Optimal strategy in a single-agent choice problem under uncertainty

Question

Consider the following single-agent choice problem under uncertainty.

Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{v})$. First, let nature draw a realisation $v$ of $V$ from $P_V$. Then, let the decision maker choose an action $y\in \mathcal{Y}$, with $\mathcal{Y}$ finite, without observing $v$. Upon the decision has been made, the decision maker gets a payoff $u(y,v)$.

For example, suppose that $\mathcal{Y}\equiv \{1,2,3\}$. $V$ is a $3\times 1$ random vector, $V\equiv (V_1,V_2,V_3)$. $P_V$ is the 3-variate standard normal distribution. $u(y,v)\equiv v_y$.

What is the definition of an optimal strategy for the decision maker in this setting?

I'm thinking about using "a sort of" Bayesian Nash equilibrium for a single-agent setting, i.e., an optimal strategy is $P_Y\in \Delta(\mathcal{Y})$ such that, $\forall y\in \mathcal{Y}$ such that $P_Y(y)>0$ and $\forall \tilde{y}\neq y$, we have that $$ \sum_{v\in \mathcal{V}} u_i(y,v)P_V(v)\geq \sum_{v\in \mathcal{V}} u_i(\tilde{y},v)P_V(v) $$ that is, in my example, $$ \sum_{v\in \mathcal{V}} (v_y -v_{\tilde{y}}) P_V(v)\geq 0 $$

But maybe a pure strategy is what people use?

Is existence and uniqueness obvious (at least in my example with a normal distribution)?

Could you also provide a reference discussing definition, existence, multiplicity?

Answer 1

Your problem can be simply expressed as $$\arg\max_{y\in\mathcal{Y}}\sum_{v\in\mathcal{V}}u_i(y,v)P_V(v)$$ Note that the subscript in $u_i$ is un-necessary. Furthermore, this is not a game so "mixed strategies" are only a solution if the maximum is not unique, but you should not worry about them (more on that later).

First, note that the maximization problem is equivalent to the inequality you present. Presenting it as a maximization problem is more akin to how you solve a single agent problem, and circumvents the need to specify the quantifiers. However, strictly speaking, it only characterizes "pure strategies", (a better term would be deterministic actions), but this is, really, without loss of generality. A probabilistic action is justified only when the maximizer is not unique, in which case the $\arg\max$ operator will already be giving you a set, and any mixture between the maximizers in that set would be optimal.

Besides being more clear, another benefit from presenting it as a maximization problem is that you can use standard theorems to guarantee existence and uniqueness, for example, if $\mathcal{Y}$ is finite then $\sum u(y,v)P_V(v)$ is continuous in $y$ (using the discrete metric) and Weierstrass ensures the existence of a solution. In general you need a $\mathcal{Y}$ to be compact and $\sum u(y,v)P_V(v)$ to be continuous in $y$. Uniqueness is a bit more tricky, but if $\sum u(y,v)P_V(v)$ is strictly concave with respect to $y$, then uniqueness is guarantied.

In your particular example given the finiteness of the action space, existence is trivial, but not uniqueness. Knowing that $P_V$ is normal is not enough since we don't know anything about the shape of $u(\cdot)$.