User:IssaRice/Minus notation in game theory
In game theory, is a shorthand for . In other words, is a tuple of strategies of all the players other than player .
One then writes things like for the payoff of the strategy profile where each player chooses .
But to my mind, there is a type error here, because 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 , each should take , but the notation for player 2 is , which is reversed. Type error!
I think usually this is not a problem, because the reader can mentally put the 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), so that one can pass a set of strategies like 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 to have domain . This is sort of inelegant because now one cannot pass to an arbitrary . (One can only pass it to .)
- One can define a function for each player that is supposed to "put back in the right spot". Formally, is defined by . Now one can write (if the th strategy needs to be put in the correct place) and (if the strategies are already in order). Note that the altered suggested in the previous bullet is, in this notation, .
- To be even more precise, given finite sets of strategies , one can start with and replace each strategy with , to number them. Then for , one starts the numbering where left off, and so on.