# An Emergency Decision Making Method for Different Situation Response Based on Game Theory and Prospect Theory

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Game Theory in Emergency Decision Making

- Players: Players are always denoted by $i=1,2,\dots ,n$ and at least $i\ge 2$; this means that there are at least two players in one game. In EDM, there are two players, who are the decision maker (DM) and the EE. Thus, in the emergency game $G=\left\{({S}_{i},{P}_{i})\right\}$, $i=1,2$, where 1 denotes the DM and 2 refers to EE.
- Strategies: Let ${S}_{i}=\left\{{{S}_{i}}_{{k}_{i}}\right\}$ be the set of action strategies of the i-th player who has ${k}_{i}$ strategies. In EDM, ${S}_{1}=\left\{{S}_{1\delta}\right\}$ refers to the set of different alternatives of DM, in which ${S}_{1\delta}$ denotes the $\delta $-th alternatives, $\delta =1,2,\cdots {k}_{1}$. ${S}_{2}=\left\{{S}_{2\theta}\right\}$ refers to the set of different situations of EE, where ${S}_{2\theta}$ denotes the $\theta $-th possible situation of EE, $\theta =1,2,\cdots {k}_{2}$.
- Payoffs: Let ${P}_{i}({S}_{i})$ be the payoffs of the i-th player, where ${P}_{1}({S}_{1})+{P}_{2}({S}_{2})=0$.

#### 2.2. Prospect Theory in Emergency Decision Making

- An editing phase, in which the gains and losses can be calculated according to the RPs provided by DM.
- An evaluation phase: in this phase, the prospect values can be obtained by a value function, then the overall prospect values will be calculated on the foundation of prospect values and the weighting vector.
- A selection phase, in which the alternative with the highest overall prospect value will be selected as the best one to deal with the given decision problem.

#### 2.3. Related Works

## 3. Emergency Decision Making Method Based on Game Theory and Prospect Theory

- Definition framework: this part introduces the basic notations and related terminology that are employed in this proposal.
- Computation of overall prospect values: in this part, the value function will be used to compute the overall prospect values according to gains and losses.
- Selecting the optimal alternative based on payoffs: the payoffs of DM including his/her psychological behavior and the payoffs of EE will be determined. Based on the payoffs, the optimal alternative will be selected to respond to corresponding emergency situation.

#### 3.1. Definition Framework

- ${S}_{1}=\left\{{S}_{1\delta}\right\}$: refers to the set of different alternatives, in which ${S}_{1\delta}$ denotes the $\delta $-th alternative, $\delta =1,2,\dots ,{k}_{1}$.
- ${S}_{2}=\left\{{S}_{2\theta}\right\}$: refers to the set of different situations, in which ${S}_{2\theta}$ denotes the $\theta $-th situations, $\theta =1,2,\dots ,{k}_{2}$.
- $X=\left\{{X}_{m}\right\}$: refers to the set of criteria, in which ${X}_{m}$ represents the m-th criterion, $m=1,2,\dots ,M$.
- ${W}_{{X}_{m}}=({w}_{{X}_{1}},\dots ,{w}_{{X}_{M}})$: refers to the weighting vector, in which ${w}_{{X}_{m}}$ represents the weight of the m-th criterion. The weighting vector is usually provided by the DM satisfying $\sum _{m=1}^{M}}{w}_{{X}_{m}}=1$, ${w}_{{X}_{m}}\in [0,1]$, $m=1,2,\dots ,M$.
- ${C}_{\delta}$: refers to the cost of the $\delta $-th available emergency alternative, $\delta =1,2,\dots ,{k}_{1}$.
- ${R}_{\theta m}=[{R}_{\theta m}^{L},{R}_{\theta m}^{H}],{R}_{\theta m}^{H}>{R}_{\theta m}^{L}$: refers to the values of RPs, in which ${R}_{\theta m}^{L}$ and ${R}_{\theta m}^{H}$ represent the lower and upper limits of RP provided by DM for the m-th criterion in the $\theta $-th situation, respectively, $m=1,2,\dots ,M$, $\theta =1,2,\dots ,{k}_{2}$.
- ${E}_{\delta m}=[{E}_{\delta m}^{L},{E}_{\delta m}^{H}],{E}_{\delta m}^{H}>{E}_{\delta m}^{L}$: refers to the value of the pre-defined effective control scope [18], in which ${E}_{\delta m}^{L}$ and ${E}_{\delta m}^{H}$ represent the lower and upper limits of losses’ protection scope from EE with respect to the $\delta $-th alternative concerning the m-th criteria, respectively. ${E}_{\delta m}$ is usually determined by the local government, $\delta =1,2,\dots ,{k}_{1}$, $m=1,2,\dots ,M$.

#### 3.2. Calculation of Gains and Losses

#### 3.3. Computation of Overall Prospect Values

#### 3.4. Selecting Optimal Alternative Based on Payoffs

#### 3.4.1. Determining the Payoffs of the Players

#### 3.4.2. Selection of the Optimal Alternative with Respect to Each Emergency Situation

- The gain and loss matrix $G{M}_{\theta}$, $L{M}_{\theta}$ can be formed on the basis of the obtained gains and losses, respectively. Then, the value matrix $V{M}_{\theta}$ and its normalized form $\overline{V{M}_{\theta}}$ can be obtained by using Equations (2) and (3), respectively. Afterwards, the overall prospect value ${O}_{\theta \delta}$ can be calculated by Equation (4).
- Based on the obtained payoffs of DM and EE, the DM can select the optimal strategies for dealing with all possible emergency situations according to Equation (7).

## 4. Case Study and Comparison

#### 4.1. Case Study

#### 4.2. Comparison with Other Methods

## 5. Conclusions and Future Works

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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EE | ||||||
---|---|---|---|---|---|---|

DM | ${S}_{21}$ | … | ${S}_{2\theta}$ | … | ${S}_{2{k}_{2}}$ | |

${S}_{11}$ | $({P}_{1}({S}_{21},{S}_{11}),{P}_{2}({S}_{21},{S}_{11}))$ | … | $({P}_{1}({S}_{2\theta},{S}_{11}),{P}_{2}({S}_{2\theta},{S}_{11}))$ | … | $({P}_{1}({S}_{2{k}_{2}},{S}_{11}),{P}_{2}({S}_{2{k}_{2}},{S}_{11}))$ | |

⋮ | ⋮ | … | ⋮ | … | ⋮ | |

${S}_{1\delta}$ | $({P}_{1}({S}_{21},{S}_{1\delta}),{P}_{2}({S}_{21},{S}_{1\delta}))$ | … | $({P}_{1}({S}_{2\theta},{S}_{1\delta}),{P}_{2}({S}_{2\theta},{S}_{1\delta}))$ | … | $({P}_{1}({S}_{2{k}_{2}},{S}_{1\delta}),{P}_{2}({S}_{2{k}_{2}},{S}_{1\delta}))$ | |

⋮ | ⋮ | … | ⋮ | … | ⋮ | |

${S}_{1{k}_{1}}$ | $({P}_{1}({S}_{21},{S}_{1{k}_{1}}),{P}_{2}({S}_{21},{S}_{1{k}_{1}}))$ | …… | $({P}_{1}({S}_{2\theta},{S}_{1{k}_{1}}),{P}_{2}({S}_{2\theta},{S}_{1{k}_{1}}))$ | …… | $({P}_{1}({S}_{2{k}_{2}},{S}_{1{k}_{1}}),{P}_{2}({S}_{2{k}_{2}},{S}_{1{k}_{1}}))$ |

**Table 2.**Positional relationship between interval values ${R}_{\theta m}$ and ${E}_{\delta m}$ [17].

Cases | Positional Relationship | |
---|---|---|

Case 1 | ${E}_{\delta m}^{H}<{R}_{\theta m}^{L}$ | |

Case 2 | ${R}_{\theta m}^{H}<{E}_{\delta m}^{L}$ | |

Case 3 | ${E}_{\delta m}^{L}<{R}_{\theta m}^{L}<{E}_{\delta m}^{H}<{R}_{\theta m}^{H}$ | |

Case 4 | ${R}_{\theta m}^{L}<{E}_{\delta m}^{L}<{R}_{\theta m}^{H}<{E}_{\delta m}^{H}$ | |

Case 5 | ${E}_{\delta m}^{L}<{R}_{\theta m}^{L}<{R}_{\theta m}^{H}<{E}_{\delta m}^{H}$ | |

Case 6 | ${R}_{\theta m}^{L}<{E}_{\delta m}^{L}<{E}_{\delta m}^{H}<{R}_{\theta m}^{H}$ |

**Table 3.**Computation formulas of gain and loss for cost criteria [17].

Cases | Gain ${\mathit{G}}_{\mathit{\delta}\mathit{m}}$ | Loss ${\mathit{L}}_{\mathit{\delta}\mathit{m}}$ | |
---|---|---|---|

Case 1 | ${E}_{\delta m}^{H}<{R}_{\theta m}^{L}$ | ${R}_{\theta m}^{L}-0.5({E}_{\delta m}^{L}+{E}_{\delta m}^{H})$ | 0 |

Case 2 | ${R}_{\theta m}^{H}<{E}_{\delta m}^{L}$ | 0 | ${R}_{\theta m}^{H}-0.5({E}_{\delta m}^{L}+{E}_{\delta m}^{H})$ |

Case 3 | ${E}_{\delta m}^{L}<{R}_{\theta m}^{L}<{E}_{\delta m}^{H}<{R}_{\theta m}^{H}$ | $0.5({R}_{\theta m}^{L}-{E}_{\delta m}^{L})$ | 0 |

Case 4 | ${R}_{\theta m}^{L}<{E}_{\delta m}^{L}<{R}_{\theta m}^{H}<{E}_{\delta m}^{H}$ | 0 | $0.5({R}_{\theta m}^{H}-{E}_{\delta m}^{H})$ |

Case 5 | ${E}_{\delta m}^{L}<{R}_{\theta m}^{L}<{R}_{\theta m}^{H}<{E}_{\delta m}^{H}$ | $0.5({R}_{\theta m}^{L}-{E}_{\delta m}^{L})$ | $0.5({R}_{\theta m}^{H}-{E}_{\delta m}^{H})$ |

Case 6 | ${R}_{\theta m}^{L}<{E}_{\delta m}^{L}<{E}_{\delta m}^{H}<{R}_{\theta m}^{H}$ | 0 | 0 |

**Table 4.**Computation formulas of gain and loss for benefit criteria [17].

Cases | Gain ${\mathit{G}}_{\mathit{\delta}\mathit{m}}$ | Loss ${\mathit{L}}_{\mathit{\delta}\mathit{m}}$ | |
---|---|---|---|

Case 1 | ${E}_{\delta m}^{H}<{R}_{\theta m}^{L}$ | 0 | $0.5({E}_{\delta m}^{L}+{E}_{\delta m}^{H})-{R}_{\theta m}^{L}$ |

Case 2 | ${R}_{\theta m}^{H}<{E}_{\delta m}^{L}$ | $0.5({E}_{\delta m}^{L}+{E}_{\delta m}^{H})-{R}_{\theta m}^{H}$ | 0 |

Case 3 | ${E}_{\delta m}^{L}<{R}_{\theta m}^{L}<{E}_{\delta m}^{H}<{R}_{\theta m}^{H}$ | 0 | $0.5({E}_{\delta m}^{L}-{R}_{\theta m}^{L})$ |

Case 4 | ${R}_{\theta m}^{L}<{E}_{\delta m}^{L}<{R}_{\theta m}^{H}<{E}_{\delta m}^{H}$ | $0.5({E}_{\delta m}^{H}-{R}_{\theta m}^{H})$ | 0 |

Case 5 | ${E}_{\delta m}^{L}<{R}_{\theta m}^{L}<{R}_{\theta m}^{H}<{E}_{\delta m}^{H}$ | $0.5({E}_{\delta m}^{H}-{R}_{\theta m}^{H})$ | $0.5({E}_{\delta m}^{L}-{R}_{\theta m}^{L})$ |

Case 6 | ${R}_{\theta m}^{L}<{E}_{\delta m}^{L}<{E}_{\delta m}^{H}<{R}_{\theta m}^{H}$ | 0 | 0 |

Alternatives | Criteria | |||
---|---|---|---|---|

${\mathit{c}}_{\mathbf{1}}$(0.5) | ${\mathit{c}}_{\mathbf{2}}$(0.25) | ${\mathit{c}}_{\mathbf{3}}$(0.25) | ${\mathit{C}}_{\mathit{\delta}}$ | |

${\mathit{E}}_{\mathit{\delta}\mathbf{1}}$ | ${\mathit{E}}_{\mathit{\delta}\mathbf{2}}$ | ${\mathit{E}}_{\mathit{\delta}\mathbf{3}}$ | ${\mathit{C}}_{\mathit{\delta}}$ | |

${S}_{11}$ | [3,5] | [200,400] | [40,50] | 10 |

${S}_{12}$ | [6,14] | [800,1200] | [50,60] | 30 |

${S}_{13}$ | [14,20] | [1200,1500] | [60,70] | 70 |

${S}_{14}$ | [18,25] | [1500,1800] | [70,80] | 130 |

Situations | Criteria | ||
---|---|---|---|

${\mathit{c}}_{\mathbf{1}}$ | ${\mathit{c}}_{\mathbf{2}}$ | ${\mathit{c}}_{\mathbf{3}}$ | |

${\mathit{R}}_{\mathit{\theta}\mathbf{1}}$ | ${\mathit{R}}_{\mathit{\theta}\mathbf{2}}$ | ${\mathit{R}}_{\mathit{\theta}\mathbf{3}}$ | |

${S}_{21}$ | [5,8] | [100,300] | [20,35] |

${S}_{22}$ | [5,12] | [300,500] | [35,45] |

${S}_{23}$ | [12,18] | [600,800] | [45,55] |

${S}_{24}$ | [18,20] | [800,100] | [55,65] |

**Table 7.**The overall prospect values ${O}_{\theta \delta}$ of the $\theta $-th alternatives in the $\delta $-th emergency situation.

${\mathit{O}}_{\mathit{\theta}\mathit{\delta}}$ | Situations | ||||
---|---|---|---|---|---|

${\mathit{S}}_{\mathbf{21}}$ | ${\mathit{S}}_{\mathbf{22}}$ | ${\mathit{S}}_{\mathbf{23}}$ | ${\mathit{S}}_{\mathbf{24}}$ | ||

Alternative | ${S}_{11}$ | −0.0710 | −0.1927 | −0.8152 | −1.000 |

${S}_{12}$ | 0.4092 | 0.2841 | −0.1008 | −0.3524 | |

${S}_{13}$ | 0.7444 | 0.6508 | 0.3419 | 0.0238 | |

${S}_{14}$ | 1.000 | 1.000 | 0.6074 | 0.2511 |

DM | EE | |||||
---|---|---|---|---|---|---|

${S}_{21}$ | ${S}_{22}$ | ${S}_{23}$ | ${S}_{24}$ | |||

${S}_{11}$ | $(-0.0071,0.0071)$ | $(-0.0193,0.0193)$ | $(-0.0815,0.0815)$ | $(-0.0100,0.0100)$ | ||

${S}_{12}$ | $(0.0136,-0.0136)$ | $(0.0095,-0.0095)$ | $(-0.0034,0.0034)$ | $(-0.0117,0.0117)$ | ||

${S}_{13}$ | $(0.0106,-0.0106)$ | $(0.0093,-0.0093)$ | $(0.0049,-0.0049)$ | $(0.0003,-0.0003)$ | ||

${S}_{14}$ | $(0.0077,-0.0077)$ | $(0.0077,-0.0077)$ | $(0.0047,-0.0047)$ | $(0.0019,-0.0019)$ |

DM | EE | |||||

${S}_{21}$ | ${S}_{22}$ | ${S}_{23}$ | ${S}_{24}$ | |||

${S}_{11}$ | $(-0.0071,0.0071)$ | $(-0.0193,0.0193)$ | $(-0.0815,0.0815)$ | $(-0.0100,0.0100)$ | ||

${S}_{12}$ | $(\underline{\mathbf{0.0136}},-0.0136)$ | $(\underline{\mathbf{0.0095}},-0.0095)$ | $(-0.0034,0.0034)$ | $(-0.0117,0.0117)$ | ||

${S}_{13}$ | $(0.0106,-0.0106)$ | $(0.0093,-0.0093)$ | $(\underline{\mathbf{0.0049}},-0.0049)$ | $(0.0003,-0.0003)$ | ||

${S}_{14}$ | $(0.0077,-0.0077)$ | $(0.0077,-0.0077)$ | $(0.0047,-0.0047)$ | $(\underline{\mathbf{0.0019}},-0.0019)$ |

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**MDPI and ACS Style**

Zhang, Z.-X.; Wang, L.; Wang, Y.-M.
An Emergency Decision Making Method for Different Situation Response Based on Game Theory and Prospect Theory. *Symmetry* **2018**, *10*, 476.
https://doi.org/10.3390/sym10100476

**AMA Style**

Zhang Z-X, Wang L, Wang Y-M.
An Emergency Decision Making Method for Different Situation Response Based on Game Theory and Prospect Theory. *Symmetry*. 2018; 10(10):476.
https://doi.org/10.3390/sym10100476

**Chicago/Turabian Style**

Zhang, Zi-Xin, Liang Wang, and Ying-Ming Wang.
2018. "An Emergency Decision Making Method for Different Situation Response Based on Game Theory and Prospect Theory" *Symmetry* 10, no. 10: 476.
https://doi.org/10.3390/sym10100476