and associated costs in a situation better than the Nash Equilibrium in the four-agent example.

\[fig: example cost global\]

Dealing with Externality
------------------------

For a multi-agent system, define the system cost as a summation of the individual costs: $$L_k := \sum_i l_{i,k}.$$ The system cost in the Nash Equilibrium is denoted $L^*_k$, which corresponds to the evaluation of $L_k$ under agent actions specified in . On the other hand, the optimal system cost is defined as $$\label{eq: system optimum}
L^o_k := \min_{u_{i,k},\forall i} L_k.$$ The optimization problem is solved in a centralized manner, which is different from how the Nash Equilibrium is obtained. To obtain the Nash Equilibrium, all agents are solving their own optimization problems independently. Although their objective functions depend on other agents’ actions, they are not jointly make the decisions, but only “infer" what others will do. By definition, $L^o_k \leq L^*_k$. In the example above, $L^*_k = 1+3\alpha \exp(-1)$ and $L^o_k = 1+\alpha \exp(-1)$. The difference $L^*_k - L^o_k$ is called the **loss of social welfare**. In the epidemic model, the loss of social welfare is due to the fact that bad consequences (i.e., infecting others) are not penalized in the cost functions of the infected 