# User:IssaRice/Minus notation in game theory

In game theory, $s_{-i}$ is a shorthand for $(s_1,\ldots,s_{i-1},s_{i+1},\ldots,s_n) \in S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n$. In other words, $s_{-i}$ is a tuple of strategies of all the players other than player $i$.

One then writes things like $v_i(s_i, s_{-i})$ for the payoff of the strategy profile where each player $i$ chooses $s_i$.

But to my mind, there is a type error here, because $v_i \colon S_1 \times \cdots \times S_n \to \mathbf R$ is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of $n=2$, each $v_i$ should take $(s_1,s_2) \in S_1\times S_2$, but the notation $(s_i, s_{-i})$ for player 2 is $(s_2,s_1)$, which is reversed. Type error!

I think usually this is not a problem, because the reader can mentally put the $s_i$ in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable),[1] so that one can pass a set of strategies like $\{s_1, \ldots ,s_n\}$ and there would be no confusion even if the ordering is different.

Here are two more ideas for dealing with this notation:

• One can change each $v_i$ to have domain $S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n$. This is sort of inelegant because now one cannot pass $s \in S_1\times \cdots \times S_n$ to an arbitrary $v_i$. (One can only pass it to $v_1$.)
• One can define a function $g_i$ for each player that is supposed to "put $s_i$ back in the right spot". Formally, $g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n$ is defined by $g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})$. Now one can write $v_i(g_i(s_i, s_{-i}))$ (if the $i$th strategy needs to be put in the correct place) and $v_i(s)$ (if the strategies are already in order). Note that the altered $v_i$ suggested in the previous bullet is, in this notation, $v_i \circ g_i$.

## References

1. To be even more precise, given finite sets of strategies $S_1,\ldots,S_n$, one can start with $S_1$ and replace each strategy $s_i$ with $(i, s_i)$, to number them. Then for $S_2$, one starts the numbering where $S_1$ left off, and so on.