## 4 Analysis of the Stability Strategy of Each Game Subject
### Stability Analysis of the Public  
The expected payoffs of the public with panic buying (\(U_{y}\)) and not panic buying (\(U_{1-y}\)) are as follows:  
\[ \begin{array}{l}U_{y}=xz[(1-\omega_{1})V_{AB}-C_{P}]+x(1-z)(V_{AB}-C_{P})+(1 -x)z[(1-\omega_{1})V_{PB}-C_{P}]\\ +(1-x)(1-z)(V_{PB}-C_{P})\\ U_{1-y}=0\end{array}  \tag{6}\]  
Thus, the average payoff of the public (\(\overline{U}_{2}\)) is as follows:  
\[\overline{U}_{2}=yU_{y}+(1-y)U_{1-y} \tag{7}\]  
According to Equation (7), the replicator dynamics equation of the public can be calculated as:  
\[F(y)=dy/dt=y(U_{y}-\overline{U}_{2})=y(1-y)[x(1-\omega_{1}z)(V_{AB}-V_{PB})+(1 -\omega_{1}z)V_{PB}-C_{P}] \tag{8}\]  
Furthermore, the first partial derivative of \(F(y)\) and the set \(G(x,z)\) are, respectively:  
\[dF(y)/dy=(1-2y)[x(1-\omega_{1}z)(V_{AB}-V_{PB})+(1-\omega_{1}z)V_{PB}-C_{P}] \tag{9}\]  
\[G(x,z)=x(1-\omega_{1}z)(V_{AB}-V_{PB})+(1-\omega_{1}z)V_{PB}-C_{P} \tag{10}\]  
When \(F(y)=0\) and \(dF(y)/dy\)<0, \(y\) is ESS. The following analysis can be obtained:  
(1) When \(x=[(1-\omega_{1}z)V_{PB}-C_{P}]/[(1-\omega_{1}z)(V_{PB}-V_{AB})]\)  
(Let \(x*=[(1-\omega_{1}z)V_{PB}-C_{P}]/[(1-\omega_{1}z)(V_{PB}-V_{AB})]\)), there is \(dF(y)/dy\equiv 0\), no matter what the value of \(y\) is, it is in a stable state. At this time, the strategy selection ratio of the public will not change with time. (2) When \(x\
eq x*\), there are two cases:  
1 Because of \(\partial G(x,z)/\partial x\)<0, we know that \(G(x,z)\) is decreasing with respect to \(x\). When \(x\)<\(x*\), there are \(dF(y)/\left.dy\right|_{y=0}\)>0 and \(dF(y)/\left.dy\right|_{y=1}\)<0, so \(y=1\) is the evolution equilibrium point. At this point, the public tends to choose to panic buy. 2 When \(x\)>\(x*\), there are \(dF(y)/\left.dy\right|_{y=0}\)<0 and \(dF(y)/\left.dy\right|_{y=1}\)>0, so \(y=0\) is the evolution equilibrium point. At this point, the public tends to choose not to panic buy. The public replication dynamic phase diagram is shown in Figure 3.  
It can be seen from Figure 3 that the volume \(V_{B_{1}}\) of \(B_{1}\) is the probability that the public chooses to panic buy, and the volume \(V_{B_{2}}\) of \(B_{2}\) is the probability that the public chooses not to panic buy, according to the calculation:  
\(V_{B_{1}}=\iint\limits_{B_{1}}\left[(1-\omega_{1}z)V_{PB}-C_{P}\right]/[(1- \omega_{1}z)(V_{PB}-V_{AB})]dzdy=\), \(V_{PB_{2}}=1-V_{B_{1}}\). \(V_{PB}/(V_{PB}-V_{AB})+C_{P}In(1-\omega_{1})/\omega_{1}(V_{PB}-V_{AB})\)  
Proposition 3: _The probability that the public chooses to panic buy is positively correlated with \(V_{PB}\) and \(V_{AB}\), and negatively correlated with \(\omega_{1}\) and \(C_{P}\)._  
Proof: According to the expression of panic buying probability \(C_{1}\), the first partial derivative of each element can be obtained.\(dV_{B_{1}}/dV_{PB}\)>0 (s.t. \(In(1-\omega_{1})\)< \(-\omega_{1}V_{AB}/C_{P}\)); \(dV_{B_{1}}/d\omega_{1}\)<0\(dV_{B_{1}}/dC_{P}\)<0. Therefore, the increase in \(V_{PB}\) and \(V_{AB}\) or the decrease in \(\omega_{1}\) and \(C_{P}\) can improve the probability of the public choosing to panic buy. \(\square\)  
Proposition 3 shows that on the one hand, the public's strategic choice is related to its trust in local governments and the cost of panic buying; the public may choose to panic buy when they believe that local governments cannot effectively deal with the COVID-19 pandemic or when they believe that panic buying is very convenient for them. On the other hand, a sharp increase in the price of necessities increases the perceived value of the public, thereby increasing the probability that the public chooses to panic purchase.