An unusual notation in game theory

Question

As a game theorist, I have come across this notation previously but never found it disturbing. However, in the game theory book by Myerson, this notation is taken to its limits, and I think that it might be mathematically questionable, unless somebody helps me to make sense of it. What am I talking about? Myerson writes (p. 154): "A pure strategy for a player in $\Gamma^{e}$ is a function that specifies a move for the player at each of his possible information states in the game. We let $C_{i}$ denote the set of such pure strategies for any player $i$ in $\Gamma^{e}$, so $$C_{i} = \times_{s \in S_{i}}D_{s}."$$ Ok, so $C_{i}$ is a set of functions that is denoted as if it was a Cartesian Product. More specifically, we denote the set of functions that assign to each element of $s \in S_{i}$ an element in $D_{s}$ by the symbol $\times_{s \in S_{i}}D_{s}$. Fine, I can live with that. A bit later, however, he writes $$\times_{s \in S^{*}} D_{s} = \times_{i \in N}\times_{s \in S_{i}} D_{s},$$ where $S^{*} = \bigcup_{i \in N} S_{i}$ (and the sets $S_{i}, i \in N,$ form a partition of $S^{*}$). I find the equality sign very irritating. Applying the same interpretation as before, on the left hand side we have the set of functions that assign to each element $s \in S^{*}$ an element in $D_{s}$. On the right hand side, however, we have the set of functions that map each player $i \in N$ to an element in $\times_{s \in S_{i}} D_{s}$, i.e. functions that assign to each player $i \in N$ a function that assigns to each element $s \in S_{i}$ an element from $D_{s}$. Are the sets on the left and the right hand side of the equality sign identical? Rather not, because the functions in the set on the left hand side have a different domain as the functions in the set on the right hand side. So, either this notation is extremely sloppy or I do not interpret it in the right way. Is there any reasonable possibility how one can interpret this so that the equality sign is legitimate?

Answer 1

You can identify $S^{*}$ with $\{(i,s) | s \in S_i \} \subset N\times S^{*}$, which gives an identification of the two sides: a function taking $s\in S^{*}$ as an argument is the same as a function taking an $(i,s)$ such that $s\in S_i$.