# Existence of optimal strategy in a choice problem with uncertainty and information structure

Consider a decision maker choosing an action, $$y$$, from the finiteset $$\mathcal{Y}$$, possibly without having complete information about the state of the world.

More precisely, let $$V$$ be a continuously distributed random variable (or vector) representing the state of the world, with support, $$\mathcal{V}$$. Let $$P_{V}\in \Delta(\mathcal{V})$$ be the probability distribution of $$V$$.

Nature draws a realisation, $$v$$, of $$V$$ from $$P_{V}$$. The decision maker is not aware of $$v$$. However, she can refine her prior, $$P_{V}$$, upon reception of a private signal which may be informative about the state of the world. In particular, let $$T$$ be a random variable (or vector) representing the private signal received the decision maker, with support $$\mathcal{T}$$. Let $$P_{T|v}\in \Delta(\mathcal{T})$$ be the probability distribution of $$T$$ conditional on $$v$$.
Nature draws a realisation, $$t$$, of $$T$$ from $$P_{T|v}$$. The decision maker observes $$t$$, uses the Bayes rule to update $$P_{V}$$ with the posterior $$P_{V|t}\in \Delta(\mathcal{V})$$, and chooses an action, $$y\in \mathcal{Y}$$. Finally, the decision maker receives a payoff, $$u(y, v)$$.

Let us now define an optimal strategy of the decision maker.

A (mixed) strategy is $$P_{Y|T}\equiv \{P_{Y|t}\in \Delta(\mathcal{Y}): t\in \mathcal{T}\}$$, collecting probability distributions of $$Y$$ conditional on every realisations $$t$$ of $$T$$.

A strategy $$P_{Y|T}$$ is optimal if it allows the decision maker to maximise her expected payoff: $$\forall t\in \mathcal{T}$$, $$\forall y\in \mathcal{Y}$$ such that $$P_{Y|t}(y)>0$$, and $$\forall \tilde{y}\in \mathcal{Y}\setminus \{y\}$$ $$\int_{ \mathcal{V}} u(y,v) P_{T|v}(t) P_{V}(\text{d}v) \geq \int_{ \mathcal{V}} u(\tilde{y}, v) P_{T|v}(t) P_{V}(\text{d} v;\theta_v).$$

Question: does an optimal strategy always exists for any information structure $$S\equiv (\mathcal{T}, P_{T|V})$$? Could you make an example on non-existence, if any?

Proof: 1st- the set $$\mathcal{Y}$$ is finite (thus compact). 2nd- for each state of the world, $$v$$, $$u(y,v)$$ is continuous in $$y$$ using the discrete metric. 3rd- Given the linearity of integrals, for each $$t\in\mathcal{T}$$ we have that $$\int_{\mathcal{V}}u(y,u)P_{T|v}(v)P_V(dv)$$ is also continuous in $$y$$. Therefore Weierstrass theorem ensures the existence of a minimum and a maximum.
Let $$y^*_t$$ be one such maximizer when the signal is $$t$$ (note it can be different for each signal realization, and that the maximizer might not be unique. In the latter case, I arbitrarily choose one for simplicity). Finally, define $$P_{Y|T}(y|t)=\mathbb{1}\{y=y^*_t\}$$. I.e. to play $$y^*_t$$ with probability one when the signal is $$t$$.
In this setup, existence might only fail if $$\mathcal{Y}$$ was not compact, or if $$u(y,v)$$ was not continuous with respect to $$y$$. Note I said might, even if these two conditions are not present, an optimal might still exist.