Information and search

Question

Consider a game where a decision maker (DM) has to choose action $l\in \mathcal{Y}$ possibly without being fully aware of the state of the world $V$.

The choice set $\mathcal{Y}$ has cardinality $L$. The state of the world is an $L\times 1 $ vector and we denote its $l$-th element by $V_l$.

The state of the world has support $\mathcal{V}$.

When DM chooses action $l\in \mathcal{Y}$, she receives the payoff $V_l$. That is, she receives a payoff equal to the $l$-th element of the vector $V$.

Let $P_V\in \Delta(\mathcal{V})$ be the DM's prior.

The DM may also process some signal (formalised by the concept of information structure) to refine his prior and get a posterior.

Let us define the concept of 1 player Bayesian Correlated Equilibrium provided in Bergemann and Morris (2013,2016,etc.).

$P_{Y,V}\in \Delta(\mathcal{Y}\times \mathcal{V})$ is a 1 player Bayesian Correlated Equilibrium if

1) $\sum_{l\in \mathcal{Y}}P_{Y,V}(l,v)=P_V(v)$ for each $v\in \mathcal{V}$

2) $\sum_{v\in \mathcal{V}}V_l P_{Y,V}(l,v)\geq \sum_{v\in \mathcal{V}}V_k P_{Y,V}(l,v)$ for each $l$ and $k\neq y$.

Bergemann and Morris show that the set of Bayesian Correlated Equilibrium is equal to set of optimal optimal behaviours under a range of information structures.

QUESTION: Consider a search model, in which we design a protocol according to which the DM discovers information about the state of the world.

For example, suppose that the DM engages in a sequential search where he discovers the $l$-th element of $V$ if and only if the maximum utility secured up to that moment is lower than a reservation value.

Can this model always be written as prior/information structure/posterior? In other words, does the framework of Bergemann and Morris nest search models?

Answer 1

The solution concept of Bayes Correlated Equilibrium applies to games, viz strategic interactions, between multiple players. Thus, in a single person decision problem its use seems to me inappropriate or at least superfluous. A lot of effort has been spent over the last 70 years (dating back at least to Blackwell 1951, 1953) to explore the notion of information and information structures in decision problems. In that literature lies your answer.

What you have described is a search problem that seems analogous to a Wald-like sequential sampling problem.

Perhaps refer also to the recently burgeoning literature on rational inattention. There, care has been taken in order to explore so-called "posterior separable" cost (of information) functions, which allows one to write the cost of information just as a function of the induced posteriors. See recent work by Caplin, Dean, and others.

Answer 2

It is true that BCE is more appropriate for games. You can certainly use it for single-agent decision problems, but it is really an overkill.

Further, BCE is a solution concept that is agnostic about what other information people receive, or how they receive it. Therefore, I will not say that BCE nests search models. If anything, the BCE of a single-agent problem simply describes the actions that such an agent could possibly choose after receiving some information (perhaps after searching).

In that sense, you can study how the BCE evolves in a search model. For example, in most search models the agent starts agnostic enough so that all actions can be a best response for some posterior belief, (so the BCE will be a relatively large set). As the player searches and learns more information, her beliefs usually concentrate somewhere. For example, the agent becomes more confident about the state of the world and in most search models a cascade forms (or the agent stops searching). A cascade is simply a situation where the agent's current prior, $\underline S$, is concentrated enough that the BCE is a singleton. That is, regardless of any possible extra information that the agent could get, there is only one rationalizable action.

I cannot see how using BCE tools to study a search model could be helpful, but this is how I would connect those two kinds of literature.