listed in \[fig: example cost\]. In the Nash Equilibrium, no agent will meet each other, since all agents except agent $1$ reduce their activity levels to zero. The actual cost (received at the next time step) equals to the expected cost (computed at the current time step).

   Agent ID   State $x_{i,k}$   Optimal $u_{i,k}$   Optimal $\mathbb{E}[l_{i,k}]$     Actual $l_{i,k}$
  ---------- ----------------- ------------------- ------------------------------- -----------------------
      1              1                  1                         1                           1
    2,3,4            0                  0                 $\alpha \exp(-1)$           $\alpha \exp(-1)$
    Total                                                                           $1+ 3\alpha \exp(-1)$

  : List of the agent decisions and associated costs in the Nash Equilibrium in the four-agent example.

\[fig: example cost\]

However, let us consider another situation where the infected agent chooses $0$ activity level and all other healthy agents choose $1$ activity level. The resulting costs are summarized in \[fig: example cost global\]. Obviously, the overall cost is reduced in the new situation. However, this better situation cannot be attained spontaneously by the agents, due to externality of the system which will be explained below.

   Agent ID   State $x_{i,k}$   Optimal $u_{i,k}$   Optimal $\mathbb{E}[l_{i,k}]$     Actual $l_{i,k}$
  ---------- ----------------- ------------------- ------------------------------- ----------------------
      1              1                  0                1+$\alpha \exp(-1)$        1+$\alpha \exp(-1)$
    2,3,4            0                  1                         0                          0
    Total                                                                           $1+ \alpha \exp(-1)$

  : List of the agent decisions 