Solutions for finite static teams

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Solutions for finite static teams

#StochasticControl >[!thm] Existence for $N$ -agent team >For an $N-$ agent static stochastic team problem satisfying the four hypotheses, there exists at least one team-optimal solution.

Cor

We can relax $(c.1)$ into $(c.1')$ and the result still holds:

$(c.1')$ Let $\mathcal{N}_{h}$ and $\mathcal{N}_{s}$ be two complementary subsets of $\mathcal{N}$ (i.e. $\mathcal{N}_{h}\cup \mathcal{N}_{s}=\mathcal{N}$ , and $\mathcal{N}_{h}\cap \mathcal{N}_{s}=\emptyset$ ) s.t. $S^{i}$ is compact $\forall i\in\mathcal{N}_{h}$ and $S^{j}\equiv \mathbb{U}^{j}$ $\forall j\in\mathcal{N}_{s}$ . Assume that $\sum_{j\in \mathcal{N}_{s}}|u^{j}|\to \infty,\ L(\xi;u^{1},\dots,u^{N})\to \infty$ a.s., for every fixed $u^{i}\in S^{i},i\in \mathcal{N}_{h}$ .

Theorem (2.4.3)

In addition to the four hypotheses, let

$S^{i}$ be a convex set for each $i\in\mathcal{N}$ , and;
$L(\xi;\mathbf{\cdot})$ be strictly convex on $\mathbf{U}$ a.s.. Then, the stochastic team problem admits a unique team-optimal solution.

Lemma (2.4.1)

Let $L:\mathbb{R}^{m_{1}}\times\dots \times \mathbb{R}^{m_{N}}\to \mathbb{R}$ be a convex (deterministic) loss function, with pbp optimal solution $\mathbf{u}^{\circ}:=(u^{1\circ},\dots,u^{N\circ})$ . If $L$ is continuously differentiable at $\mathbf{u}^{\circ}$ , then $\mathbf{u}^{\circ}$ is globally (team) optimal.

Intuition

So all we need is our loss function to be differentiable at our pbp solution for it to also be team optimal.

Now, using this lemma and the definition of a Stationary Team Policy we can show the following:

Theorem (2.4.4)

For an $N$ -agent static stochastic team problem, let

The hypotheses $(c.3)$ and $(c.4)$ be satisfied;
$S^{i}$ be an Open convex subset of a finite dimensional Vector Space for each $i\in\mathcal{N}$ ;
$L(\xi;\cdot)$ be convex and continuously differentiable on $\mathbf{S}:=S^{1}\times\dots \times S^{N}$

Under these conditions, if the policy $\underline{\gamma}^{\circ}$ , taking values in $\mathbf{S}$ , is stationary, it is team-optimal.

Linked from

Stationary Team Policy