= \arg\min \mathbb{E}[\sum_{t=k}^{\infty} \gamma^{t-k} l_{i,k} ],$$ where $\gamma\in[0,1]$ is a discount factor. The objective function depends on all agents’ current and future actions. It is difficult to directly obtain an analytical solution of . Later we will use multi-agent reinforcement learning to obtain a numerical solution.

In this section, to simplify the problem, we consider a single stage game where the agents have zero discount of the future, i.e., $\gamma = 0$. Hence the objective function is reduced to $$u_{i,k} = \arg\min \mathbb{E}[l_{i,k} ],$$ which only depends on the current actions of agents. According to the state transition probability in \[table:dynamics\], the expected cost is $$\label{eq: expected cost}
\mathbb{E}[l_{i,k} ]  = \left\{\begin{array}{ll}
1-\Pi_{j\in\mathcal{I}_k} (1-\min\{u_i,u_j\}) + \alpha_i p(u_{i,k}) & \text{if }x_{i,k} = 0\\
1 + \alpha_i p(u_{i,k}) & \text{if }x_{i,k} = 1
\end{array}\right..$$

Nash Equilibrium
----------------

![image](e_curve.pdf){width="\linewidth"}

According to , the expect cost for an infected agent only depends on its own action. Hence the optimal choice for an infected agent is $u_{i,k} = \bar u := \arg\min_u p(u)$. Then the optimal choice for a healthy agent satisfies: $$\begin{aligned}
u_{i,k} &=& \arg\min_u [1-\Pi_{j\in\mathcal{I}_k} (1-\min\{u,\bar u\}) + \alpha_i p(u)],\\
&=&\arg\min_u [1-(1-\min\{u,\bar u\})^{m_k} + \alpha_i p(u)].\label{eq: healthy agent cost}\end{aligned}$$ Note that the term 