I’ve been wanting to do something like this for a while, so it’s good to see it properly worked out here.
If you wanted to expand this you could look at games which weren’t symmetrical in the players. So you’d have eight variables, W, X, Y and Z, and w, x, y and z. But you’d only have to look at the possible orderings within each set of four, since it’s not necessarily valid to compare utilities between people. You’d also be able to reduce the number of games by using the swap-the-players symmetry.
Oh, I hadn’t seen it before, but this file on wikipedia seems like it might be roughly that expanded version? InfoDirect link to pdf I haven’t looked closely at it though. The pdf doesn’t render in firefox for me, but does render in evince, my external pdf viewer.
I don’t in general think there’s anything wrong with comparing utilities between people in these things—that’s what I’m doing when I talk about whether 2W>X+Y - but it would be simpler not to do so. Still, even then I think extending to all 8 would give far too many possibilities to be manually tractable—I make it 4!⋅4!/2=288.
But it wouldn’t be too hard to write a program to classify them according to Nash equilibria, if someone wanted to do that. That might be a decent start.
I’ve been wanting to do something like this for a while, so it’s good to see it properly worked out here.
If you wanted to expand this you could look at games which weren’t symmetrical in the players. So you’d have eight variables, W, X, Y and Z, and w, x, y and z. But you’d only have to look at the possible orderings within each set of four, since it’s not necessarily valid to compare utilities between people. You’d also be able to reduce the number of games by using the swap-the-players symmetry.
Oh, I hadn’t seen it before, but this file on wikipedia seems like it might be roughly that expanded version? Info Direct link to pdf I haven’t looked closely at it though. The pdf doesn’t render in firefox for me, but does render in evince, my external pdf viewer.
I don’t in general think there’s anything wrong with comparing utilities between people in these things—that’s what I’m doing when I talk about whether 2W>X+Y - but it would be simpler not to do so. Still, even then I think extending to all 8 would give far too many possibilities to be manually tractable—I make it 4!⋅4!/2=288.
But it wouldn’t be too hard to write a program to classify them according to Nash equilibria, if someone wanted to do that. That might be a decent start.