Talk:Symmetric game

144 ordinally distinct 2x2 games?

Can someone link a reference to this statement? I've tried to count the number of ordinally distinct 2x2 games and I didn't obtain the result 144. Maybe I am wrong but even in this case, I think that a reference or more explanations would be helpful for a potential reader...
--91.183.237.30 (talk) 14:12, 15 August 2013 (UTC)

Possibly false common-interest claim

It is claimed that « Partha Dasgupta and Eric Maskin give the following definition, which has been repeated since in the economics literature

$U_{i}(a_{1},\ldots ,a_{i},\ldots ,a_{N})=U_{\pi (i)}(a_{\pi (1)},\ldots ,a_{\pi (i)},\ldots ,a_{\pi (N)}).$

However, this is a stronger condition that implies the game is not only symmetric in the sense above, but is a common-interest game, in the sense that all players' payoffs are identical. » But if we take 2 players, arbitrary $i,\pi$, and $j=\pi (i)=\pi ^{-1}(i)$, as all permutations are involutions in this case, the above yields

$U_{\pi (i)}(a_{1},\ldots ,a_{i},\ldots ,a_{N})=U_{j}(a_{1},\ldots ,a_{i},\ldots ,a_{N})=U_{\pi (j)}(a_{\pi (1)},\ldots ,a_{\pi (i)},\ldots ,a_{\pi (N)})=U_{i}(a_{\pi (1)},\ldots ,a_{\pi (i)},\ldots ,a_{\pi (N)}).$

Thus for 2 players both definitions are equivalent, but there are usual-sense symmetric 2-player games which are not common-interest games - eg prisoner's dilemma-type games. In fact for any number of players the definition of Dasgupta and Maskin does not imply that a game satisfying it is a common-interest game. There is a (noncanonical) correspondence between symmetric games in the usual sense and Dasgupta-Maskin symmetric games: as a symmetric $n$-game is determined by giving one $n$-tuple of payoffs for some $n$-tuple of strategies in each orbit under the action of the symmetric group, we can pick the $n$-tuple of payoffs of one representative $n$-tuple of strategies for each $S_{n}$-orbit in the original game - these are the noncanonical choices - then generate $n$-tuples of payoffs by letting the symmetric group act normally on the index in $U_{i}$ and act with the inverse action on $n$-tuples of strategies - one can also consider other actions of the symmetric group there. This yields (in a noncanonical way) a Dasgupta-Maskin symmetric game for each usual-sense symmetric game and vice versa. This construction also works for symmetric games under subgroups of $S_{n}$ - non fully symmetric games. A Dasgupta-Maskin symmetric game is what can be called a "contravariant symmetric game", "inverse-symmetric game", or just "DM game". I have not read the Dasgupta-Maskin article but i guess that for their purposes the "orientation" of the symmetric group action does not affect analysis, and for 2 players orientation is trivial.

Note that the above correspondence does not pass to Nash equilibria: already in the 2-player case instead of keeping the same utility functions - the choice we can define as canonical, only available in the case of 2 players - $U_{i}(a_{1},a_{2})$ for $i=1,2$ when $a_{1}\neq a_{2}$, under the correspondence, we can permute $U_{1}(a_{1},a_{2})\leftrightarrow U_{2}(a_{1},a_{2})$. In the case of prisoner's dilemma-type games this leaves the global optimum invariant but now being also a Nash equilibrium. It is not clear how Nash equilibria should behave when actions on tuples of (indices of) strategies are varied, keeping the action on (indices of) utilities fixed, to create new games from a symmetric one.

Further, one could consider the usual constructions in representation theory of finite groups: induction, restriction,... and study their interplay with concepts in game theory, and their complexity theory properties. It seems that equilibria could be refined, classified, and constructed thanks to the presence of symmetries. It does not look like there would be interesting applications in social sciences, but in representation theory and group theory game theory could provide interesting invariants, as game theory is about "complicated" fixed points and that may be an underexplored area in group theory.

[This whole talk section has grown somewhat inordinately as it seems necessary to clarify the notion of symmetry for a game.] Plm203 (talk) 18:50, 1 September 2024 (UTC)