In the article entitled Non-Cooperative Game written by Nash in 1951, he discussed about the symmetries of games. Due to my lack of basic knowledge in permutations and symmetries, I looked up some basic concepts of them, but they're still quite unclear to me.
In this article, it says
A symmetry of a game will be a permutation of its pure strategies which satisfies certain conditions, given below. If two strategies belong to a single player, they must go into two strategies belonging to a single player.
What does this symmetry look like? Here's my interpretation:
A symmetry, say $\phi$, is a bijection with the set of "all" pure strategies (regardless of players), say D, as its domain to D itself. And this bijection satisfies that if two strategies, $x$ and $y$, belong to a single player $i$, then $\phi(x)$ and $\phi(y)$ are strategies of a single player $j$, who could or could not be $i$.
Is there any problem in my interpretation?
And here's one more little problem:
The article continues saying that if $\phi$ is the permutation of the pure strategies it induces a permutation $\psi$ of the players. What does it mean by the word $``$induces$"$? And what does $\psi$ look like?
Here is the link to this article:
Yes, you're correct. Symmetry, in this context, means just that. It's little more than notation.
The bijection $\phi$ decomposes into a composition (the product $\Pi$ denotes composition)
$$ \phi = T \circ(\Pi_i \phi_i), $$
where each $\phi_i$ only permutes player $i$'s pure strategies and $T$ is a permutation on the set of players.
The map $T$ is the permutation on players induced by $\phi$. It is the induced quotient map, in mathematical language. Nash requires his symmetry to respect the equivalence classes labeled by players, so this symmetry descends to a symmetry of players.