correlated equilibrium,Nash equilibrium, equivalent definition, unclear comment in the Tirole's Fudenberg's book

Question

Here on page 58, there is a sentence in the first paragraph which is completely unclear to me: "(The only reason this isn't completely obvious is that there may be several information sets $h_i$ where player $i$ played $s_i$, in which case his information set has been reduced to $s_i$ alone.)"

First, "reduced" mean loosing information or on the other hand specifying more precisely to $s_i$?

What is the precise meaning of the sentence quoted? I.e. why different information sets $h_i$ where $i$ played $s_i$ lead to ""new?"" information set $s_i$?

Answer 1

The previous sentence reads

Given $\mathfrak s=(\mathfrak s_1,\dots, \mathfrak s_n)$, equilibrium with respect to $(Ω,\{H_i\},\tilde p)$, set $p(s)=\sum\tilde p(ω)$ where the sum runs over all $ω\in Ω$ such that $$\mathfrak s_i(ω)=s_i $$ for every $i\in I$.

So, all states of the world $ω\in Ω$ where $\mathfrak s$ was recommended are now identified as one state. E.g. assume $n=2$ players and two states $ω_1$ and $ω_2$, where the players are recommended under $\mathfrak s$ to play $s_1(ω_1)=s_1(ω_2)=s_1$ and $s_2(ω_1)=s_2(ω_2)=s_2$. So, the players know in which state they are and then "hear" their recommendation.

Now, under the transformation, you identify these two states as one, called $(s_1,s_2)$. This occurs with probability $\tilde p(ω_1)+\tilde p(ω_2)$ and the players do not get informed about the exact $ω$ that has occured (so they lose information) but only about the recommendation.