Notation Mistake in Game Theory Textbook?

Question

I am currently reading Fudenberg and Tirole's "Game Theory" and I wondering if the following is a mistake in notation. I am currently reading about multistage games with observable actions, the author defines a strategy to be a sequence of functions $\{s_i^k\}_{k=0}^K$ where the first stage has an "empty history" and so each strategy must begin with a function $s_i^0$ whose domain is $\emptyset$.

As we know there is a unique map from the empty set into any other set, so this doesn't make any sense mathematically as it doesn't allow for what the author intended wherein each player should be able to start with whatever action they want for the first stage of the game.

Is there something I am missing here?

Answer 1

Looking at Fudenberg and Tirole's book, there is no contradiction in their notation. Here are the relevant quotes from section 3.2.1 (if I missed anything, please let me know):

We let $h_0=\varnothing$ be the "history" at the start of the play. [...]

Continuing iteratively, we define $h^{k+1}$, the history at the end of stage $k$, to be the sequence of actions in the previous periods, $h^{k+1}=(a^0,a^1,\dots,a^k)$. [...]

If we let $H^k$ denote the set of all stage-$k$ histories, and let $A_i(H^k)=\bigcup_{h^k\in H^k}A_i(h^k)$, a pure strategy for player $i$ is a sequence of maps $\{s^k_i\}_{k=0}^K$ where each $s^k_i$ maps $H^k$ to the set of player $i$'s feasible actions, $A_i(H^k)$ [...] .

Now, let us unpack these definitions. We see that $s^0_i$ is a function from $H^0$ to $A_i(H_0)$. What is $H^0$? It is the set of stage-zero histories. There is only one possible zero-stage history; in the first sentence I quoted, the only possible initial history is defined to be $\varnothing$. Therefore, $H^0=\{h^0\}=\{\varnothing\}$, the set whose unique element is $\varnothing$. Finally, we conclude that $s^0_i$ is simply a function from a set of size one, $H^0$, to the set $A_i(H^0)$. This is exactly what is intended; choosing a function from a set of size one to a $A_i(H^0)$, is equivalent to choosing a particular element of $A_i(H^0)$, which is just choosing an initial action.

Basically, it seems like you confused $H^0$, the set of zero-stage histories, with $h^0$, the unique element of the set of zero-stage histories. $\varnothing$ is not the domain of $s^0_i$, it is the input into $s^0_i$.