Confusion on Strategy Sets in a simultaneous move game

I have a confusion on how to define strategy sets;

I understand that in dynamic games strategy sets are defined as $\textbf{sets of functions}$ - see for example my previous post, link below:

Define and characterize equilibria of the following game

However I cannot square this with what would be a strategy set in a simultaneous move game; as an illustration consider the following simple game:

There are two players competing for a prize of $1$. Each player $i$ choses effort level $x_i \in \mathbb{R}$. When the efforts levels are $(x_1,x_2)$, the probability that player 1 wins the prize is given by $p(x_1,x_2)$ and the probability that player 2 wins the prize is given by $1-p(x_1,x_2)$. Player $i$'s cost of exerting effort $x_i$ is equal to $x_i$.

Players choose their effort levels simultaneously.

Now looking standard textbooks I suspect that the strategy sets of both players would be defined as: $S_1 = \mathbb{R}$, $S_2 = \mathbb{R}$ Since both players will choose an effort level $x\in \mathbb{R}$.

However given that a $\textbf{strategy}$ for player $i$ is defined as a complete contingent plan, it would be natural for me to define a strategy as a function:
$s_i: \mathbb{R} \to \mathbb{R}$,

that is for each possible effort level $x_j \in \mathbb{R}$ that player $j$ chooses, player $i$ will choose an effort level $x_i \in \mathbb{R}$. But then since $S_i$ is the $\textbf{set of}$ strategies it would appropriate to define the strategy set of this game for player i as:

$S_i = \{s_i : \mathbb{R} \to \mathbb{R}\}$ , that is the set of all possible functions $s_i$.

But then this definition would conflict with the original one I gave at the beginning.

So which of the two (if any) would be the correct "strategy set" of this simultaneous move game? Thank You!

You write " it would be natural for me to define a strategy as a function: $$s_i: \mathbb{R} \to \mathbb{R}."$$ There is some murkiness here, because what is the first $\mathbb{R}$ supposed to be? The strategy set of the other player? But that is not observed by player $i$, so it cannot factor into her decision. As a function the strategy would look actually look like $$s_i: \left\{I_i\right\} \to \mathbb{R},$$ where $I_i$ is the single information set of player $i$, so $\left\{I_i\right\}$ is a singleton set. There is a bijective mapping between the vector space of these strategies and the space $\mathbb{R}$ so it does no harm to define $S_i$ as equal to $\mathbb{R}$.
• thank you, but I don't understand: if the "strategy space is just the action space" why is it valid to specify $S_i$ as a function? (I mean even if we can specify it as a function, then the domain of this function will be constant..and surely cannot be R as I specified above); since each player has only one information set then an example of a strategy for player i and j would simply be: si=1, si=1 "no matter what the other player strategy is" and the respective sets of strategies would be $S_i=\mathbb{R},S_j=\mathbb{R}$ Oct 25 '15 at 20:10
• Ok thank You very much; now it is much clearer. So, now would you agree that $\textbf{an example of a strategy for player i}$ would be given by $s_i = k$ where $k \in \mathbb{R}$? Oct 25 '15 at 21:00