The concept of information structure in incomplete information games

Question

I would like your help to understand the concept of information structure in the incomplete information game at p.6-7 of this paper.

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Let me summarise the game as described in the paper.

There are $N\in \mathbb{N}$ players, with $i$ denoting a generic player.

There is a finite set of states $\Theta$, with $\theta$ denoting a generic state.

A basic game $G$ consists of

• for each player $i$, a finite set of actions $A_i$, where we write $A\equiv A_1\times A_2\times ... \times A_N$, and a utility function $u_i: A\times \Theta \rightarrow \mathbb{R}$.

• a full support prior $\psi\in \Delta(\Theta)$.

An information structure $S$ consists of

• for each player $i$, a finite set of signals $T_i$, where we write $T\equiv T_1\times T_2\times ... \times T_N$.

• a signal distribution $\pi: \Theta \rightarrow \Delta(T)$.

A decision rule of the incomplete information game $(G,S)$ is a mapping $$ \sigma: T\times \Theta\rightarrow \Delta(A) $$

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My question:

I interpret $\pi(t|\theta)$ as a probability that, conditional on the realisation $\theta$ of the state, the players receive as signals $t_1,...,t_N$, respectively. According to the given information structure, signals are more or less informative.

If this interpretation is correct, then I'm confused about the first sentence at p.7 of the linked paper: "If there is complete information, i.e., if $\Theta$ is singleton, [...]".

My first guess was that complete information is characterised by specifying $S$ and not by restricting the support of the state. In other words, I thought that complete information corresponds to the information structure $\bar{S}\equiv (\bar{T},\bar{\pi})$ such that

• for each player $i$, $\bar{T}_i=\Theta$.

• $\bar{\pi}(\theta|\theta)=1$.

Where am I wrong? Why the author defines complete information as having $|\Theta|=1$?

Answer 1

The two formulations are equivalent in the sense that if every type of player always learns the state of the world, there is no uncertainty and there is really no need to carry around the realized state as a variable. By assuming $|\Theta|=1$ you simplify the notation without losing generality (of course you lose some information, but this is irrelevant for the purposes of that paper). If you want, you can assume that the authors' statements are true state-by-state.

Answer 2

What you are defining is the complete information structure $\bar{S}$ with $\bar{T}_{i}=\Theta$ for all $i$ and $$ \bar{\pi}(t|\theta) = \begin{cases} \ 1, & \text{if} \;\; t_{i}= \theta \text{ for all} \; i \\ 0, & \text{otherwise}, \end{cases} $$ for all $\theta \in \Theta$.

Note that this is not the same as the case in which $\Theta$ is a singleton, but it is rather an extreme information structure. The same authors define this here (Bayes Correlated Equilibrium and the Comparison of Information Structures - Bergemann & Morris), it beats me why they didn't include this detail in the paper you are looking at.

Answer 3

Maybe a bit too late - but here goes.

Remember the objective of their paper: They want to characterize the set of Bayes Nash Equilibrium of a standard incomplete information game as a function of all possible information structures. To this effect they separate their model into 2 parts: an information structure, and the basic game (which is simply a standard incomplete-information game).

In their model $\theta\in\Theta$ is the only payoff relevant unknown state of the world. The signals provide information about $\theta$, but are not payoff relevant. As a result, incompleteness of information in the basic game (Which is the game of interest) is defined in terms of $\Theta$.

The key distinction is whether agents know $\theta$ when choosing their actions. If $\Theta$ is singleton, the agents know the payoff relevant state when choosing their actions. On the other hand, suppose $|\Theta|>1$. A perfectly revealing information structure spits out signals that reveal the state in this situation, but the actions are still taken before $\theta$ is actually realised. Due to the one-one relation between the signal realisations and the state under perfect information structure, the agents act as if the base game is a complete-information game.

P.S. They could have equivalently defined complete information as a degenerate prior with $|\Theta|>1$. But this raises the inconvenient issues of 0-probability events - something you dont want to get into when your paper is one big exercise in Bayes Rule.