Sequential Game Model for Urban Emergency Human–Machine Collaborative Decision-Making

Abstract

Decision-making algorithms based on big data, artificial intelligence and other technologies are increasingly being applied to urban emergency decision-making, and urban smart emergency response is gradually appearing to be transformed from traditional empirical decision-making to human–machine collaborative decision-making. This paper explores the motivations for cooperative decision-making between leaders (human) and followers (machines) in urban emergency management in the presence of science and technology input spillovers. It focuses on the impact of human–machine cooperative decision-making on urban emergency response capacity, science and technology inputs and total urban emergency response benefits and discusses how to maximize the total benefits of urban emergency response under different levels of spillovers. In this paper, a three-stage dynamic game model is constructed: leaders and followers decide whether to establish a cooperative decision in the first stage; decide the level of science and technology inputs in the second stage; and compete for sequential decisions in the third stage. It was found that, firstly, unlike the case of static games, in sequential games, leaders and followers develop a willingness to cooperate in decision-making only when the spillover coefficients are in the lower range. Second, cooperative human–machine decision-making may diminish the importance of human experience in urban emergency management. Finally, the effectiveness of collaborative human–machine decision-making in urban emergencies deserves further research. The research in this paper provides recommendations for smart urban emergency management.

1. Introduction

With the increasing participation of big data and AI technology in the urban emergency decision-making process, emergency management shows the characteristics of high dynamics, precision and real time, which provide a base for improving the scientific nature of urban emergency management and the efficiency of services [1]. Traditionally, urban emergency decision-making is an empirical decision, which deals with limited experience and large amounts of evidence through the human brain [2,3]. However, a good decision maker needs to gain relevant experience from a large number of emergency response efforts [4], which takes a long time and is costly [5]. Urban public crisis events are characterized by complexity, randomness, suddenness and non-linearity, which means that it is difficult to fully control the emergency situation through human experience [6].

Therefore, the change from limited empirical decision-making with human brain processing to collaborative human–machine decision-making is particularly necessary in urban emergency management [7,8]. In human–machine collaborative decision-making, machine algorithms can provide key information or optimization suggestions for decision-making in real time [9], resulting in a decision output that combines algorithms and human judgement [10]. In this process, a large number of new questions remain to be explored, such as how human decision-making perceives and coordinates machine decision-making and how machine decision-making affects human decision-making [11].

Currently, research on the application of AI in urban emergency management has focused on the transformation, which is the transformation of the management mode at the macro level [12]. However, research on the micro-level human–machine collaborative decision-making mechanism is still relatively limited [13]. In contrast, mathematical modelling with game ideas for collaborative decision-making problems is a viable research direction [14]. Game theory is an effective mathematical tool that is used to describe the behavior of multiple decision makers in their strategic interactions. In recent years, many scholars exploring urban emergency decision-making have adopted this approach. Shan et al. [15] propose a constrained strategy game approach to analyze the collaborative decision-making behavior of multiple actors in urban public crises. Cahlíková et al. [16] research the task management and control problem based on the dynamic game strategy and design a fast optimal search algorithm to give the optimal decision for urban governance. Pan et al. [17] propose a matrix game dimensionality reduction method to provide an effective solution to the problem of smart urban emergency strategy.

On the other hand, research on collaborative human–machine decision-making for public policy has received sustained focus and different levels of development in recent years [18]. The analysis can be divided into three areas for different research perspectives: human–machine collaboration, machine behavior and empirical decision-making (as shown in Figure 1). Human–machine collaboration is a complex, huge system [19], which includes people, machines and the environment, and there are high-dimensional dynamic interactive behaviors between people and people, people and machines and machines and machines [20]. Presently, research on human–machine collaboration focuses on the macro-level, such as human–machine collaboration mode and human–machine relationship [21]. In particular, Grimmelikhuijsen [22] proposes that, when humans and machines cooperate in an optimal way, decision-making can be improved in a complementary manner. Höchtl et al. [23] consider the weights required for human and machine inputs in a collaborative human–machine decision-making process, assign weights to the work undertaken between humans and machines, construct perceived preferences for partnerships described in numerical terms and draw conclusions for application in managerial decision-making. However, existing research focuses on the human state, human psychological changes and human cognitive ability in human–machine collaborative decision-making [24] and lacks accurate measurement methods [25], feasible calculation methods [26] and dynamic modelling evaluation [27].

Concerning machine behavior, its evolutionary path is a gradual change from solely machine performance to self-learning behavior obtained through operating experience [28]. Relevant research has shown that humans are increasingly involved in machine interaction activities that continue to create and shape the behavior of intelligent machines [29,30,31,32]. This includes the use of cognitive models by humans to cope with abnormal machine behavior and the importance of the degree of human–machine coordination. At the same time, there has been much focus on machine intelligence changing human behavior [33], including the impact of endogenous properties of individual machines on human behavior [34], characteristics of the operation of groups of machines [35] and the effect of the mechanism of machine operation on humans [36,37]. This research provides further evidence that machine behavior is critical to the potential role of human–machine collaborative decision-making.

Experiential decision-making is the process of human understanding and internalization of rules and characteristics to form an accumulation of knowledge, including psychological [38], cognitive, emotional [39], social relations [40], cultural practices [41] and many other dimensions. However, due to the limitations of human access to information, judgements based only on limited information and limited cognitive abilities are unreliable. Therefore, a large number of scholars focus on the use of machine algorithms to assist humans in completing empirical decision-making, which can largely compensate for the above shortcomings through the powerful data collection and computational capabilities of machine algorithms [42,43,44,45]. Machine algorithm-assisted decision-making can be divided into two categories: automated assistance [46] and augmented assistance [47]. Automation assistance is the reduction in human effort by machine algorithms to complete a task. Enhanced assistance means that algorithms provide analyses, predictions and recommendations to the decision maker, which are handled by the decision maker to form a conclusion. All of the above studies have shown that empirical decision-making has limitations and that machine algorithms assisting empirical decision-making can overcome the shortcomings.

Based on the above research, this paper attempts to explore a dynamic decision-making approach. When facing urban emergency events, this research designs a sequential decision space, considering the following three states: independent decision-making, assisted decision-making and coordinated decision-making. Finally, the possibility of collaborative decision-making between human and machine in an urban emergency is analyzed through the sequential game model. The following three main issues are resolved:

(1)

Are there cooperative decision-making motives for humans and machines in urban emergency management?

(2)

Under what conditions are cooperative human–machine decisions triggered?

(3)

How does human–machine collaborative decision-making affect the total science and technology inputs and total benefits of urban emergency management?

In order to solve the above problems, this paper constructs a three-stage dynamic game model for equilibrium analysis and further verifies the robustness of the conclusions through arithmetic examples.

The remainder of this paper is organized as follows. Section 2 uses a three-stage dynamic game model to describe the human–machine game process in urban emergency decision-making. Section 3 explores the conditions, effects and contributions of human–machine cooperative decision-making. Section 4 summarizes the research findings and proposes corresponding policies.

2. Methodology

This section utilizes a three-stage dynamic game model to describe the human–machine game process in urban emergency decision-making. A basic description of the model and model solving are included. This study is based on the idea of sequential game following the time sequence to analyze the decision-making behavior of human and machine. We consider that participants act sequentially, and each participant may know the decision choices of previous participants when making a decision (declare the condition in the model).

2.1. Basic Description of the Model

The two types of participants in urban emergency management, human and machine, are denoted as $h,m$, respectively. The decision sets of the two types of participants are $q_h$ and $q_m$; then, the total decision volume of urban emergency management is $Q=q_h+q_m$.

Assumption 1.

Decision space $s_i(i=h,m)$, the inverse demand function for participant $i$ is $p_i\left(Q\right)=s_i-Q$ according to the inverse demand function $P=g\left(Q\right)$,${V(s}_i)>0$,$Q<s_i$ [48].

Assumption 2.

Both the human and the machine have a fixed decision cost of $0$ and an initial marginal decision cost of $\overline{c_i}(0<\overline{c_i}<s_i,i=h,m)$. Because of the presence of decision spillovers, the S&T inputs of man and machine are $x_i(i=h,m)$, which not only reduces their own decision costs but also reduces the costs of the other aspects of the management process, which are $x_j,(j\ne i)$ Thus, the marginal decision costs of the two types of participants $i$ can be expressed as follows:

$$c_i=\overline{c_i}-x_i-\alpha x_j,(i=h,m)$$

(1)

where $\alpha \in \left[\mathrm{0,1}\right]$ is the spillover coefficient, which represents the marginal cross-over effect of S&T inputs in the governance process on participant $i$. Non-negative decision costs require $x_i+\alpha x_j\le \overline{c_i}$. The cost of the S&T input for participant $i$ is ${\displaystyle \frac{\delta }{2}}{x}_i^2,\left(\delta >0\right)$, where $\delta$ is the S&T input coefficient, and a smaller value of $\delta$ indicates a better S&T input effect. The cost of S&T inputs is characterized by diminishing marginal returns, that is, the cost per unit of S&T inputs increases with the increase in decision-making inputs. Therefore, scientific and technological progress needs to invest resources but can effectively improve the efficiency of decision-making [49,50].

Assumption 3.

The benefit of urban emergency management is $B=b_h+b_m$, that is, the benefit of decision-making by human and machine decreases the total decision-making cost and other management costs, with the following formula:

$$b_i=\left(s_i-q_i-q_j\right)\times q_i-\left(\overline{c_i}-x_i-\alpha x_j\right)\times q_i-{\displaystyle \frac{\delta }{2}}{x}_i^2\left(i=h,m,j\ne i\right)$$

(2)

In the formula, the first term $\left(s_i-q_i-q_j\right)\times q_i$ denotes the decision-making benefit of the governance participant; the second term $\left(\overline{c_i}-x_i-\alpha x_j\right)\times q_i$ denotes the total decision-making cost of the governance participant; and the third term ${\displaystyle \frac{\delta }{2}}{x}_i^2$ denotes the S&T input cost of the governance participant. In addition, the relative efficiency ${\mu }_i=s_i-\overline{c_i},\left(i=h,m\right)$ of participant $i$ is defined such that the larger the decision space or the lower the initial marginal decision cost of the participant, the larger ${\mu }_i$ is [51].

In urban emergencies, the game in which humans and machines participate in governance is a three-stage complete information dynamic game (as shown in Figure 2). Participants’ decision-making includes whether to establish collaborative governance, the level of S&T input and the amount of decision-making. In the first stage, the human and the machine decide whether to establish cooperative governance or not, defining independent governance denoted by $N$ and cooperative governance denoted by $C$. In the second stage, human and machine simultaneously determine their respective S&T inputs $x_i$. In the third stage, humans compete with machines for the amount of Stackelberg decisions.

Assumption 4.

Without loss of generality, the human is the “Leader”, and the machine is the “Follower”; the leader prefers to determine the amount of decision $q_h$, and the follower, after observing $q_h$, determines the amount of decision $q_m$, and eventually, they realize the equilibrium benefit.

Assumption 5.

Humans are rational but constrained decision makers with full autonomy to accept or reject the machine’s suggestions. And although the model does not include ethics or accountability as independent variables, human decision-making will reflect these considerations through their strategic choices.

2.2. Model Solution

In this paper, we use reverse reasoning to find the Stackelberg equilibrium, which means that each participant has chosen the optimal strategy, and no better result can be obtained by changing the strategy of any party. Starting from the final stage of the game, the optimal decisions for each stage are derived step by step.

2.2.1. Stage 3: Single Decision-Making

The leader decides first, and its decision quantity is $q_h$, and we predict the followers’ response to the leader’s decision quantity to be $q_m\left(q_h\right)$. First, solve for the follower optimal reaction function $q_m^{*}\left(q_h\right)$. Suppose $s_h=s_m=s$, $\overline{c_h}=\overline{c_m}=c$. For the follower’s benefit function Equation (2), the first-order derivative ${\displaystyle \frac{\partial b_m}{\partial q_m}}=0$ is obtained with respect to $q_m$:

$$q_m^{*}\left(q_h\right)={\displaystyle \frac{s-q_h-c_m}{2}}$$

(3)

We substitute Equation (3) into the leader’s benefit function Equation (2). The equilibrium decision quantity of the leader is obtained from the first-order derivative ${\displaystyle \frac{\partial b_h}{\partial q_h}}=0$:

$$q_h^{*}={\displaystyle \frac{s+c_m-2c_h}{2}}$$

(4)

Substituting Equation (4) into Equation (3) thus obtains the equilibrium decision quantity of the followers:

$$q_m^{*}={\displaystyle \frac{s-{3c}_m+2c_h}{4}}$$

(5)

$q_h^{*}$ and $q_m^{*}$ show that the number of decisions made by the participating governors is not only affected by their own decision costs but also by the governance costs of other governors. However, the effect of the two is opposite; the smaller $c_h$ is, the larger $q_h^{*}$ is, and the smaller $c_m$ is, the smaller $q_h^{*}$ is. This indicates that self-influence is greater than cross-influence.

Substituting $q_h^{*}$ and $q_m^{*}$ into Equation (2), the equilibrium benefits of the leader and follower are denoted, respectively, as follows:

$$b_h={\displaystyle \frac{{(s+c_m-2c_h)}^2}{8}}-{\displaystyle \frac{\delta }{2}}{x}_h^2$$

(6)

$$b_m={\displaystyle \frac{{(s-3c_m+2c_h)}^2}{8}}-{\displaystyle \frac{\delta }{2}}{x}_m^2$$

(7)

2.2.2. Stage 2: Assisted Decision-Making

In this stage, leaders and followers simultaneously decide on their respective assisted decision-making styles, that is, the level of S&T inputs. Since the objective functions of the participating governors are different in the case of machine-assisted [52,53] and expert experience-assisted decision-making [54,55], the two cases need to be considered separately.

Case 1: Machine-assisted decision-making. Human decision makers use machine-assisted decision-making to improve the benefits of decision-making. They maximize their own decision-making as their goal. Substituting Equation (1) into Equation (6), the leader’s decision gain function is expressed as follows:

$$b_h^N={\displaystyle \frac{{\left[s-c+\left(2\alpha -1\right)x_m+\left(2-\alpha \right)x_h\right]}^2}{8}}-{\displaystyle \frac{\delta }{2}}{x}_h^2$$

(8)

Case 2: Expert experience-assisted decision-making. In this case, the machine completes the decision-making process by including expert experience to improve the machine’s decision-making gain. They have the goal of maximizing the benefit of their decisions. Substituting Equation (1) into Equation (7), the follower’s decision benefit function is expressed as follows:

$$b_m^N={\displaystyle \frac{{\left[s-c+\left(3\alpha -1\right)x_h+\left(3-2\alpha \right)x_m\right]}^2}{16}}-{\displaystyle \frac{\delta }{2}}{x}_m^2$$

(9)

Solve the equilibrium level of the S&T input of leaders and followers according to the first-order derivative conditional joint equation $\left\{\begin{array}{c}{\displaystyle \frac{\partial b_h^N}{\partial x_h}}=0\\ {\displaystyle \frac{\partial b_m^N}{\partial x_m}}=0\end{array}\right.$.

$$x_h^{N^{*}}={\displaystyle \frac{\left(6-2{\alpha }^3-{\alpha }^2-13\alpha -2\alpha \delta -4\delta \right)\mu }{2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6}}$$

(10)

$$x_m^{N^{*}}={\displaystyle \frac{\left(6-2{\alpha }^3+9{\alpha }^2-13\alpha +2\delta \alpha -3\delta \right)\mu }{2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6}}$$

(11)

Equations (10) and (11) show that both the level of decision spillovers and decision efficiency affect the equilibrium level of S&T inputs.

According to Equations (10) and (11), the amount of decision-making between leader and follower under assisted decision-making is solved.

$$q_h^{N^{*}}={\displaystyle \frac{\left(4{\alpha }^2-10\alpha -4\delta +6\right)\mu \delta }{2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6}}$$

(12)

$$q_m^{N^{*}}={\displaystyle \frac{\left(2{\alpha }^2-6\alpha -2\delta +4\right)\mu \delta }{2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6}}$$

(13)

Substituting Equations (12) and (13) into Equations (6) and (7) obtains the corresponding equilibrium benefits.

$$b_h^{N^{*}}={\displaystyle \frac{{\left(2{\alpha }^2-5\alpha -2\delta +3\right)}^2\left(-{\alpha }^2+4\alpha +4\delta -4\right){\mu }^2\delta }{2{[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6]}^2}}$$

(14)

$$b_m^{N^{*}}={\displaystyle \frac{{({\alpha }^2-3\alpha -\delta +2)}^2(-{4\alpha }^2+12\alpha +8\delta -9){\mu }^2\delta }{2{[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6]}^2}}$$

(15)

This paper draws on the consumer surplus theory proposed by Alfred Marshall to measure the additional benefits gained in urban emergency response decisions. The additional benefit AB is obtained from Equations (12) and (13).

$${AB}^{N^{*}}={\displaystyle \frac{1}{2}}\left(q_h^{N^{*}}+q_m^{N^{*}}\right)$$

(16)

$${AB}^{N^{*}}={\displaystyle \frac{{({3\alpha }^2-8\alpha -3\delta +5)}^2{\mu }^2{\delta }^2}{{[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6]}^2}}$$

(17)

The total benefit TB of urban emergency management is jointly determined by ${AB}^{N^{*}}$, $b_h^{N^{*}}$ and $b_m^{N^{*}}$.

$${TB}^{N^{*}}=b_h^{N^{*}}+b_m^{N^{*}}+{AB}^{N^{*}}$$

(18)

$$\begin{gathered} {TB}^{N^{*}}={\displaystyle \frac{{\left({2\alpha }^2-5\alpha -2\delta +3\right)}^2\left(-{\alpha }^2+4\alpha +4\delta -4\right){\mu }^2{\delta }^2}{2{\left[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6\right]}^2}} \\ +{\displaystyle \frac{{\left({\alpha }^2-3\alpha -\delta +2\right)}^2\left(-4{\alpha }^2+12\alpha +8\delta -9\right){\mu }^2{\delta }^2}{2{\left[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6\right]}^2}} \\ +{\displaystyle \frac{4{(3{\alpha }^2-8\alpha -3\delta +5)}^2\delta {\mu }^2{\delta }^3}{2{\left[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6\right]}^2}} \end{gathered}$$

(19)

2.2.3. Stage 1: Human–Machine Collaborative Decision-Making

The literature has pointed out that the essence of human–machine collaborative decision-making is the degree of cooperation between humans and machines through communication, sharing of knowledge and technology reflecting the human–machine decision-making process, while the endogenous spillover coefficients are also key. Therefore, considering the urban emergency human–machine collaborative decision-making problem with spillover effects is more relevant to the situation in reality. In the case of human–machine collaboration, the sharing of decision-making information between the two parties can ensure the successful implementation of collaborative decision-making. Then, the leader and the followers have the same goal, which is to maximize the total benefit. From Equations (8) and (9), we obtain the benefit of collaborative decision-making:

$$\begin{gathered} B^C=b_h^N+b_m^N={\displaystyle \frac{{\left[s-c+\left(2\alpha -1\right)x_m+\left(2-\alpha \right)x_h\right]}^2}{8}}-{\displaystyle \frac{\delta }{2}}{x}_h^2 \\ +{\displaystyle \frac{{\left[s-c+\left(3\alpha -1\right)x_h+\left(3-2\alpha \right)x_m\right]}^2}{16}}-{\displaystyle \frac{\delta }{2}}{x}_m^2 \end{gathered}$$

(20)

Similar to the solution process used in assisted decision-making (see Appendix A), the relevant results are shown in

Table 1. Equilibrium results for Stage 1.

	Equilibrium
S&T input	$x_h^{C^{*}}={\displaystyle \frac{[4{\alpha }^2-4{\alpha }^3-4+2\delta +\alpha \left(4+\delta \right)]\mu }{4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4}}$
	$x_m^{C^{*}}={\displaystyle \frac{[4{\alpha }^2-4{\alpha }^3-4+\delta +2\alpha \left(2+\delta \right)]\mu }{4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4}}$
Decision-making quantities	$q_h^{C^{*}}={\displaystyle \frac{2\mu \delta \left(-5{\alpha }^2+10\alpha +2\delta -5\right)}{4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4}}$
	$q_m^{C^{*}}={\displaystyle \frac{2\mu \delta \left(-3{\alpha }^2+6\alpha +\delta -3\right)}{4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4}}$
Benefits	$b_h^{C^{*}}={\displaystyle \frac{{\mu }^2\delta \left[-16{\alpha }^6+32{\alpha }^5+4{\alpha }^4\left(27\delta +4\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$ $-{\displaystyle \frac{{\mu }^2\delta \left[8{\alpha }^3\left(49\delta +8\right)+4\alpha \left(8-3978{\delta }^3\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$ $+{\displaystyle \frac{{\mu }^2\delta \left[4\left(-4+29\delta -21{\delta }^2+4{\delta }^3\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$
	$b_m^{C^{*}}={\displaystyle \frac{{\mu }^2\delta \left[-16{\alpha }^6+32{\alpha }^5+8{\alpha }^4\left(11\delta +2\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$ $-{\displaystyle \frac{{\mu }^2\delta \left[8{\alpha }^3\left(37\delta +8\right)+{\alpha }^2\left(-52{\delta }^2+408\delta +16\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$ $+{\displaystyle \frac{{\mu }^2\delta \left[4\alpha \left(23{\delta }^2-70\delta +8\right)+8{\delta }^3-49{\delta }^2+80\delta \right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$
The additional benefit	${AB}^{C^{*}}={\displaystyle \frac{1}{2}}\left(q_h^{C^{*}}+q_m^{C^{*}}\right)$
Total Benefits	${TB}^{C^{*}}={\displaystyle \frac{\left(A+B\right){\mu }^2\delta +4{({-8\alpha }^2+16\alpha +3\delta -8)}^2{\mu }^2{\delta }^3}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$

.

In the human–computer collaborative decision-making stage, leaders and followers decide whether to establish a cooperative decision-making relationship by comparing $b_i^{N^{*}}$ with $b_i^{C^{*}}i=h,m$. When $b_i^{C^{*}}\ge b_i^{N^{*}}$, establish cooperative decision-making; when $b_i^{C^{*}}<b_i^{N^{*}}$, the two parties are unable to establish cooperative decision-making and prefer independent decision-making (Algorithm 1).

Algorithm 1: Human–machine collaborative decision-making algorithm

3. Results

This section first explores the conditions under which leaders and followers will establish a collaborative decision-making relationship and, on this basis, analyses the impact of human–machine collaborative decision-making on the level of S&T inputs. Finally, it focuses on the impact of human–machine collaborative decision-making on urban emergency management.

It is first necessary to solve for the S&T input cost coefficient $\delta$ equation as follows.

The second-order derivative of Equation (8) with respect to $x_h$ is obtained:

$${\displaystyle \frac{{\partial }^2{b}_h^N}{\partial x_h^2}}={\displaystyle \frac{{\alpha }^2-4\alpha -4\delta +4}{4}}$$

(21)

Making (21) less than zero obtains $\delta >{\displaystyle \frac{1}{4}}\left({\alpha }^2-4\alpha +4\right)$; from $\alpha ϵ\left[\mathrm{0,1}\right]$, $\delta >1$; similarly, for (9), taking the second-order derivative with respect to xm and making it less than 0 obtains $\delta >1.1$.

According to the Hession matrix’s method for determining extreme points, the second-order derivative is positive, and the function has a local minimum. Firstly, the first-order principal sub-equation of Equation (21) is solved so that it is less than 0. The second-order principal sub-equation is greater than 0, and the range of values of $\delta$ can be obtained.

$$\left\{\begin{array}{c}{\displaystyle \frac{3}{2}}-{\displaystyle \frac{5}{2}}\alpha +{\displaystyle \frac{11}{8}}{\alpha }^2-\alpha <0\\ {\displaystyle \frac{11}{8}}-{\displaystyle \frac{5}{2}}\alpha +{\displaystyle \frac{3}{2}}{\alpha }^2-\alpha <0\end{array}\right.$$

(22)

From $\alpha ϵ\left[\mathrm{0,1}\right]$, we have $\delta >1.5$.

Second-order principal cubic formula is as follows:

$$\left({\displaystyle \frac{3}{2}}-{\displaystyle \frac{5}{2}}\alpha +{\displaystyle \frac{11}{8}}{\alpha }^2-\alpha \right)\left({\displaystyle \frac{11}{8}}-{\displaystyle \frac{5}{2}}\alpha +{\displaystyle \frac{3}{2}}{\alpha }^2-\alpha \right)-{\left({\displaystyle \frac{23}{8}}\alpha -{\displaystyle \frac{5}{4}}{\alpha }^2-{\displaystyle \frac{5}{4}}\right)}^2>0$$

(23)

solve for $\delta >2.7$. According to the four cases in which a takes the value of $\delta$,$\delta >2.7$.

3.1. Establishment of Human–Machine Collaborative Decision-Making

Whether or not to establish human–computer collaborative decision-making in the urban emergency management process requires a comparison of the equilibrium benefits in the case of cooperative vs. independent decision-making. Let $∆b_i=b_i^{C^{*}}-b_i^{N^{*}}\left(i=h,m\right)$, and denote the difference in equilibrium returns when making cooperative and independent decisions. Substituting Equations (14) and (24) into $∆b_h=b_h^{C^{*}}-b_h^{N^{*}}$ results in the following:

$$∆b_h={\displaystyle \frac{{\mu }^2\delta \left[-16{\alpha }^6+32{\alpha }^5+4{\alpha }^4\left(27\delta +4\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$$

(24)

$$-{\displaystyle \frac{{\mu }^2\delta \left[8{\alpha }^3\left(49\delta +8\right)+4\alpha \left(8-3978{\delta }^3\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$$

$$+{\displaystyle \frac{{\mu }^2\delta \left[4\left(-4+29\delta -21{\delta }^2+4{\delta }^3\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$$

$$-{\displaystyle \frac{{({\alpha }^2-3\alpha -\delta +2)}^2(-{4\alpha }^2+12\alpha +8\delta -9){\mu }^2\delta }{2{[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6]}^2}}$$

$$={\displaystyle \frac{{{\mu }^2\delta \left(5+5\alpha -2{\alpha }^2\right)}^2\left(-24-8\alpha +2{\alpha }^2\right)}{{\left(66+73\alpha -28{\alpha }^2+7{\alpha }^3-2{\alpha }^4\right)}^2}}$$

$$+{\displaystyle \frac{\left(16+112\alpha +128{\alpha }^2-204{\alpha }^3+56{\alpha }^4+4{\alpha }^5-2{\alpha }^6\right){\mu }^2\delta }{{\left(10+40\alpha -25{\alpha }^2+{\alpha }^4\right)}^2}}$$

Equations (15) and (25) into $∆b_m=b_m^{C^{\mathrm{*}}}-b_m^{N^{\mathrm{*}}}$ results in the following:

$$∆b_m={\displaystyle \frac{{\mu }^2\delta \left[-16{\alpha }^6+32{\alpha }^5+8{\alpha }^4\left(11\delta +2\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$$

(25)

$$-{\displaystyle \frac{{\mu }^2\delta \left[8{\alpha }^3\left(37\delta +8\right)+{\alpha }^2\left(-52{\delta }^2+408\delta +16\right)\right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$$

$$+{\displaystyle \frac{{\mu }^2\delta \left[4\alpha \left(23{\delta }^2-70\delta +8\right)+8{\delta }^3-49{\delta }^2+80\delta \right]}{2{\left[4{\alpha }^4-{\alpha }^2\left(23\delta +8\right)+40\alpha \delta +8{\delta }^2-23\delta +4\right]}^2}}$$

$$-{\displaystyle \frac{{\left(2{\alpha }^2-5\alpha -2\delta +3\right)}^2\left(-{\alpha }^2+4\alpha +4\delta -4\right){\mu }^2\delta }{2{[2{\alpha }^4-7{\alpha }^3+{\alpha }^2\left(6\delta +4\right)+17\delta +\alpha \left(-20\delta +7\right)-6]}^2}}$$

$$={\displaystyle \frac{{{\mu }^2\delta \left(-2-3\alpha +{\alpha }^2\right)}^2\left(-46-24\alpha +8{\alpha }^2\right)}{{\left(66+73\alpha -28{\alpha }^2+7{\alpha }^3-2{\alpha }^4\right)}^2}}$$

$$+{\displaystyle \frac{\left(4+48\alpha +102{\alpha }^2-156{\alpha }^3+46{\alpha }^4+4{\alpha }^5-{2\alpha }^6\right){\mu }^2\delta }{{\left(10+40\alpha -25{\alpha }^2+{\alpha }^4\right)}^2}}$$

There is a willingness to cooperate when $∆b_i\ge 0$ and no willingness to cooperate when $∆b_i<0$. There is cooperative decision-making behavior between man and machine when and only when $∆b_h\ge 0$ and $∆b_m\ge 0$, and b varies with a as shown in Figure 3, and the range of variation in $\alpha$ is obtained, which leads to Proposition 1. According to the conditions under which the human–machine establishes cooperative decision-making, the joint equation $\left\{\begin{array}{c}0\le \alpha \le 1\\ ∆b_h\ge 0\\ ∆b_m\ge 0\end{array}\right.$ and ${\mu }^2\delta$ constant greater than zero; solve using mathematics to obtain $\alpha \in \left[0.03,0.33\right]$.

Proposition 1.

Spillover coefficients are closely related to the human–machine decision-making process. Cooperative human–machine decision-making is reached when $\alpha \in \left[0.03,0.33\right]$, and human and machine choose to make independent decisions when $\alpha ϵ\left[\mathrm{0,0.03}\right]\cup \left[\mathrm{0.33,1}\right]$.

Static decision-making, in which both players in a decision decide the amount of the decision at the same time, has been studied by scholars. They argue that, when the spillover coefficient is large ($\alpha >0.5$), decision makers tend to collaborate in decision-making from a decision cost reduction perspective. When the spillover coefficient is small ($\alpha \le 0.5$), the decision maker does not cooperate. In contrast to this literature, this paper studies sequential decision-making competition. Both parties will engage in cooperative decision-making behavior only if the spillover coefficient is in a much smaller range. In other cases, however, leaders and followers will not agree. Thus, it can be found that both static and dynamic decisions may reach cooperation, but the key to whether or not they are cooperative decisions is the spillover coefficient. In urban emergency management processes, the marginal cross-effects of other cost inputs on decision makers are determined. In such cases, there is often only one decision sequence that gives decision makers the willingness to cooperate. Then, urban emergency managers can choose the appropriate form of decision-making competition based on the level of spillover and obtain more opportunities for decision-making cooperation.

Why did this result? Leaders and followers face two situations when making decisions: on the one hand, by establishing cooperative decision-making, humans and machines can share technology and experience, adjusting the level of decision-making of managers and reducing the cost of decision-making of managers, and cooperative decision-making is more favorable from this point of view. On the other hand, cooperative decision-making strengthens the first-mover advantage of humans in decision-making competition; only if the level of technological spillovers is in the range of $\alpha \in \left[\mathrm{0.03,0.33}\right]$, the decision-making influence of leaders is greater than that of followers, and both have the willingness to cooperate and are able to achieve cooperative on decision-making behavior.

3.2. Impact of Cooperative Human–Machine Decision-Making on the Level of S&T

Based on the analysis of human–machine collaborative decision-making in the urban emergency management process, this subsection explores the impact of human–machine collaborative decision-making on the level of S&T in emergency management. First, compare the level of S&T inputs of leaders and followers in the same scene. The trends of S&T input levels $x_i^C$ and $x_i^N(i=h,m)$ with $\alpha$ for cooperative and independent decision-making between humans and machines are shown in Figure 4, which leads to Proposition 2.

Proposition 2.

The level of S&T input of leaders is higher than the level of S&T input of followers in both cooperative and independent decision-making.

Proposition 2 shows that the act of decision-making itself has two attributes, whether it is machine-assisted human decision-making or expert experience-assisted machine decision-making or human–machine cooperative decision-making. On the one hand, whether a leader or a follower reduces their own decision-making costs when completing a decision, they are also reducing the decision-making costs of the other one to some extent, that is, the spillover effect. On the other hand, a reduction in the cost of decision-making for both leaders and followers indirectly reduces the competitiveness of the other’s decisions. However, spillover effects can lead to “free-riding” behavior by decision makers. One side only maintains a lower level of decision-making and profits from the other side’s efforts. This is the competition effect. Due to the limitations of the size of the capital invested in the machine upfront, urban emergency management prefers that the machine assist the human to complete the decision. Additionally, the competition effect reduces the competitiveness of machine decisions, which forces a reduction in machine decision-making inputs to alleviate the negative impact of the human–machine competition effect. Under the combined influence of the two attributes, followers have lower levels of decision-making. In contrast, leaders have less resource constraints and are more likely to innovate actively and thus have higher levels of decision-making.

As shown in Figure 5, in the case of independent decision-making, the decision level of both human and machine decreases with the increase in the spillover coefficient $\alpha$. In the case of cooperative decision-making, the decision level of the leader decreases and then increases as $\alpha$ increases, and the decision level of the follower increases as $\alpha$ increases. This shows that the level of decision-making between humans and machines is affected by a combination of spillover and competitive effects. When $i(i=h,m)$ makes decisions independently, the competitive effect of decision-making behavior weakens $i$’s willingness to make decisions, and the larger the value of $\alpha$, the more pronounced the competitive effect. The lower the level of decision-making in $i$, the more cooperation can internalize the spillover effects of decision-making behavior. $\alpha$ the larger the value of a, the stronger the positive effect is. Therefore, followers continue to increase their level of decision-making as the value of $\alpha$ increases. Unlike followers, leaders themselves have the advantage of being more flexible in their actions. The positive effect of cooperative decision-making becomes significant only when the value of $\alpha$ is high enough. Therefore, leaders’ decision-making level shows a tendency to decrease first and then increase.

In addition to this, this paper also focuses on the cost of S&T inputs for the same decision maker in different cases. In different cases, the trend of S&T input level $x_h^C$, $x_m^C$, $x_h^N$, $x_m^N$ with $\alpha$ for leaders and followers is shown in Figure 5. This leads to inference 1.

Corollary 1.

(1) Anytime the leader’s level of S&T input is higher in cooperative decision-making than in independent decision-making.

(2) When $a\ge 0.2$, followers have higher levels of S&T inputs in cooperative decision-making than in independent decision-making.

(3) When $a<0.2$, followers have lower levels of S&T inputs in cooperative decision-making than in independent decision-making.

3.3. Impact of Cooperative Human–Machine Decision-Making on $X,\mathit{AB}\mathit{and}\mathit{TB}$

The above focuses on the impact of collaborative decision-making on one participant. This subsection explores the impact of human–machine collaborative decision-making on the total S&T input level ($X$), additional benefits ($AB$) and total benefits ($TB$) of emergency management in urban emergency management. Where $X^z=x_h^{z^{*}}+x_m^{z^{*}}$, analyze the trend of $X^z$, ${AB}^z$, and ${TB}^z$ $\left(z=N,C\right)$ with a as shown in Figure 6, which leads to Proposition 3.

Proposition 3.

(1) When $\alpha <0.6$, the level of total S&T inputs for collaborative human–machine decision-making is lower than the level of total S&T inputs for independent decision-making.

(2) When $\alpha \ge 0.6$, the level of total S&T inputs for collaborative human–machine decision-making is lower than the level of total S&T inputs ($X$) for independent decision-making.

(3) When $\alpha <0.56$, the additional benefit of collaborative human–machine decision-making (AB) is lower than the additional benefit of independent decision-making ($AB$).

(4) When $\alpha \ge 0.56$, the additional benefit of collaborative human–machine decision-making (AB) is lower than the additional benefit of independent decision-making ($AB$).

(5) The total benefit ($TB$) when humans and machines make collaborative decisions is lower than when they make independent decisions.

Proposition 3 shows that human–machine collaborative decision-making reduces the level of total S&T input when $\alpha \in \left[0,0.6\right]$. A small spillover coefficient implies that the spillover effect is relatively small and that most of the benefits that participants gain from the decision belong to themselves. Thus, if humans and machines make decisions independently, each participant tends to do a lot of resource allocation and information gathering. In contrast, with human–machine collaborative decision-making, the participants aim to maximize the total benefit. Therefore, participants chose to reduce S&T inputs to obtain higher total benefits, which resulted in lower total S&T total inputs for urban emergency management. When $\alpha \in \left[0.6,1\right]$, human–machine collaborative decision-making can increase the level of total S&T input. A large spillover coefficient implies that participants receive greater benefits from each other’s S&T inputs. In the case of independent decision-making, participants tended to “free-rider” on competitors’ S&T inputs. They only maintain lower levels of S&T input levels. Human–machine cooperation is able to internalize these externalities, with participants increasing their S&T inputs in order to increase total benefits, such that the total S&T input level is relatively high.

In conclusion, when the spillover coefficient is small, cooperative decision-making reduces the decision-making ability of the participants, which reduces the effect of competition between man and machine. However, this implies additional costs in other areas of urban emergency management, which reduces the additional benefits and total benefits. When the spillover coefficient is large enough, collaborative decision-making improves the decision-making ability of the participants. This results in a significant decrease in decision-making costs for the participants, and then, additional benefits are increased; however, collaborative human–machine decision-making does not maximize the total benefits of emergency management. This is due to the fact that the increase in additional benefits is less than the decrease in decision benefits, so total benefits do not improve. This shows that it is not enough to rely only on human–machine collaborative decision-making to complete urban emergency management, and it is more necessary for regulatory agencies to improve the management process.

3.4. Discussion of the Difference in $\mu$ Between Humans and Machine

The above discussion is based on the fact that participants’ decision cost functions are symmetric. However, in the real emergency management process, the participants’ decision space and initial marginal decision costs are asymmetric. Especially in sequential games, leaders have a larger decision space and lower initial marginal decision costs. Based on this, this subsection discusses the case where the decision-making efficiencies $\mu$ are not equal. The following arithmetic example analyzes the impact of human–machine collaborative decision-making on the total S&T input level, total number of decisions, decision benefits, additional benefits and total benefits. Let $s_h=120,s_m=100,\overline{c_h}=40,\overline{c_h}=50,{\mu }_h=20,{\mathrm{and}\mu }_m=50;$ the trend of total S&T input ($X$), total decision ($Q$), decision benefit ($B$), additional benefit ($AB$) and total benefit ($TB$) with a is shown in Figure 7.

From Figure 7, it can be seen that, when $\alpha$ takes a large value, human–machine collaborative decision-making can increase the total S&T inputs, number of total decisions and decision-making benefits in urban emergency management as well as increase the additional benefits and achieve the maximization of the total benefits of urban emergency. This is consistent with the conclusions of Section 3.3. It also proves the stability of the conclusions of this article. This research can provide a theoretical basis for policies related to urban emergency management. Due to the limitations of collaborative human–machine decision-making in urban emergency management, the necessary intervention of the supervisory authorities is required. When the level of spillovers is low, regulators need to be concerned about the phenomenon of “free-riding” in collaborative human–machine decision-making or when the level of S&T inputs from one side of the human–machine equation is too high, leading to a decrease in the total benefits of urban emergency management. When spillover levels are high, regulators should take appropriate measures to stimulate innovative human–machine collaborative decision-making to better cope with complex and changing urban emergencies.

4. Conclusions

With the development of AI technology, decision makers have put forward higher intelligent requirements for urban emergency risk management [56,57]. Humans are increasingly dependent on machine intelligence to make the right choices. The evolution of machine intelligence is changing from detecting and identifying urban risks to sensing and understanding human needs. Then, machine decision-making in urban emergencies is also becoming particularly important. However, both human experience decision-making and intelligent decision-making by machines need to face high S&T cost inputs and uncontrollable risk of decision-making errors. How to reduce S&T input costs, control the risk of decision-making errors and improve the early warning capability for emergencies in the urban emergency decision-making process are particularly important. Collaborative human–machine decision-making for urban emergencies can combine the empirical decision-making ability of humans with the progressive process of real-time machine algorithms, which becomes an important way to solve the above problems.

Based on this, a three-stage dynamic game model is constructed to study the cooperative decision-making problem between the leader (human) and the follower (machine) in the urban emergency response process. On the one hand, it focuses on the phenomenon of cooperative decision-making between humans and machines and analyzes the motivations for cooperative decision-making between leaders and followers. On the other hand, the competing amount of sequential decision-making by decision makers is considered, thus compensating for the shortcomings of the single conclusion of the existing research results. This study found, firstly, that spillover level is a key factor in the collaborative decision-making of leaders and followers. Unlike static games, in sequential games, leaders and followers develop a willingness to cooperate in decision-making only when the spillover coefficient is in the lower range and the profit of cooperative human–machine decision-making is higher than the profit of independent decision-making. Second, we report the importance of collaborative decision-making’s ability to integrate the decision-making strengths of leaders and followers. From this perspective, collaborative decision-making changes the position of machine intelligence in urban emergency decision-making from machine-assisted to human–machine collaboration. Third, cooperative human–machine decision-making may not achieve the maximum total benefit of urban emergency management. Cooperative decision-making may result in the loss of additional benefits for one party, and the decision may turn out to be ineffective.

In Tokyo, earthquake early warning systems are a prime example of human–computer collaboration. Traditionally, these systems relied on expert decision-making to enable early deployment of resources. However, with the widespread adoption of intelligence-assisted human decision-making, smart devices now play a pivotal role in detecting seismic activity. As soon as shaking is detected, AI-powered sensors automatically trigger alerts and notify emergency management systems within seconds. These systems then support human controllers in making critical decisions, such as suspending subway operations or distributing warning messages to the public. AI-based sensors and real-time monitoring systems are now capable of detecting various anomalies, including smoke, gas leaks and earthquake tremors. When such events are detected, automated alerts are instantly broadcasted via multiple channels, including NHK, cell phones and social media. In addition, these systems can provide real-time updates on the status of shelters, ensuring that residents receive timely information. These advancements exemplify the effectiveness of human–machine collaboration, where AI aids in rapid data analysis and decision-making, while human operators oversee the implementation of emergency measures.

There are also some shortcomings in this paper. Firstly, only two types of decision makers, human and machine, are considered, while in reality, urban emergency involves the participation of multiple subjects such as government, enterprises, citizens and NGOs, and the agents of collaborative decision-making should be more refined. Secondly, comparing the Stackelberg model, Duopoly model and Bertrand model may be able to obtain more valuable conclusions and provide suggestions for collaborative decision-making between humans and machines in urban emergencies.