Game Theory: Can someone explain the notation used in the definition of "best response"

Question

I am reading a paper which states that that the best response correspondence of a player is mapping:

$B_i(s_{-i}): S_{-i} \Rightarrow S_i$ such that $B_i(s_{-i}) \in arg\ max_{s_i \in S_i} u_i(s_i, s_{-i})$

In particular:

• $s_{-i}$ denotes vector of all actions for all players except for $i$

• $S_{-i}$ denotes set of all action profiles for all players except for $i$

Can someone state in plain words as to what this mapping implies? In particular:

• Why is the set of all actions for players except for $i$ is used as argument for $B_{i}$?
• What does the $\Rightarrow $ represent?
• Why is $(s_i, s_{-i})$ used as argument for the utility function $u_i$

Answer 1

Presumably $B_i(s_{-i})$ is a best response (or possibly the set of best responses) by player $i$ when the others play $s_{-i}$. In a collection of game theory notation the set is called $BR_i(s_{-i})$. As it is the response to a particular play $s_{-i}$ by the others, it is reasonable for that to be an argument.

I suspect $\Rightarrow$ may just be a substitute for $\to$, so $B_i$ sends an element of $S_{-i}$ to an element of $S_i$.

$u_i(s_i,s_{-i})$ is simply the utility outcome for $i$ when player $i$ uses $s_i$ and the other players use $s_{-i}$, and hence has those as arguments. It might be possible to read this value in the pay-off matrix.

Answer 2

• $B_i$ gives the set of player i's optimal (i.e. utility-maximising) actions in response to a given profile of opponents' actions.
• ⇒ denotes a (set-valued) correspondence rather than a (single-valued) function.
• The utility function $u_i$ is one of the primitives of a normal-form game, mapping action profiles (such as $(s_i,s_{-i})$ into real numbers.