Example of information structure in a one-player Bayes Correlated Equilibrium

Question

Model

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

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

When DM chooses action $y\in \mathcal{Y}$ and the state of the world is $v\in \mathcal{V}$, she receives the payoff $u(y,v)$.

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

The DM also processes some signal $T$ with support $\mathcal{T}$ distribution $P_{T|V}$ to refine his prior and get a posterior on $V$, denoted by $P_{V|T}$, via the Bayes rule.

Let $S\equiv \{\mathcal{T}, P_{T|V}\}$ be called "information structure".

A strategy for the DM is $P_{Y|T}$. Such a strategy is optimal if it maximises his expected payoff, where the expectation is computed using the posterior, $P_{V|T}$.

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Question

Suppose that the DM discovers that the state of the world, $v$, is a number in $[a,b]\subset \mathcal{V}$. Can this be written as an information structure (i.e., a distribution of a signal)?

Answer 1

The signal structure must specify what the DM learns in all possible states of the world, so I think your question should read:

"Suppose that the DM discovers whether the state of the world $v\in V$ is in $[a,b]\subset V$ or not. Can this be written as an information structure?"

or should read:

"Suppose that the DM discovers the state of the world $v\in V$ whenever it is in $[a,b]\subset V$ and learns the minimum otherwise. Can this be written as an information structure?".

From the comments of the question, I think you are more interested in the first one. Regardless, a couple of comments are important:

1. You want the DM to assign positive probability to an interval, and there are many ways to do so. There are some restrictions that are coming from your prior, but they are relatively mild. The most important one is that $P_V(v)>0$ for all $v\in [a,b]$. That is, the prior must assign positive probability to all numbers in the interval.

2. Even when you decide how much probability you want to assign to each point (assuming it is feasible given the restrictions mentioned above), there are many signal structures that can achieve this (Thus, I will be providing just a couple of examples).

Example 1:

Suppose $V=\mathbb{R}$ (the real line), and $P_V$ is the normal standard distribution over the reals, suppose that you are interested in finding a signal such that the posterior $P_{V|T}$ is the normal standard distribution truncated to $[a,b]$. Then the signal structure can simply be as follows:

Let $\mathcal{T}=\{blue,red\}$ $$P_{T|V}(blue|v)=\left\{\begin{array}{c c} 1 & ; v\in[a,b]\\ 0 & ; v\notin [a,b] \end{array}\right. \ \ \ \forall v\in V $$

Of course $P_{T|V}(red|v)=1-P_{T|V}(blue|v)$.

Notice that if the DM gets a signal "blue", they infer that the state is in $[a,b]$, and because the signal "blue" is equally likely to be received for any state in $[a,b]$, then the posterior has the same shape as the prior (i.e. it will be the truncated normal). Contrast this with the following signal: (for simplicity, let's assume that $[a,b]=[1,2]$:

$$P_{T|V}(blue|v)=\left\{\begin{array}{c c} \frac{1+v}{3} & ; v\in[a,b]\\ 0 & ; v\notin [a,b] \end{array}\right. \ \ \ \forall v\in V $$

and $P_{T|V}(red|v)=1-P_{T|V}(blue|v)$

Now, receiving the signal "blue" also makes the DM infer that the state is in $[a,b]$, but the signal structure sends message "blue" more often when the state is closer to $b$, so the posterior will assign a higher probability to states closer to $b$ than what the truncated normal would do, and less probability to the states closer to $a$.

Example 2:

If instead, what you wanted from your signal was the second interpretation. Namely, "Suppose that the DM discovers the state of the world, $v\in V$, whenever it is in $[a,b]\subset V$ and learns the minimum otherwise. Can this be written as an information structure?"

You can simply let $T=\mathbb{R}$ and define the signal structure as follows: (for simplicity, assume again that $[a,b]=[1,2]$)

$$P_{T|V}(t|v)=\left\{\begin{array}{c l} 1 & ; \ t=v \ \& \ v\in[a,b]\\ 1 & ; \ t=0 \ \& \ v\notin[a,b]\\ 0 & ; \ otherwise. \end{array}\right. \ \ \ \forall v\in V $$ (Here the message "0" can be any other random number that is not in $[a,b]$.)

Therefore the DM receives signal "0" whenever the state is outside the interval [a,b] (so they learn the minimum possible) or they learn the state of the world (by getting message "v") when the state is in the interval.