[Math] Game Theory: Can someone explain the notation used in the definition of “best response”

game theorynotation

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 $u_i$

Best Answer

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.

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