# A Game-Theoretic Analysis for Distributed Honeypots
## 4 Game Theory Analysis
### Bayesian Equilibriums When \(p<2N/(r+2N)\)  
Then, we illustrate if there exist a dominant strategy for visitors in the case of \((\pi_{11},\pi_{11})\) and \(p<2N/(r+2N)\). The posteriori probabilities are set to \(\{P^{{}^{\prime}}(\theta_{111}|\pi_{11})=q,P^{{}^{\prime}}(\theta_{110}|\pi_{1 1})=1-q\}\).  
The payoff equations for an attacker can be calculated as:  
\[\mu_{\theta_{21}(\pi_{21})}=P^{{}^{\prime}}(\theta_{111}|\pi_{11})(-\eta c/N-b )+P^{{}^{\prime}}(\theta_{110}|\pi_{11})(\gamma a/N-b)=q(-\eta c/N-b)+(1-q)( \gamma a/N-b) \tag{5}\]  
\[\mu_{\theta_{21}(\pi_{20})}=P^{{}^{\prime}}(\theta_{111}|\pi_{11})\times 0+P^ {{}^{\prime}}(\theta_{110}|\pi_{11})\times 0=0 \tag{6}\]  
Similarly to Equations (3) and (4), we assume \(\mu_{\theta_{21}(\pi_{21})}=\mu_{\theta_{21}(\pi_{20})}\). Then, we have \(\gamma a-bN/\gamma a+\eta c=q\). When \(\gamma a-bN/\gamma a+\eta c>q\), it is obviously that the strategy \(\mu_{\theta_{21}(\pi_{21})}\) gains more profits than \(\mu_{\theta_{21}(\pi_{20})}\). We can infer that if \(q<(\gamma a-bN)/(\gamma a+\eta c)\), the strategy \(\pi_{21}\) will dominate in the view of an attacker. When \(\gamma a-bN/\gamma a+\eta c<q\), we obtain \(\mu_{\theta_{21}(\pi_{21})}<\mu_{\theta_{21}(\pi_{20})}\) where the attacker would like to choose the strategy \(\pi_{20}\) to abandon visiting the server.  
The payoff equations for a user can be calculated as:  
\[\mu_{\theta_{20}(\pi_{21})}=P^{{}^{\prime}}(\theta_{111}|\pi_{11})(-a)+P^{{}^ {\prime}}(\theta_{110}|\pi_{11})(a)=q(-a)+(1-q)(a) \tag{7}\]  
\[\mu_{\theta_{20}(\pi_{20})}=P^{{}^{\prime}}(\theta_{111}|\pi_{11})\times 0+P^ {{}^{\prime}}(\theta_{110}|\pi_{11})\times 0=0 \tag{8}\]  
Assuming that \(\mu_{\theta_{20}(\pi_{21})}=\mu_{\theta_{20}(\pi_{20})}\), we have \(1/2=q\). When \(q<1/2\), the strategy \(\pi_{21}\) will be better for the user. Otherwise, the strategy \(\pi_{20}\) is a better choice. Since our system should provide the user with normal services, the strategy \(\pi_{20}\) (i.e., the user does not visit the server) is inconsistent with the reality, which should be aborted.  
Equilivirus of visitors are illustrated in Table 9. Based on the dominant strategy \((\pi_{11},\pi_{11})\) and \(p<2N/(r+2N)\), the best access condition for an attacker and a user are individually \(q<\gamma a-bN/\gamma a+\eta c\) and \(q<1/2\), where  
\[\eta c\geq\gamma a-2 bN \tag{9}\]  
is inferred.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Player** & **Condition** & **Dominant Strategy** & **Equilibrium** \\ \hline _Realservice_ & \(p<2N/(r+2N)\) & \(\pi_{11}\) & \((\pi_{11},\pi_{11})\) \\ _Honeypot_ & \(p<2N/(r+2N)\) & \(\pi_{11}\) & \\ _Realservice_ & \(p>2N/(r+2N)\) & \(\pi_{10}\) & \\ _Honeypot_ & \(p>2N/(r+2N)\) & \(\pi_{11}\) & \((\pi_{10},\pi_{11})\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: List of equilibriums for server.  
In general, there are two Bayesian equilibriums for all players, shown in Table 10. In the condition of \(p<2N/(r+2N)\), \(q<1/2\) and \(\eta c\geq\gamma a-2 bN\), a Bayesian equilibrium is formed under the strategy set \(((\pi_{11},\pi_{11}),(\pi_{21},\pi_{21}))\). The other is obtained when \(\gamma a-bN/\gamma a+\eta c<q<1/2\) in the strategy set \(((\pi_{11},\pi_{11}),(\pi_{20},\pi_{21}))\). Such a strategy set illustrates an ideal circumstance in our life, indicating that the attacker will not launch an attack and the legal user will access the server.