**2. Preliminaries**

In this section, GT and PT will be briefly reviewed so that unfamiliar readers can understand our proposed method easily. In addition, some related works to illustrate the importance and necessity of this research are reviewed.

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

GT is a useful tool to solve decision making problems in which the situations either have conflict or cooperation and sometimes both [23]. These situations may happen when there are two or more players (DMs) involved in a same system and they attempt to achieve their own objectives using the same resources [28]. As a branch of mathematical analysis, GT provides a scientific process to choose the best strategies for each possible situation throughout the game [19]. Such a characterization of GT is suitable for EDM problems, in which the DM usually needs to have a corresponding response with respect to different emergency situations.

Generally, if a game has *n* players, it will be denoted as *G* = {(*Si*; *Pi*), *i* = 1, 2, ... , *<sup>n</sup>*}, where *Si* and *Pi* denote the strategies and payoffs of the *i*-th player, respectively. In the game process of EDM problems, there are usually two players, i.e., the EE and DM, in which the EE is a special player because it is unconscious about the benefits or costs. Thus, the game between EE and DM can be denoted as *G* = {(*Si*; *Pi*)}, where *i* = 1, 2.

The game can be classified according to the relationship among the players [29]: if the relationship among the players is competitive, the game is a noncooperation game; otherwise, if the players are cooperative, it is a cooperation game. Obviously, the relationship between EE and DM in the game is noncooperation, so the game in EDM problems can be assumed as a typical noncooperation game, the zero-sum game, i.e., *P*1 + *P*2 = 0, which means if the DM gains (*i*, the EE loses (*i*, otherwise, the EE gains (*i*, while the DM loses (*i*.

Three basic notions of GT for the EDM problem are briefly introduced as follows:


The game can also be classified according to the action sequence among players [29]: if the players take the action simultaneously or the players do not know the exact information of the other player's action, the game is a static game; if not, the game is a dynamic one. The dynamic one is also called the extensive from game (EFG) [29]. Obviously, in the EDM problems, the player EE always takes the action firstly, so the game between DM and EE is an EFG problem. However, in the real world, because of the imprecise and incomplete information of the EE, which strategy the EE will take the DM does not know. Thus, when this situation occurs, the EFG problem can be regarded as the static one, and its game tree is shown in Figure 1.

**Figure 1.** The game tree between an emergency event (EE) and decision maker (DM).

Based on the presentation mentioned above, since the EDM problem is a static game, therefore the payoff matrix of EE and DM can be simplified into Table 1 according to Figure 1.


**Table 1.** The payoff matrix of emergency event (EE) and decision maker (DM).

Based on Figure 1 and Table 1, the game process between EE and DM can be described as shown in Figure 2. In our proposal, we assume that the EE chooses its strategy randomly.

**Figure 2.** The game process between EE and DM.

The assumption presented in current EDM studies based on GT [20–24] in which the DM is completely rational is not fully reasonable. Due to the importance of the psychological behavior of DM, it will be taken into account in the phase of determining payoffs and will be introduced in detail in the third section of this proposal.

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

As was mentioned in the Introduction, DM's psychological behavior is a key and important factor in the EDM process especially, when DM is under pressure. However, such an important issue is neglected in the current EDM approaches based on GT; thus, it will be taken into account in this proposal by using PT.

PT is a useful tool to consider human being's psychological behavior issues, which was firstly presented in Kahneman and Tversky's study in 1979 [10] and was developed by them in 1992 [30] as an economic behavior theory. In the proposal of Kahneman and Tversky, they provided a simple and clear computation process to describe the psychological behavior using reference points (RPs), losses, gains and overall prospect values, which are important concepts in PT. Since PT has a simple calculation process and a clear logic, it has been widely applied in the field of decision making to solve the problems considering human being's psychological behavior [13,15,30–32]. Therefore, the PT will be utilized to address the DM's psychological behavior in our proposal.

Generally, in the process of decision making, PT was distinguished as three phases [30]:


3. 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.

According to PT, human beings are usually more sensitive to losses than the same gains, and their psychological behavior shows risk-seeking for losses and risk-aversion for gains [26]. Thus, PT can be depicted by means of an S-shaped value function that shows a concave shape in the loss domain and a convex shape in the gain domain, respectively (see Figure 3). The value function of PT is related to RPs and expressed by a power law presented as below [10]:

$$w(\mathbf{x}) = \begin{cases} \mathbf{x}^a, & \mathbf{x} \ge 0 \\ -\lambda(-\mathbf{x})^\beta, & \mathbf{x} < 0 \end{cases} \tag{1}$$

where *α* is the parameter with respect to gains, while *β* is the parameter associated with losses, 0 ≤ *α*, *β* ≤ 1. *x* means gains with *x* ≥ 0, and losses with *x* < 0. *λ* denote the parameter of risk aversion, *λ* > 1. The values of parameters *α*, *β* and *λ* are determined through experiments [26,33–35].

**Figure 3.** S-shaped value function of prospect theory (PT).

### *2.3. Related Works*

In order to demonstrate the importance and necessity of this study, several important studies in the literature are briefly reviewed that are close to our research.

The DM's psychological behavior has been addressed in existing EDM studies by different researchers. For example, Fan et al. [14] proposed a risk decision analysis method for emergency response that addressed DM's psychological behavior in the decision process by employing PT. Wang et al. [16] developed an EDM method that considered not only DM's psychological behavior in the decision process by using PT, but also the dynamic evolution feature of EE. Due to the uncertainty information about EEs in real-world situations, it is usually a big challenge for DM to estimate possible losses by using crisp values that are employed in existing EDM studies [14,16,36]. Wang et al. [18] presented an EDM method based on PT considering DM's psychological behavior with interval values, which not only extended the scope of PT for dealing with interval values, but also made the EDM method close to the real world. With the increasing complexity of EEs in the real world, one DM alone [14,16,18,36] cannot make comprehensive judgments and proper decisions; therefore, Wang et al. [17] proposed a group EDM method for emergency situations by using group wisdom to support DM making a decision that takes into account experts' psychological behavior in the decision process by using PT. Due to the fact that there are various types of information about EEs in the real world, such as crisp values [14,16,36], interval values [18], linguistic information [37], and so on, none of the proposals considers various types of information at the same time; to do that, Wang et al. [38] proposed a group EDM method for not only considering various types of information at the same time, but also together with experts' psychological behavior and hesitation in qualitative contexts. Motivated

by [38], Zhang et al. [39] presented an EDM method based on PT and hesitant fuzzy sets considering not only experts' psychological behavior, but also experts' hesitation in quantitative contexts.

Despite existing EDM studies based on PT having achieved fruitful results [14,16–18,36–40], they neglect an important fact that different emergency situations should be handled by using different measures because of the limited resources and dynamic evolution of EEs.

Nevertheless, to address such an important issue in the real world, GT has been employed in existing EDM studies. For example, Yang and Xu [20] proposed an engineering model based on sequential games considering different situations coping with a flood eruption EDM problem. Chen et al. [41] provided a game theory-based approach for evaluating possible terrorist attacks and corresponding deployment of emergency responses. Gupta et al. [23] proposed a game-theoretic EDM method for considering the optimal allocation solutions of resources to different situations of the EEs, particularly when the available resources are limited. Cheng and Zheng [42] proposed a game-theoretical analysis method considering possible solutions of emergency evacuation for different emergency cases. Rezazadeh et al. [43] presented a security risk assessment method based on game theory for considering the possible terrorist attacks on oil and gas pipelines. Gao et al. [44] proposed an approach for considering different scenarios coping with corporate environment risk based on game theory. Wu [45] presented two game theoretic models for search-and-rescue resource allocation and selection of an acceptable plan for different districts after devastating tsunamis.

Although the existing EDM studies based on GT have obtained remarkable results regarding the different situations coping with the problems of EEs, they build on an assumption that DM is totally rational in the decision process. However, different behavior studies [19,26,27] have proven that DM has limited rationality and his/her psychological behavior can affect the decision behavior directly, especially under a risk and uncertainty environment, and must be considered because of its importance in the decision process.

To overcome the limitations pointed out above and highlight the significance and importance of our research, this study combines the merits of PT and GT to propose a novel EDM method based on GT and PT that considers not only the different situations of coping with problems, but also DM's psychological behavior in the EDM process, which is introduced in detail in Section 3.

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

As previously mentioned, the proposed EDM method based on GT and PT is introduced in this section. The general framework of our proposal is illustrated in Figure 4, and it consists of three main phases:


**Figure 4.** The general framework of the proposed method.
