_{i}^{u}$ and$${b}_{3}=POW{A}_{{\widehat{W}}^{o}}({a}_{0}^{m},{a}_{1}^{m},\cdots ,{a}_{i}^{m})$$

**Theorem**

**1.**

- $0={a}_{0}^{m}\le {b}_{3}\le {a}_{i}^{m}\le 1$;
- For fixed${s}_{i}$and${W}^{o}$, if the probability of${a}_{i}^{m}$is closer to 1, then${b}_{3}$is closer to${a}_{i}^{m}$; if the probability of${a}_{0}^{m}$is closer to 1, then${b}_{3}$is closer to${a}_{0}^{m}$;
- Let${p}_{k}=r$$(k=0,1,2,\cdots ,i)$, where$0<r<1$. For a fixed${s}_{i}$, if$\lambda \to 0$, then${b}_{3}\to {a}_{0}^{m}$; if$\lambda \to 1$, then${b}_{3}\to {a}_{i}^{m}$.

**Proof of Theorem**

**1.**

- Since $\mathrm{min}\{{a}_{0}^{m},{a}_{1}^{m},\cdots ,{a}_{i}^{m}\}={a}_{0}^{m}=0$, $\mathrm{max}\{{a}_{0}^{m},{a}_{1}^{m},\cdots ,{a}_{i}^{m}\}={a}_{i}^{m}\le 1$ and ${b}_{3}$ is derived by the operator $POW{A}_{{\widehat{W}}^{o}}$, ${b}_{3}$ is between the minimum and maximum aggregated values (i.e., ${a}_{0}^{m}$ and ${a}_{i}^{m}$).
- For convenience, the probability of ${a}_{i}^{m}$ is denoted by ${p}_{0}$. For a fixed weight vector ${W}^{o}={({w}_{1}^{o},{w}_{2}^{o},\cdots ,{w}_{n}^{o})}^{T}$, the closer the value of ${p}_{0}$ is to 1, the larger the value of ${\widehat{w}}_{0}^{o}$, which will result in the value of ${\widehat{w}}_{0}^{o}{a}_{i}^{m}$ being closer to ${a}_{i}^{m}$. Hence, for fixed ${s}_{i}$ and ${W}^{o}$, if ${p}_{0}$ is closer to 1, then ${b}_{3}$ is closer to ${a}_{i}^{m}$. It can be deduced that the property “for fixed ${s}_{i}$ and ${W}^{o}$, if ${p}_{i+1}$ is closer to 1, then ${b}_{3}$ is closer to ${a}_{0}^{m}$” also holds.
- Since ${p}_{k}=r$ $(k=0,1,2,\cdots ,i)$, where $0<r<1$, it holds that ${\widehat{w}}_{k}^{o}={w}_{k}^{o}$. In this case, if $\lambda \to 0$, then ${\widehat{w}}_{1}^{o}\to 0,{\widehat{w}}_{2}^{o}\to 0,\cdots ,{\widehat{w}}_{i}^{o}\to 0$ and ${\widehat{w}}_{i+1}^{o}\to 1$, which indicates that ${b}_{3}\to {\widehat{w}}_{i+1}^{o}{a}_{0}^{m}\to {a}_{0}^{m}$. The property “if $\lambda \to 1$, then ${b}_{3}\to {a}_{i}^{m}$” can be proven similarly.

- Fuzzy envelope for $P{T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{least}\text{}{s}_{i})$. ${T}_{{G}_{H}}(\mathrm{at}\text{}\mathrm{least}\text{}{s}_{i})$ can be transformed into $\tilde{A}=\{{a}_{i}^{l},{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{\tau}^{m},{a}_{\tau}^{u}\}$. The parameters ${b}_{1}$, ${b}_{3}$, ${b}_{4}$ and ${b}_{2}$ are determined as ${b}_{1}=\mathrm{min}\{{a}_{i}^{l},{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{\tau}^{m},{a}_{\tau}^{u}\}={a}_{i}^{l}$, ${b}_{3}={b}_{4}=\mathrm{max}\{{a}_{i}^{l},{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{\tau}^{m},{a}_{\tau}^{u}\}={a}_{\tau}^{u}$,$${b}_{2}=POW{A}_{{\widehat{W}}^{p}}({a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{\tau}^{m})$$

**Theorem**

**2.**

- $0\le {a}_{i}^{m}\le {b}_{2}\le {a}_{\tau}^{m}=1$;
- For fixed${s}_{i}$and${W}^{p}$, if the probability of${a}_{\tau}^{m}$is closer to 1, then ${b}_{2}$ is closer to ${a}_{\tau}^{m}$ ; if the probability of ${a}_{i}^{m}$ is closer to 1, then ${b}_{2}$ is closer to ${a}_{i}^{m}$;
- Let${p}_{k}=r$$(k=0,1,2,\cdots ,\tau -i)$, where$0<r<1$. For a fixed${s}_{i}$, if$\lambda \to 0$, then${b}_{2}\to {a}_{i}^{m}$; if$\lambda \to 1$, then${b}_{2}\to {a}_{\tau}^{m}$.

**Proof of Theorem**

**2.**

- Since $\mathrm{min}\{{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{\tau}^{m}\}={a}_{i}^{m}\ge 0$, $\mathrm{max}\{{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{\tau}^{m}\}={a}_{\tau}^{m}=1$ and ${b}_{2}$ is obtained by the $POW{A}_{{\widehat{W}}^{p}}$ operator, ${b}_{2}$ is between the minimum and maximum aggregated values.
- For convenience, the probability of ${a}_{\tau}^{m}$ is denoted by ${p}_{0}$. For a fixed weight vector ${W}^{p}={({w}_{1}^{p},{w}_{2}^{p},\cdots ,{w}_{n}^{p})}^{T}$, the closer the value of ${p}_{0}$ is to 1, the larger the value of ${\widehat{w}}_{0}^{p}$, which will cause the value of ${\widehat{w}}_{0}^{p}{a}_{\tau}^{m}$ to be closer to ${a}_{\tau}^{m}$. Hence, for fixed ${s}_{i}$ and ${W}^{p}$, if ${p}_{0}$ is closer to 1, then ${b}_{2}$ is closer to ${a}_{\tau}^{m}$. It also can be inferred that the property “for fixed ${s}_{i}$ and ${W}^{p}$, if ${p}_{\tau -i+1}$ is closer to 1, then ${b}_{2}$ is closer to ${a}_{i}^{m}$”.
- Since ${p}_{k}=r$ $(k=0,1,2,\cdots ,\tau -i)$, where $0<r<1$, it holds that ${\widehat{w}}_{k}^{p}={w}_{k}^{p}$. In this case, if $\lambda \to 0$, then ${\widehat{w}}_{1}^{p}\to 0,{\widehat{w}}_{2}^{p}\to 0,\cdots ,{\widehat{w}}_{\tau -i}^{p}\to 0$ and ${\widehat{w}}_{\tau -i+1}^{p}\to 1$, which shows that ${b}_{2}\to {\widehat{w}}_{\tau -i+1}^{p}{a}_{i}^{m}\to {a}_{i}^{m}$. The property “if $\lambda \to 1$, then ${b}_{2}\to {a}_{\tau}^{m}$” can also be proven.

- Fuzzy envelope of $P{T}_{{G}_{H}}(\mathrm{between}\text{}{s}_{i}\text{}\mathrm{and}\text{}{s}_{j})$. ${T}_{{G}_{H}}(\mathrm{between}\text{}{s}_{i}\text{}\mathrm{and}\text{}{s}_{j})$ can be transformed into $\tilde{A}=\{{a}_{i}^{l},{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{j}^{m},{a}_{j}^{u}\}$. The parameters ${b}_{1}$ and ${b}_{4}$ are determined as ${b}_{1}=\mathrm{min}\{{a}_{i}^{l},{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{j}^{m},{a}_{j}^{u}\}={a}_{i}^{l}$ and ${b}_{4}=\mathrm{max}\{{a}_{i}^{l},{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{j}^{m},{a}_{j}^{u}\}={a}_{j}^{u}$; For determination for the parameters ${b}_{2}$ and ${b}_{3}$, it is required to consider the parity of $i+j$.
- (i)
- If $i+j$ is odd, then$${b}_{2}=POW{A}_{{\widehat{W}}^{p}}({a}_{i}^{m},{a}_{i+1}^{m}\cdots ,{a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m})\text{}\mathrm{and}\text{}{b}_{3}=POW{A}_{{\widehat{W}}^{o}}({a}_{{\scriptscriptstyle \frac{i+j+1}{2}}}^{m},{a}_{{\scriptscriptstyle \frac{i+j+1}{2}}+1}^{m}\cdots ,{a}_{j}^{m})$$
- (ii)
- If $i+j$ is even, then$${b}_{2}=POW{A}_{{\widehat{W}}^{p}}({a}_{i}^{m},{a}_{i+1}^{m}\cdots ,{a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m})\text{}\mathrm{and}\text{}{b}_{3}=POW{A}_{{\widehat{W}}^{o}}({a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m},{a}_{{\scriptscriptstyle \frac{i+j}{2}}+1}^{m}\cdots ,{a}_{j}^{m})$$

**Theorem**

**3.**

- ${a}_{i}^{m}\le {b}_{2}\le {b}_{3}\le {a}_{j}^{m}$;
- For fixed${s}_{i}$,${s}_{j}$,${W}^{o}$and${W}^{p}$, it holds that
- (i)
- If the probability of${a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$(or${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$) is closer to 1, then ${b}_{2}$ is closer to ${a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$ (or ${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$);
- (ii)
- If the probability of${a}_{i}^{m}$is closer to 1, then ${b}_{2}$ is closer to ${a}_{i}^{m}$;
- (iii)
- If the probability of${a}_{{\scriptscriptstyle \frac{i+j+1}{2}}}^{m}$(or${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$) is closer to 1, then${b}_{3}$is closer to${a}_{{\scriptscriptstyle \frac{i+j+1}{2}}}^{m}$(or${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$);
- (iv)
- If the probability of${a}_{j}^{m}$is closer to 1, then ${b}_{3}$ is closer to ${a}_{j}^{m}$;

- For fixed${s}_{i}$and${s}_{j}$, if both arguments${a}_{i}^{m},{a}_{i+1}^{m}\cdots ,{a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$(or${a}_{i}^{m},{a}_{i+1}^{m}\cdots ,{a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$) and arguments${a}_{{\scriptscriptstyle \frac{i+j+1}{2}}}^{m},{a}_{{\scriptscriptstyle \frac{i+j+1}{2}}+1}^{m}\cdots ,{a}_{j}^{m}$(or${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m},{a}_{{\scriptscriptstyle \frac{i+j}{2}}+1}^{m}\cdots ,{a}_{j}^{m}$) have the same probabilities, respectively, then it holds that
- (i)
- If$\lambda \to 0$, then${b}_{2}\to {a}_{i}^{m}$; if$\lambda \to 1$, then${b}_{2}\to {a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$(or${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$);
- (ii)
- If$\lambda \to 0$, then${b}_{3}\to {a}_{{\scriptscriptstyle \frac{i+j+1}{2}}}^{m}$(or${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$); if$\lambda \to 1$, then${b}_{3}\to {a}_{j}^{m}$.

**Proof of Theorem**

**3.**

- Since $\mathrm{min}\{{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{j}^{m}\}={a}_{i}^{m}\ge 0$, $\mathrm{max}\{{a}_{i}^{m},{a}_{i+1}^{m},\cdots ,{a}_{j}^{m}\}={a}_{j}^{m}\le 1$ and ${b}_{2}$ is obtained by the $POW{A}_{{\widehat{W}}^{p}}$ operator, ${b}_{3}$ is obtained by the $POW{A}_{{\widehat{W}}^{o}}$ operator, ${b}_{2}$ and ${b}_{3}$ are between the minimum and maximum aggregated values.
- For convenience, the probability of ${a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$ (or ${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$) is denoted by ${p}_{0}$. For fixed weight vectors ${W}^{p}$, the closer the value of ${p}_{0}$ is to 1, the larger the value of ${\widehat{w}}_{0}^{p}$, which will cause the value of ${\widehat{w}}_{0}^{p}{a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$(or ${\widehat{w}}_{0}^{p}{a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$) to be closer to ${a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$ (or ${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$). Hence, for fixed ${s}_{i}$, ${s}_{j}$ and ${W}^{p}$, if ${p}_{0}$ is closer to 1, then ${b}_{2}$ is closer to ${a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$ (or ${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$). Similarly, (ii)–(iv) also can be proven.
- Since all the arguments have the same probabilities, if $\lambda \to 0$, then ${\widehat{w}}_{1}^{p}\to 0,$ ${\widehat{w}}_{2}^{p}\to 0,\cdots ,{\widehat{w}}_{\frac{j-i-1}{2}}^{p}(or\text{}{\widehat{w}}_{\frac{j-i}{2}}^{p})\to 0$ and ${\widehat{w}}_{\frac{j-i+1}{2}}^{p}\text{}(or\text{}{\widehat{w}}_{\frac{j-i+2}{2}}^{p})\to 1$, which shows that ${b}_{2}\to {\widehat{w}}_{\frac{j-i+1}{2}}^{p}{a}_{i}^{m}\text{}(or\text{}{\widehat{w}}_{\frac{j-i+2}{2}}^{p}{a}_{i}^{m})\to {a}_{i}^{m}$. If $\lambda \to 1$, then ${\widehat{w}}_{1}^{p}\to 1,{\widehat{w}}_{2}^{p}\to 0,\cdots ,$ ${\widehat{w}}_{\frac{j-i-1}{2}}^{p}(or\text{}{\widehat{w}}_{\frac{j-i}{2}}^{p})\to 0$, ${\widehat{w}}_{\frac{j-i+1}{2}}^{p}\text{}(or\text{}{\widehat{w}}_{\frac{j-i+2}{2}}^{p})\to 0$, which indicates that ${b}_{2}\to {\widehat{w}}_{1}^{p}{a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$ (or ${\widehat{w}}_{1}^{p}{a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$)$\to {a}_{{\scriptscriptstyle \frac{i+j-1}{2}}}^{m}$ (or ${a}_{{\scriptscriptstyle \frac{i+j}{2}}}^{m}$). (ii) can be also proven similarly.

**Example**

**2.**

- Fuzzy envelope for PLTS$P{L}_{S}^{1}=\{{s}_{1}(1)\}$can be obtained as$env(P{L}_{S}^{1})=Tr(0,0.17,0.17,0.33)$.
- Fuzzy envelope for PLTS$P{L}_{S}^{2}=\{{s}_{0}(0.2),{s}_{1}(0.3),{s}_{2}(0.4)\}$is obtained as follows:${b}_{1}={b}_{2}=\mathrm{min}\{{a}_{0}^{l},{a}_{0}^{m},{a}_{1}^{m},{a}_{2}^{m},{a}_{2}^{u}\}={a}_{0}^{l}=0$,${b}_{4}=\mathrm{max}\{{a}_{0}^{l},{a}_{0}^{m},{a}_{1}^{m},{a}_{2}^{m},{a}_{2}^{u}\}={a}_{2}^{u}=0.5$. Since${s}_{i}={s}_{2}$, according to [43], $\lambda =2/6$, then$${b}_{3}=\frac{1}{\frac{2}{6}\times 0.4+\frac{2}{6}(1-\frac{2}{6})\times 0.3+{(1-\frac{2}{6})}^{2}\times 0.2}(\frac{2}{6}\times 0.4\times {a}_{2}^{m}+\frac{2}{6}(1-\frac{2}{6})\times 0.3\times {a}_{1}^{m}+{(1-\frac{2}{6})}^{2}\times 0.2\times {a}_{0}^{m})=0.192.$$
- Fuzzy envelope for PLTS$P{L}_{S}^{3}=\{{s}_{4}(0.5),{s}_{5}(0.2),{s}_{6}(0.3)\}$is obtained as follows:${b}_{1}=\mathrm{min}\{{a}_{4}^{l},{a}_{4}^{m},{a}_{5}^{m},{a}_{6}^{m},{a}_{6}^{u}\}={a}_{4}^{l}=0.5$,${b}_{3}={b}_{4}=\mathrm{max}\{{a}_{4}^{l},{a}_{4}^{m},{a}_{5}^{m},{a}_{6}^{m},{a}_{6}^{u}\}={a}_{6}^{u}=1$. Since${s}_{i}={s}_{4}$, according to [43], $\lambda =4/6$, then$${b}_{2}=\frac{1}{{(\frac{4}{6})}^{2}\times 0.3+\frac{4}{6}(1-\frac{4}{6})\times 0.2+(1-\frac{4}{6})\times 0.5}({(\frac{4}{6})}^{2}\times 0.3\times {a}_{6}^{m}+\frac{4}{6}(1-\frac{4}{6})\times 0.2\times {a}_{5}^{m}+(1-\frac{4}{6})\times 0.5\times {a}_{4}^{m})=0.818.$$
- Fuzzy envelope for PLTS$P{L}_{S}^{4}=\{{s}_{3}(0.2),{s}_{4}(0.3),{s}_{5}(0.5)\}$is obtained as follows:${b}_{1}=\mathrm{min}\{{a}_{3}^{l},{a}_{3}^{m},{a}_{4}^{m},{a}_{5}^{m},{a}_{5}^{u}\}={a}_{3}^{l}=0.33$,${b}_{4}=\mathrm{max}\{{a}_{3}^{l},{a}_{3}^{m},{a}_{4}^{m},{a}_{5}^{m},{a}_{5}^{u}\}={a}_{5}^{u}=1$. Since${s}_{i}={s}_{3}$and${s}_{j}={s}_{5}$, according to [43], ${\lambda}_{1}=4/5$ (for determining ${b}_{2}$) and ${\lambda}_{2}=1/5$ (for determining ${b}_{3}$), then ${b}_{2}=\frac{1}{\frac{4}{5}\times 0.3+(1-\frac{4}{5})\times 0.2}(\frac{4}{5}\times 0.3\times {a}_{4}^{m}+(1-\frac{4}{5})\times 0.2\times {a}_{3}^{m})=0.646$ and ${b}_{3}=\frac{1}{\frac{1}{5}\times 0.5+(1-\frac{1}{5})\times 0.3}(\frac{1}{5}\times 0.5\times {a}_{5}^{m}+(1-\frac{1}{5})\times 0.3\times {a}_{4}^{m})=0.717$.

#### 4.3. A New Cosine Similarity Measure for PLTSs

**Definition**

**9**

**.**Let ${S}^{\prime}=\{{s}_{-\tau},\cdots ,{s}_{-1},{s}_{0},{s}_{1},\cdots ,{s}_{\tau}\}$ be a subscript-symmetric linguistic term set. Given any two HFLTSs ${H}_{{S}^{\prime}}^{i}=\left\{{s}_{{\gamma}_{l}^{i}}\right|{s}_{{\gamma}_{l}^{i}}\in {S}^{\prime},\text{}l=1,2,\cdots ,\#{H}_{{S}^{\prime}}^{i}\}$ $(i=1,2)$, the cosine similarity measure between them is formulated as

**Definition**

**10.**

**Example**

**3.**

## 5. A Probabilistic Linguistic Matrix Game Based on Fuzzy Envelope and PT

#### 5.1. Model Formulation

#### 5.2. Solving the Models

- (i)
- Define the reference point

- (ii)
- Calculate the gain and loss of $T{r}_{ij}$ on the negative and positive ideal PLTS

- (iii)
- Compute the overall prospect value of payoff value at situation $({\xi}_{i},{\varsigma}_{j})$ given by player ${P}_{I}$

#### 5.3. Framework

## 6. Results

#### 6.1. An Example from the Development Strategy of SNNR

_{0}: extremely poor (EP), s

_{1}: very poor (VP), s

_{2}: poor (P), s

_{3}: medium (M), s

_{4}: good (G), s

_{5}: very good (VG), s

_{6}: extremely good (EG)}be a LTS. The payoff matrix values with probabilistic linguistic information are shown in Table 1. These evaluation values in the payoff matrix are given by the invited team of senior experts after field research, consulting relevant historical materials and combining the current national policies.

#### 6.2. Solving the Case by the Proposed Method

#### 6.3. Sensitivity Analysis for the Parameter $\eta $

- According to Table 2 and Figure 5, Figure 6 and Figure 7, we can receive the following observations:
- (i)
- The mixed strategies, gain-floor and loss-ceiling for players will change with the change of the parameter $\eta $, which reflects the flexibility of the proposed method.
- (ii)
- For the management department, when $\eta $ takes a value between 0.3 and 1, the probability ranking of each strategy in the selected mixed strategy keeps constant totally, that is, ${\xi}_{5}\ge {\xi}_{4}\ge {\xi}_{1}\ge {\xi}_{3}\ge {\xi}_{2}$. The stability of the probability ranking shows that the management department should put ${\xi}_{5}$: Livestock farming in the first place and ${\xi}_{2}$: Manufacturing should be the last consideration when formulating the development strategy for SNNR.
- (iii)
- For Nature, no matter how the parameter $\eta $ changes, Nature should put ${\varsigma}_{2}$: Biological diversity in first place since the probability of strategy ${\varsigma}_{2}$: Biological diversity in the selected mixed strategy is always greater than 0.5.
- (iv)
- The gain-floor and loss-ceiling for players are equal, whichis consistent with the results obtained in [13], and the values of these increase with the increase of the parameter $\eta $.

- Strategic interventions

#### 6.4. Comparative Analyses

#### 6.4.1. Comparison with the Method without Considering Players’ Psychological Factor

- For player ${P}_{I}$, the probability ranking of each pure strategy in the selected mixed strategy ${\mathit{y}}^{\ast}={(0.2183,0,0.1429,0.6321,0.0068)}^{T}$ obtained by the method [13] is ${\xi}_{4}\ge {\xi}_{1}\ge {\xi}_{3}\ge {\xi}_{5}\ge {\xi}_{2}$, which is totally different from the probability ranking obtained by the proposed method (see Table 2 and Figure 5). That is to say, the probability ranking seems to change markedly if the game process does not include the psychological behavior of players. In addition, the probability of ${\xi}_{5}$: Livestock farming is largest when formulating the development strategy for SNNR, which is more in line with reality.
- For player ${P}_{II}$, the probability ranking of each pure strategy in the selected mixed strategy ${\mathit{z}}^{\ast}={(0.2839,0.5053,0.0525,0.1582)}^{T}$ obtained by the method [13] is ${\varsigma}_{2}\ge {\varsigma}_{1}\ge {\varsigma}_{4}\ge {\varsigma}_{3}$, which is slightly different from the probability ranking obtained by the proposed method (see Table 2 and Figure 6). Although the pure strategies with the largest probability obtained by the two methods are the same (i.e., ${\varsigma}_{2}$: Biological diversity), if the psychological behavior of the players without considering in the game process, the ranking of probability will change.
- According to Table 2 and Figure 7, the obtained gain-floor of player ${P}_{I}$ and the loss-ceiling of player ${P}_{II}$ by the method [13] are less than those that acquired by the proposed method when the parameter $\eta $ is not smaller than 0.8. Besides, the proposed method is also more flexible due to the consideration of players’ risk attitude.

#### 6.4.2. Comparison with Triangular Fuzzy Envelope

- It can be seen from Table 3 and Figure 8 that the probability ranking obtained by the triangular fuzzy envelope method is completely different from that obtained by trapezoidal fuzzy envelope method. Moreover, the pure strategy with the highest probability is ${\xi}_{4}$: Planting, and the probabilities of selecting pure strategies ${\xi}_{2}$: Manufacturing and ${\xi}_{5}$: Livestock farming are equal to 0. The probability of ${\xi}_{3}$: Tourism is also equal to 0 when the parameter $\eta \ge 0.5$, which appears to be inconsistent with reality.
- According to Table 3 and Figure 9, the probability ranking of each pure strategy in the selected mixed strategy ${\mathit{z}}^{\ast}$ is ${\varsigma}_{2}>{\varsigma}_{1}>{\varsigma}_{4}>{\varsigma}_{3}$ when $\eta <0.5$. The probability ranking is ${\varsigma}_{2}>{\varsigma}_{1}>{\varsigma}_{4}={\varsigma}_{3}$ when $\eta \ge 0.5$. The pure strategies with the largest probability obtained by the two methods are ${\varsigma}_{2}$: Biological diversity, which is in line with the concept of sustainable development. However, as shown in Figure 9, the probability of selecting pure strategy ${\varsigma}_{3}$: Capacity of water storage is always equal to 0 no matter how the parameter $\eta $ changes, and the probability of ${\varsigma}_{4}$: Lakes and wetland area is also equal to 0 when $\eta \ge 0.5$, which does not conform to the actual situation evidently.
- In the light of Table 3 and Figure 10, we can find that when $\eta $ varies from 0 to 1, the variation tendency of the gain-floor ${U}_{2}^{\ast}$ of player ${P}_{I}$ obtained by the triangular fuzzy envelope method [5] is consistent with that obtained by the proposed method in this paper. However, when $0\le \eta \le 0.6$, the gain-floor ${U}^{\ast}$ (loss-ceiling ${V}^{\ast}$) is smaller than the gain-floor ${U}_{2}^{\ast}$ (loss-ceiling ${V}_{2}^{\ast}$). When $0.7\le \eta \le 1$, the gain-floor ${U}^{\ast}$ is greater than the gain-floor ${U}_{2}^{\ast}$. As mentioned earlier, the larger the value of the parameter $\eta $, the more optimistic the player, the better the result will be, that is, the greater the gain-floor and loss-ceiling. In reality, most players usually tend to ponder and solve the problem with an optimistic attitude. Hence, the greater the value of $\eta $, the higher the importance of the negative ideal PLTS, and the result obtained by the proposed method is better than the triangular fuzzy envelope method. The proposed method is more applicable for a situation in which the players are optimistic.

#### 6.4.3. Comparison with Hesitant Fuzzy Linguistic Information

- From Table 4, Figure 11 and Figure 12, the probability rankings of each pure strategy in the selected mixed strategies for player ${P}_{I}$ obtained by the HFLTSs and PLTSs are almost the same. The probability rankings of each pure strategy in the selected mixed strategies for player ${P}_{II}$ obtained by the HFLTSs and PLTSs are exactly the same. This seems to indicate the effectiveness of the proposed method in this paper.
- According to Table 4 and Figure 13, it is not hard to discover that the variation tendency of the gain-floor ${U}_{3}^{\ast}$ of player ${P}_{I}$ obtained by the HFLTSs with the parameter $\eta $ varying from 0 to 1 is also consistent with that obtained by the proposed method in this paper. However, the gain-floor ${U}^{\ast}$ (loss-ceiling ${V}^{\ast}$) is always larger than the gain-floor ${U}_{3}^{\ast}$ (loss-ceiling ${V}_{3}^{\ast}$), which reveals the superiority of the method proposed in this paper.

## 7. Conclusions

- From the perspective of decision-maker and Nature, we propose a new PLMG method to solve decision-making problems. In order to defuzzify the probabilistic linguistic information, this paper proposes a fuzzy envelope of PLTS by using a trapezoidal fuzzy membership function. The parameters of the trapezoidal fuzzy membership function are decided by applying the improved POWA operator. The proposal of the improved POWA operator makes it unnecessary to normalize the PLTS in advance when determining the fuzzy envelope of the PLTS. Therefore, the new fuzzy envelope has a strong ability in polymerizing the original linguistic terms and avoiding the loss of the initial information.
- Since each player has a different perception of gain and loss, for depicting the psychological behavior of decision-makers regarding losses and gains, the PT is creatively introduced into the PLMG method based on the predefined cosine distance measure. By comparing with the method without considering psychological factors, it is confirmed that the player’s psychological behavior does lead to different game results, which is consistent with reality. Thus, it is necessary to incorporate the psychological behavior of players into the actual game process.
- The sensitivity analysis and comparative analyses with other methods indicate the flexibility and superiority of the proposed method. A DSS is developed based on the proposed method to illustrate its practical value.

**Remark**

**3.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Gain-floor ${U}^{\ast}$ and loss-ceiling ${V}^{\ast}$ with parameter $\eta $ varying from 0 to 1.

**Figure 10.**Gain-floors ${U}^{\ast}$ and ${U}_{2}^{\ast}$ with parameter $\eta $ varying from 0 to 1.

**Figure 13.**The gain-floor ${U}^{\ast}$ and ${U}_{3}^{\ast}$ with parameter $\eta $ varying from 0 to 1.

Payoff Values | ${\mathit{\varsigma}}_{1}$ | ${\mathit{\varsigma}}_{2}$ | ${\mathit{\varsigma}}_{3}$ | ${\mathit{\varsigma}}_{4}$ |
---|---|---|---|---|

${\xi}_{1}$ | 20%EP, 30%VP, 40%P | 50%G, 20%VG, 30%EG | 30%M, 50%G, 20%VG | VG |

${\xi}_{2}$ | 10%M, 40%G, 50%VG | 30%VP, 70%P | 50%P, 10%M, 20%G | 30%P, 40%M, 30%G |

${\xi}_{3}$ | P | 40%G, 50%VG, 10%EG | 10%G, 90%VG | 10%VP, 30%P, 50%M |

${\xi}_{4}$ | 10%G, 20%VG, 70%EG | 20%M, 30%G, 30%VG | EG | 40%G, 60%VG |

${\xi}_{5}$ | 60%M, 40%G | M | 40%P, 60%M | 20%M, 80%G |

$\mathit{\eta}$ | ${\mathit{y}}^{\ast}$ | ${\mathit{z}}^{\ast}$ | ${\mathit{U}}^{*}$ | ${\mathit{V}}^{*}$ | $\mathbf{E}({\mathit{y}}^{\ast},{\mathit{z}}^{\ast})$ |
---|---|---|---|---|---|

0 | ${(0.0205,0,0.0330,0.5073,0.4392)}^{T}$ | ${(0.0353,0.6703,0.2029,0.0915)}^{T}$ | −0.0951 | −0.0951 | $\{{s}_{2.7939},{s}_{0.7804},{s}_{0.3104}\}$ |

0.1 | ${(0.0239,0,0.0303,0.4892,0.4566)}^{T}$ | ${(0.0421,0.6665,0.1906,0.1007)}^{T}$ | −0.0471 | −0.0471 | $\{{s}_{2.7809},{s}_{0.7759},{s}_{0.3017}\}$ |

0.2 | ${(0.0276,0,0.0276,0.4710,0.4738)}^{T}$ | ${(0.0499,0.6618,0.1774,0.1108)}^{T}$ | 0.0009 | 0.0009 | $\{{s}_{2.7691},{s}_{0.7723},{s}_{0.2928}\}$ |

0.3 | ${(0.0316,0,0.0249,0.4525,0.4910)}^{T}$ | ${(0.0590,0.6560,0.1631,0.1218)}^{T}$ | 0.0488 | 0.0488 | $\{{s}_{2.7583},{s}_{0.7695},{s}_{0.2838}\}$ |

0.4 | ${(0.0359,0,0.0222,0.4340,0.5079)}^{T}$ | ${(0.0696,0.6491,0.1475,0.1338)}^{T}$ | 0.0967 | 0.0967 | $\{{s}_{2.7486},{s}_{0.7680},{s}_{0.2746}\}$ |

0.5 | ${(0.0406,0,0.0196,0.4153,0.5245)}^{T}$ | ${(0.0822,0.6408,0.1303,0.1467)}^{T}$ | 0.1445 | 0.1445 | $\{{s}_{2.7398},{s}_{0.7681},{s}_{0.2653}\}$ |

0.6 | ${(0.0459,0,0.0170,0.3966,0.5405)}^{T}$ | ${(0.0972,0.6312,0.1111,0.1605)}^{T}$ | 0.1924 | 0.1924 | $\{{s}_{2.7318},{s}_{0.7702},{s}_{0.2561}\}$ |

0.7 | ${(0.0518,0,0.0145,0.3779,0.5557)}^{T}$ | ${(0.1154,0.6200,0.0896,0.1750)}^{T}$ | 0.2402 | 0.2402 | $\{{s}_{2.7241},{s}_{0.7750},{s}_{0.2469}\}$ |

0.8 | ${(0.0587,0,0.0122,0.3593,0.5699)}^{T}$ | ${(0.1377,0.6073,0.0650,0.1900)}^{T}$ | 0.2881 | 0.2881 | $\{{s}_{2.7161},{s}_{0.7832},{s}_{0.2382}\}$ |

0.9 | ${(0.0668,0,0.0099,0.3410,0.5823)}^{T}$ | ${(0.1658,0.5931,0.0365,0.2046)}^{T}$ | 0.3360 | 0.3360 | $\{{s}_{2.7065},{s}_{0.7961},{s}_{0.2300}\}$ |

1 | ${(0.0767,0,0.0078,0.3233,0.5923)}^{T}$ | ${(0.2022,0.5777,0.0026,0.2175)}^{T}$ | 0.3840 | 0.3840 | $\{{s}_{2.6929},{s}_{0.8153},{s}_{0.2230}\}$ |

$\mathit{\eta}$ | ${\mathit{y}}^{\ast}$ | ${\mathit{z}}^{\ast}$ | ${\mathit{U}}_{2}^{*}$ | ${\mathit{V}}_{2}^{*}$ |
---|---|---|---|---|

0 | ${(0.0595,0,0.0768,0.8637,0)}^{T}$ | ${(0.0643,0.9120,0,0.0236)}^{T}$ | −0.0314 | −0.0314 |

0.1 | ${(0.0726,0,0.0775,0.8500,0)}^{T}$ | ${(0.0706,0.9100,0,0.0194)}^{T}$ | 0.0067 | 0.0067 |

0.2 | ${(0.0861,0,0.0777,0.8362,0)}^{T}$ | ${(0.0771,0.9084,0,0.0145)}^{T}$ | 0.0448 | 0.0448 |

0.3 | ${(0.1002,0,0.0775,0.8224,0)}^{T}$ | ${(0.0837,0.9072,0,0.0090)}^{T}$ | 0.0830 | 0.0830 |

0.4 | ${(0.1148,0,0.0767,0.8085,0)}^{T}$ | ${(0.0906,0.9065,0,0.0029)}^{T}$ | 0.1212 | 0.1212 |

0.5 | ${(0.1772,0,0,0.8228,0)}^{T}$ | ${(0.0974,0.9026,0,0)}^{T}$ | 0.1595 | 0.1595 |

0.6 | ${(0.1931,0,0,0.8069,0)}^{T}$ | ${(0.1041,0.8959,0,0)}^{T}$ | 0.1979 | 0.1979 |

0.7 | ${(0.2093,0,0,0.7907,0)}^{T}$ | ${(0.1110,0.8890,0,0)}^{T}$ | 0.2363 | 0.2363 |

0.8 | ${(0.2258,0,0,0.7742,0)}^{T}$ | ${(0.1180,0.8820,0,0)}^{T}$ | 0.2749 | 0.2749 |

0.9 | ${(0.2426,0,0,0.7574,0)}^{T}$ | ${(0.1251,0.8749,0,0)}^{T}$ | 0.3134 | 0.3134 |

1 | ${(0.2598,0,0,0.7402,0)}^{T}$ | ${(0.1324,0.8676,0,0)}^{T}$ | 0.3521 | 0.3521 |

$\mathit{\eta}$ | ${\mathit{y}}^{\ast}$ | ${\mathit{z}}^{\ast}$ | ${\mathit{U}}_{3}^{*}$ | ${\mathit{V}}_{3}^{*}$ |
---|---|---|---|---|

0 | ${(0.0180,0,0.0315,0.5073,0.44432)}^{T}$ | ${(0.0359,0.6622,0.2027,0.0992)}^{T}$ | −0.0955 | −0.0955 |

0.1 | ${(0.0203,0,0.0287,0.4893,0.4617)}^{T}$ | ${(0.0426,0.6576,0.1907,0.1092)}^{T}$ | −0.0475 | −0.0475 |

0.2 | ${(0.0228,0,0.0259,0.4711,0.4802)}^{T}$ | ${(0.0503,0.6519,0.1778,0.1200)}^{T}$ | 0.0004 | 0.0004 |

0.3 | ${(0.0254,0,0.0232,0.4526,0.4988)}^{T}$ | ${(0.0592,0.6450,0.1640,0.1318)}^{T}$ | 0.0482 | 0.0482 |

0.4 | ${(0.0283,0,0.0204,0.4339,0.5174)}^{T}$ | ${(0.0696,0.6369,0.1490,0.1445)}^{T}$ | 0.0960 | 0.0960 |

0.5 | ${(0.0315,0,0.0177,0.4149,0.5359)}^{T}$ | ${(0.0817,0.6273,0.1328,0.1582)}^{T}$ | 0.1438 | 0.1438 |

0.6 | ${(0.0351,0,0.0150,0.3958,0.5541)}^{T}$ | ${(0.0960,0.6161,0.1150,0.1729)}^{T}$ | 0.1915 | 0.1915 |

0.7 | ${(0.0391,0,0.0124,0.3766,0.5720)}^{T}$ | ${(0.1132,0.6032,0.0953,0.1884)}^{T}$ | 0.2392 | 0.2392 |

0.8 | ${(0.0437,0,0.0098,0.3573,0.5893)}^{T}$ | ${(0.1340,0.5883,0.0732,0.2045)}^{T}$ | 0.2869 | 0.2869 |

0.9 | ${(0.0491,0,0.0073,0.3380,0.6056)}^{T}$ | ${(0.1598,0.5715,0.0481,0.2206)}^{T}$ | 0.3346 | 0.3346 |

1 | ${(0.0556,0,0.0048,0.3190,0.6206)}^{T}$ | ${(0.1926,0.5526,0.0190,0.2357)}^{T}$ | 0.3824 | 0.3824 |

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Li, S.; Tu, G.
Probabilistic Linguistic Matrix Game Based on Fuzzy Envelope and Prospect Theory with Its Application. *Mathematics* **2022**, *10*, 1070.
https://doi.org/10.3390/math10071070

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Li S, Tu G.
Probabilistic Linguistic Matrix Game Based on Fuzzy Envelope and Prospect Theory with Its Application. *Mathematics*. 2022; 10(7):1070.
https://doi.org/10.3390/math10071070

**Chicago/Turabian Style**

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2022. "Probabilistic Linguistic Matrix Game Based on Fuzzy Envelope and Prospect Theory with Its Application" *Mathematics* 10, no. 7: 1070.
https://doi.org/10.3390/math10071070