Write search model as a signal density inducing posterior belief

Question

Consider a decision maker (DM) who as to choose an action from the finite set $\mathcal{Y}$ with cardinality $L. $

The payoff the he gets depends on the action chosen and the state of the world. The state of the world is denoted by $z\equiv (z_1,...,z_L)\in \mathcal{Z}^L\subseteq \mathbb{R}^L$. The payoff when the DM chooses action $y\in \mathcal{Y}$ and the state of the world is $z\in \mathcal{Z}^L$ is denoted by $u(y,z_y)$.

Before choosing the DM does not observe the state of the world. His prior on $z$ is denoted by $F_Z\in \Delta(\mathcal{Z}^L)$ and has marginals that are equivalent across $y$.

Before choosing the DM can discover $z$ by searching sequentially and paying cost $c_y$ when uncovering $z_y$ for each $y\in \mathcal{Y}$.

When search stops, the DM chooses action $y$ maximising his payoff among the searched actions.

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Question

Can we find a conditional signal density $P_{T}(\cdot|z)\in \Delta(\mathcal{T})$ such that the problem above can be rewritten as $$ \max_{y\in \mathcal{Y}}\sum_{z\in \mathcal{Z}} u(y,z_y) \underbrace{\frac{P_T(t|z) F_Z(z)}{\sum_{z\in \mathcal{Z}}P_T(t|z) F_Z(z)}}_{\text{Posterior belief by Bayes rule}} $$ for each $t\in \mathcal{T}$?

Answer 1

I am not entirely sure about what you mean with "Before choosing the DM can discover $z$ by searching sequentially and paying cost $c_y$ when uncovering $z_y$ for each $y\in Y$."

However, it sounds like you want your DM to choose both a stoping time, $t$, and an action, $y$. Since the DM is searching sequentially, the decision to continue searching or not will depend on how searching affects their beliefs and the cost of searching, which you are not fully specifying. Searching typically gives you access to some signal that will affect your beliefs, and the cost of searching is usually defined regardless of the result of the signal. You have not specified such searching signal, and it sounds like you only pay a cost for searching if you uncover state $z_y$. However, let me get to your question.

At a high level, you cannot re-write your problem through a signal density, $P_T(\cdot|z)$, because the re-write does not include the searching costs.

Argument for the statemet: Suppose there is a well-defined search problem and the decision-maker chooses $y\in Y$ only after the search phase has concluded (If these assumptions are not true, my argument is just strengthen). Given an optimal search strategy (that can only depend on the prior $F_Z$, the search costs $c_z$, and (perhaps) the interim beliefs during the search process), for each state of the world the DM will (probabilistically) end with some sequence of signals (one for each period they searched). Denote each possible sequence with a different element in $T$ and let $P_T(t|z)$ be the probability that the DM ends with sequence $t$ when the state is $z$. However, each sequence of signals will, in general, have a different cost. This cost cannot be, in general, captured by the probability density.

If searching was free, then sure, your problem is one of choosing $y$ to maximize the expected utility where expectations are taking with respect to some posterior beliefs that arise after searching, and the density $P_T(\cdot|z)$ is the object that summarizes the searching process that is assumed to be well-defined.