# Terminology and formalization for “anonymous” game forms

I have an intuitive notion of what an "anonymous" (extensive) game form $\Gamma$ is. In my mind an "anonymous" game is a game in which the players' identity does not matter. This includes the (again intuitive) ideas that

1) The players' strategy spaces are "identical" (e.g. a normal form game in which players have the same action spaces, or a centipede game in which the starting player is determined randomly).

2) There is no discrepancy in the order in which players play (at least stochastically). For instance, the probability that any player play first (or be one of the players who play first simultaneously) should be the same, and similarly for the second, third, ... play.

3) For every (strategy) profile $s = \{s_1,\dots,s_n\}$ in the space of profiles and every permutation $\pi : \{1,\dots,n\} \leftrightarrow \{1,\dots,n\}$, the outcome of $\pi(s) = \{s_{\pi(1)},\dots,s_{\pi(n)}\}$ is the same as the outcome of $s$, up to permutation (something like $\Gamma_i(\pi(s)) = \pi(\Gamma_i(s))$, e.g. an ultimatum game in which the identity of the player who makes the offer is determined randomly).

With the help of @ml0105, I realized that 1) and 3) identify the class of symmetric games (in fact, I think 3) is more restrictive than needed for a game to be symmetric, but let's leave it there). Hence I guess my question boils down to :

is there a common name for symmetric sequential games that also satisfy 2)?

Note : I am aware that the term "anonymous game" is already in use in the literature (e.g. Daskalakis and Papadimitriou (2007), Computing Equilibria in Anonymous Games, or https://en.wikipedia.org/wiki/Succinct_game#Anonymous_games) and defines a different class of game than the one I tried to intuitively describe about.

• You mean a symmetric game? – ml0105 Jan 6 '16 at 23:53
• @ml0105: I thought of symmetric games as being the class I had in mind but I think there are a couple of differences. At least two problems/differences I can think of 1) I am not sure of how symmetric games are defined for extensive form games (and haven't been able to find a formal definition thereof, any reference is welcome). 2) I am interested in game forms here, not in games. In particular, this means the outcome must be independent of the identity of the players, but the utilities/payoffs may depend on the identity of the players. Does that make sense? – Martin Van der Linden Jan 7 '16 at 0:40
• 1) Every dynamic game can be represented as a normal form game. Regardless, symmetry is what you would expect it to be. In fact, based on what you're describing, symmetry is condition (2) in your OP. 2) Perhaps just the class of symmetric sequential games? – ml0105 Jan 7 '16 at 2:23
• @ml0105 : thanks for following up. 1) I realized I was not clear on the definition of symmetry. The formal definition from Dasgupta and Maskin (1986) reproduced on en.wikipedia.org/wiki/Symmetric_game helped. 2) I could not find a good definition of "symmetric sequential game" (googeling proved of little help). Do you have a reference for it or is it a term you are proposing? 3) Anyhow, I should clarify my question. I feel like "symmetric sequential game" is close to were I want to get at, but not quite there, although for reasons which are hard (if not impossible) to guess from what I wrote. – Martin Van der Linden Jan 7 '16 at 4:47
• not quite it, but distantly related is Thompson's "interchange of moves" condition from his paper "Equivalence of games in extensive form". – HRSE Jan 7 '16 at 5:11