Research on fuzzy dynamic route choice model and algorithm of wargame

Abstract

This paper aims at automatic route choice in wargame deduction, that is, using computer algorithms to mimic the route choice of human players. Route choice is the process by which a player of a game chooses the optimal route from a set of candidate routes. Essentially, it can be viewed as a multi-attribute decision making problem. However, due to the uncertain and dynamic decision making environment in wargaming, the existing decision making theory and methods face some challenges in addressing such problems. We summarize these challenges in two aspects. On the one hand, from a psychological point of view, the process of route choice by a human player is a dynamic decision making process. However, most of the multi-attribute decision making methods widely used today were developed under static conditions and do not accurately describe the decision making behaviors of individuals in dynamic and real-world settings. On the other hand, in the real dynamic decision making environment of wargaming, there is a large amount of ambiguous and uncertain information. How to imitate human thinking to quantify the information is crucial for making scientific decisions and for automating decision making. Based on the above two aspects of the analysis, in this paper, we first construct a fuzzy dynamic route choice model of wargame. The model simulates the decision making process of humans from two stages of “information preprocessing” and “information processing”. Then we propose a fuzzy dynamic route choice algorithm of wargame. Furthermore, the model is applied to a practical case of route choice of wargame, and the effectiveness and advantages of the model are illustrated through a comparative analysis. Finally, the model is further quantitated and analyzed in combination with the actual case, and the approximation of the model to human decision making psychology is illustrated from both a simulation and a mathematical justification perspective. These results demonstrate that the proposed model can not only handle a large amount of fuzzy and uncertain information, but also simulate the decision making psychology of wargame players. This research result is expected to provide a new effective method for automatic route choice in wargame simulations.

Appendices

Appendix A

Results analysis

This result also has profound implications for actual economic activities. For instance, in market competition, new products will seize more market shares of similar products.

In fact, we can prove this conclusion quantitatively by referring to the proof process in Refs. [12] and [14]. In Equation (2), the weight vector ${\textbf{W}}\left( t\right)$ changes over time and obeys a stationary random process. It is assumed that the attention weight is identically distributed. Therefore, the mean value and covariance ${\textbf{W}}\left( t\right)$ can be calculated by:

$$\begin{aligned}{} & {} {\mathbb {E}}\left( {\textbf{W}}\left( t \right) \right) = {\textbf{w}}h, \end{aligned}$$

(9)

$$\begin{aligned}{} & {} \text {Cov}\left( {{\textbf{W}}\left( t \right) } \right) = {\mathbb {E}}\left( \left( {{\textbf{W}}\left( t \right) - {\textbf{w}}{h}} \right) {\left( {{\textbf{W}}\left( t \right) - {\textbf{w}}{h}} \right) }^{\prime } \right) = \psi \cdot h, \end{aligned}$$

(10)

in which $\psi$ can be got from $\psi = \mathrm{{diag}}\left( {\textbf{w}} \right) - {\textbf{w}} \cdot {\textbf{w}}'$. Since the time step h is any sufficiently small quantity set in advance, both the mean and covariance are constants independent of time t. The parameters of valence and preference can be regarded as linearly weighted transformation of the original decision information matrix under attention weight information. Therefore, these two parameters also obey stationary random process.

$$\begin{aligned} {\mathbb {E}} \left( {{\textbf{V}}\left( t \right) } \right) & = {\mathbb {E}} \left( {\textbf{CMW}\left( t \right) } \right) \nonumber \\ & = {{\textbf{C}}}{{\textbf{M}}}{\mathbb {E}} \left( {W\left( t \right) } \right) = CM{\textbf{w}}\cdot h = \mathbf {\mu }h \cdot h, \end{aligned}$$

(11)

and

$$\begin{aligned}&{\text {Cov}}\left( {{\textbf{V}}\left( t \right) } \right) = \mathrm{{Cov}}\left( {\textbf{CMW}\left( t \right) } \right) \nonumber \\&= {{\textbf{K}}} \cdot {\text {Cov}}\left( {{\textbf{W}}\left( t \right) } \right) \cdot {{\textbf{K}}}' = {{\textbf{K}}} \cdot \left( {\psi \cdot h} \right) \cdot {{\textbf{K}}} = h \cdot \mathbf {\Phi }. \end{aligned}$$

(12)

Formula (1) can be rewritten as follows:

$$\begin{aligned}&{{\textbf{P}}}\left( t \right) = {{{\textbf{S}}}{{\textbf{P}}}}\left( t - h\right) + {{\textbf{V}}}\left( t \right) = {{\textbf{S}}}{{\textbf{P}}}\left( nh - h\right) + {{\textbf{V}}}\left( nh \right) \nonumber \\&= \mathop \sum \limits _{i = 0}^{n - 1} {{{\textbf{S}}}^i}{{\textbf{V}}}\left( nh - ih\right) + {{{\textbf{S}}}^n}{{\textbf{P}}}\left( 0 \right) . \end{aligned}$$

(13)

Since the eigenvalue of the feedback matrix ${\textbf{S}}$ is less than 1, then the diffusion process defined by (8) is convergent and stable, and the mean value of ${\textbf{P}}\left( t \right)$ is:

$$\begin{aligned} \xi \left( t \right)&= {\mathbb {E}}\left( {{{\textbf{P}}}\left( t \right) } \right) = {\mathbb {E}}\left( {{{\textbf{S}}}{{\textbf{P}}}\left( {t - h} \right) + {\textbf{V}}\left( t \right) } \right) \nonumber \\&= {\mathbb {E}}\left( {\mathop \sum \limits _{i = 0}^{n - 1} {{\textbf{S}}^i}{\textbf{V}}\left( {nh - ih} \right) + {{\textbf{S}}^n}P\left( 0 \right) } \right) \nonumber \\&= {\left( {{\textbf{I}} - {\textbf{S}}} \right) ^{ - 1}}\left( {{\textbf{I}} - {{\textbf{S}}^n}} \right) \mathbf {\mu } \cdot h + {{\textbf{S}}^n}{\textbf{P}}\left( 0 \right) . \end{aligned}$$

(14)

Considering the eigenvalues of ${\textbf{S}}$ less than 1, when $t\rightarrow 1$, $\xi \left( t \right) \rightarrow {\left( {{\textbf{I}} - {\textbf{S}}} \right) ^{ - 1}}\mathbf {\mu } \cdot h$.

In our manuscript, ${\textbf{S}} = \left[ \begin{array}{l} \mathrm{{0}}\mathrm{{.5}}\quad \quad \alpha \quad \quad {\hspace{1.0pt}} \beta \\ \alpha \quad \quad 0.5\quad \quad \gamma \\ \beta \quad \quad \gamma \quad \quad 0.5 \end{array} \right]$, in which, the first, second, and third lines correspond to the alternatives A, B and C respectively. By Equation (4), $\alpha = - 0.1{e^{ - \sqrt{2} \theta }}$, $\beta = - 0.1{e^{ - \sqrt{2} \left| {{m_1} - {m_2}} \right| }}$, $\gamma = - 0.1{e^{ - \sqrt{2} \sqrt{{{\left( {{m_1} - {m_2}} \right) }^2} + {\theta ^2}} }}$ when ${\textbf{M}}_B = \left[ {{m_1} + \theta ,{m_2} + \theta } \right] \quad \left( {\theta > 0} \right)$. Then

$$\begin{aligned}&{\left( {{\textbf{I}} - {\textbf{S}}} \right) ^{ - 1}}\mathrm{{ = }}\frac{1}{{0.5\left( {0.25 - {\gamma ^2}} \right) + \alpha \left( { - 0.5\alpha - \gamma \beta } \right) - \beta \left( {\alpha \gamma + 0.5\beta } \right) }}\nonumber \\&\times \left[ \begin{array}{l} 0.25 - {\gamma ^2}\quad \quad \beta \gamma + 0.5\alpha \quad \quad \alpha \gamma + 0.5\beta \\ \beta \gamma + 0.5\alpha \quad \quad 0.25 - {\beta ^2}\quad \quad \alpha \beta + 0.5\gamma \\ \alpha \gamma + 0.5\beta \quad \quad \alpha \beta + 0.5\gamma \quad \quad 0.25 - {\alpha ^2} \end{array} \right] . \end{aligned}$$

(15)

Based on Equation (6),

$$\begin{aligned} \mathbf {\mu } = \textbf{CMw} = \left[ \begin{array}{l} \mathrm{{1}}\quad \quad - \frac{1}{2}\quad \quad - \frac{1}{2}\\ - \frac{1}{2}\quad \quad 1\quad \quad - \frac{1}{2}\\ - \frac{1}{2}\quad \quad - \frac{1}{2}\quad \quad 1 \end{array} \right] \left[ \begin{array}{l} {m_{1\quad }}\quad \quad {m_2}\\ m + {\theta _{1\quad }}{m_2} + \theta \\ {m_2}\quad \quad \quad {m_{1\quad }} \end{array} \right] \left[ \begin{array}{l} {w_1}\\ {w_2} \end{array} \right] . \end{aligned}$$

(16)

Because both attributes are equally important in the simulation experiment, ${w_1} = {w_2}$. Then by matrix operations, we can simply represent $\mathbf {\mu }$ as $\mathbf {\mu } = \left[ \begin{array}{l} - \lambda \\ 2\lambda \\ - \lambda \end{array} \right] ,\left( {\lambda > 0} \right)$. From Equation (9), we can get that

$$\begin{aligned}&\xi \left( \infty \right) = {\left( {{\textbf{I}} - {\textbf{S}}} \right) ^{ - 1}}\mu \cdot h\nonumber \\&\mathrm{{ = }}h\frac{1}{{0.5\left( {0.25 - {\gamma ^2}} \right) + \alpha \left( { - 0.5\alpha - \gamma \beta } \right) - \beta \left( {\alpha \gamma + 0.5\beta } \right) }}\nonumber \\&\quad \left[ \begin{array}{l} 0.25 - {\gamma ^2}\quad \quad \beta \gamma + 0.5\alpha \quad \quad \alpha \gamma + 0.5\beta \\ \beta \gamma + 0.5\alpha \quad \quad 0.25 - {\beta ^2}\quad \quad \alpha \beta + 0.5\gamma \\ \alpha \gamma + 0.5\beta \quad \quad \alpha \beta + 0.5\gamma \quad \quad 0.25 - {\alpha ^2} \end{array} \right] \left[ \begin{array}{l} - \lambda \\ 2\lambda \\ - \lambda \end{array} \right] \nonumber \\&= h\frac{1}{{0.5\left( {0.25 - {\gamma ^2}} \right) + \alpha \left( { - 0.5\alpha - \gamma \beta } \right) - \beta \left( {\alpha \gamma + 0.5\beta } \right) }}\nonumber \\&\quad \left[ \begin{array}{l} \lambda \left( { - 0.25 + {\gamma ^2} + 2\beta \gamma + \alpha - \alpha \gamma - 0.5\beta } \right) \\ **********************\\ \lambda \left( { - 0.25 + {\alpha ^2} + 2\alpha \beta + \gamma - \alpha \gamma - 0.5\beta } \right) \end{array} \right] . \end{aligned}$$

(17)

Furthermore,

$$\begin{aligned}{} & {} {\xi _C}\left( \infty \right) - {\xi _A}\left( \infty \right) \nonumber \\{} & {} \quad = \frac{{\lambda \left( {\alpha - \gamma } \right) \left( {\alpha + \gamma + 2\beta - 1} \right) }}{{0.5\left( {0.25 - {\gamma ^2}} \right) + \alpha \left( { - 0.5\alpha - \gamma \beta } \right) - \beta \left( {\alpha \gamma + 0.5\beta } \right) }}. \end{aligned}$$

(18)

From $\alpha = - 0.1{e^{ - \sqrt{2} \theta }}$, $\beta = - 0.1{e^{ - \sqrt{2} \left| {{m_1} - {m_2}} \right| }}$, $\gamma = - 0.1{e^{ - \sqrt{2} \sqrt{{{\left( {{m_1} - {m_2}} \right) }^2} + {\theta ^2}} }}$, we can get that $- 0.1< \alpha< \beta< \gamma < 0$. So the difference ${\xi _C}\left( \infty \right) - {\xi _A}\left( \infty \right)$ is positive. Then $P\{ A|\{ A,C\} \} = \{ C|\{ A,C\} \}$ and $P\{ A|\{ A,B,C\} \} = \{ C|\{ A,B,C\} \}$ are established. That is to say that the similarity effect occurs.

In this section, we first discuss the influence of the preference intensity threshold and time pressure on human decision making psychology in combination with the model. Thereinto, the threshold reflects the personality, experience and so on of decision makers in real life, while time pressure reflects the time factor in the real decision making environment. Secondly, we discuss the influence of mixed threshold and time pressure on decision making psychology. Finally, we prove that our model satisfies the similarity effect in decision making from both simulation results and theory. In conclusion, to a certain extent, our model can approximate the decision making psychology of human beings in real environment.