dynamics and cost function;

-   All agents are homogeneous in the sense that they share the same dynamics and cost functions. And agents know that all agents are homogeneous, hence they know others’ dynamics and cost functions;

-   At time $k$, agents can measure $x_{j,k}$ for all $j$. But they cannot measure $u_{j,k}$ until time $k+1$. Hence, the agents are playing a simultaneous game. They need to infer others’ decisions when making their own decisions at any time $k$.

Cost Function
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We consider two conflicting interests for every agent:

-   Limit the activity level to minimize the chance to get infected;

-   Maintain a certain activity level for living.

We define the run-time cost for agent $i$ at time $k$ as $$l_{i,k} = x_{i,k+1} + \alpha_i p(u_{i,k}),$$ where $x_{i,k+1}$ corresponds to the first interest, $p(u_{i,k})$ corresponds to the second interest, and $\alpha_i > 0$ adjusts the preference between the two interests. The function $p(u)$ is assumed to be smooth. Due to our homogeneity assumption on agents, they should have identical preferences, i.e., $\alpha_i = \alpha$ for all $i$.

Agent $i$ chooses its action at time $k$ by minimizing the expected cumulative cost in the future: $$\label{eq: repeated game}
u_{i,k} 