The N-Agent Cost Functional

Let $N$, $l$, $\tilde{l}$, $m$ be natural numbers with $N\ge1$, $l\ge2$, $\tilde{l}\ge1$, $m\ge1$, and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$. Adopt the setting of the controlled $N$-agent dynamics: a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate bound $B$, an observation-rate family $\tilde{\beta}$ on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, and an observation-driven control policy $h$ with horizon $T$, control dimension $m$ and $\tilde{l}$ channels which is $\mathcal{A}$-valued. Let $(L,G)$ be population cost data on $l$ states with control dimension $m$.

Let $(\sigma^i,\Upsilon^\upsilon,\alpha)$ be a solution of the controlled $N$-agent dynamics on $[0,T]$ for these data, with empirical state measure $\Sigma_t$; such a solution exists, and the expectation below is well defined in $(-\infty,+\infty]$ and does not depend on which solution is taken, by clauses (ii) and (v) of the existence and uniqueness theorem. The $N$-agent cost of the policy $h$ is

the integral and the expectation being those of clause (v).