> I find it difficult to map the 'symmetric game' definition "the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them" onto a turn based game,

Yeah so probably the better way to think about it might be with the payoff matrix. Because symmetry is actually about the strategy. That's why there are the notes about the laddering in Go. But the payoff of a symmetric game is actually when A = -A^T. So if we have a 2x2 game a symmetric zero-sum one is where the payoff matrix might look like [[0, 1], [-1, 0]] Where we're like an inverse-identity matrix (actually anti-symmetric) but the diagonals are opposite. Maybe it is best to think about this from a geometric perspective, this symmetry here (in this specific example) is a rotation matrix. That's what it does when applied to another matrix. Recall our standard form is [[cos(theta), -sin(theta)],[sin(theta), cos(theta)]]. Pretty easy to get our matrix from there if you remember that cos(90)=cos(180)=0 and sin(90)=1 but sin(180)=-1. So our angle of rotation is 180 degrees (or pi radians). You could also see that if we made the two columns vectors we'd see they pointed in opposite directions. That's the symmetry! Okay, yeah, maybe that's confusing lol. But I find it helpful to see matrices as transforms and I wish this was stated a bit more clearly and often.

So now that we maybe understand that, symmetry is about a __strategy__, not a player. Because our payoff matrix is strategy based. For example, our strategy for rock-paper-scissors is to pick each outcome 1/3 of the time, which gives us this symmetric payoff. But if we pick rock every time we don't get that payoff, right? So it's actually not about who goes first or second but also includes the strategy aspect.

At least that's my understanding which a lot is prompted by this conversation (thanks!)

The reason I'm finding the go argument hard is thinking of a basic "entropy" based strategy (it'll serve you well in boardgames, especially when sight reading). The idea is if you don't know the best move, play the move that gives you the most future moves. It'll trick you into thinking that this strategy is actually simple, it isn't. So in the game of Go, this isn't reasonably different from making a random move! Because there are just so many. And realistically your strategy is going to be the composition of many different strategies. Like you said, pull out the decision tree but we can actually abstract this a bit more and have a decision strategy tree that's a superset to our strategy that's a response (e.g. a ladder is set up so we play the laddering strategy). The reason I'm not buying the argument isn't about the logic, it is about the possible move sets. Even with super-ko (the board cannot return to a state it has previously been at any time in the game (must be fun to keep track of...)). So forgetting about all the extras that are played in go, passing shouldn't result in a meaningful change in the number of possible strategies. But this argument might actually be an argument in favor of symmetry, not against it. Coming back to Chess, we know that game __is not__ symmetric. Why? Because white has different strategies than black. If instead the first "move" is to flip a coin and that decides who is white and who is black, then the game actually becomes symmetric. Kinda wild...

I didn't read this, but a glance suggests that black dominates in smaller games

http://erikvanderwerf.tengen.nl/pubdown/thesis_erikvanderwer...