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),[1] 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,
.
References
- ↑ 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.