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I would like your help to understand the concept of expansion of an information structure in the incomplete information game at p.6-9 this paper.


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 $$ \sigma: T\times \Theta\rightarrow \Delta(A) $$


Expansion:

Consider two information structures, $S^1\equiv (T^1, \pi^1)$ and $S^2\equiv (T^2, \pi^2)$. We say that $S^*\equiv (T^*, \pi^*)$ is a combination of $S^1$ and $S^2$ if

  • $T_i^*=T_i^1\times T_i^2$ $\forall i$.

  • $\pi^*:\Theta \rightarrow \Delta(T^1\times T^2)$ has $\pi^1$ and $\pi^2$ as marginals.

An information structure $S^*$ is an expansion of an information structure $S^1$ if there exists an information structure $S^2$ such that $S^*$ is a combination of $S^1$ and $S^2$.


My question:

  • The game, as it is described by the authors, seems to assume that, before receiving the signal $T_i$, each player $i$ knows nothing about what will be the realisation of the state. I call this as the baseline level of information assumed.

    (For example, in other contexts, one may assume that the state is a vector of size $N\times 1$ and, before receiving the signal $T_i$, each player $i$ knows the realisation of the $i$th component of such a vector. This would correspond to another kind of baseline level of information)

  • Let $\underline{S}$ denote the information structure that is totally uninformative, i.e., it does not add anything to the baseline level of information assumed (also called DEGENERATE at p.26 of the linked paper). In other words, $\underline{S}$ consists of

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

    (b) a signal distribution $\pi: \Theta \rightarrow \Delta(T)$ such that $\pi(\cdot|\theta)=\tilde{\pi}$ $\forall \theta \in \Theta$ for some $\tilde{\pi}\in \Delta(T)$. In other words, the conditional probability is equal to the unconditional one and our belief on the probability distribution of the state is not updated.

    Notice that there are many ways to characterise the uninformative information structure (just by varying $T$ and $\tilde{\pi}$).

  • Let $\mathcal{S}$ denote the collection of all possible information structures. More precisely,

    $$ \mathcal{S}\equiv \{S| T \text{ is a separable metric space}, \text{ $\pi:\Theta \rightarrow \Delta(T)$ is a probability measure on $(T,\mathcal{B}(T))$}\} $$ where $\mathcal{B}(\cdot)$ denotes Borel sigma algebra.

    Note that $\mathcal{S}$ contains all possible ways to characterise the uninformative information structure.

  • Question: Can we show that, for a given $\underline{S}$, each $S\in \mathcal{S}$ is an expansion of $\underline{S}$ (including $S=\underline{S}$)?

    This seems to me to hold at the light of Theorem 1 combined with reading at p.26 of the paper "Now consider the case where the original information structure is degenerate (there is only one signal which represents the prior over the states of the world). In this case, the set of Bayes correlated equilibria correspond to joint distributions of actions and states that could arise under rational choice by a decision maker with any information structure"

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You have to be careful what you can say is that if $\underline{S}$ be given by $(T^1,\pi^1)$ and $S$ is any information structure, say given by $(T^2,\pi^2)$, then there exists an expansion $S^*$ of $\underline S$ given by $(T^1\times T^2, \pi^1\cdot \pi^2)$ where $\pi^1\cdot \pi^2$ is the product distribution of $\pi^1$ and $\pi^2$. Note that I have constructed the expansion, thus shown its existence.

The information structures, $S$ and $S^*$ are different since their support is in different spaces, $T^2$ vs $T^1\times T^2$. However, they are informationally equivalent. The equivalence here is in terms of the distribution of posterior beliefs induced by $S$ and $S^*$. Alternatively, if $\succsim$ is the Blackwell partial order, then $S\succsim S^*$ and $S^*\succsim S$. Your intuition, however, is correct, any information structure will be "essentially" an expansion of $\underline S$.

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