Here is slight modification of the example Christoph presented in the last MEDS lunch (I don’t remember the attribution). I am going to describe three games, and, because for my main point I need to think of the game as taking place in the laboratory, I will call the game `an experiment’ and the players `subjects’, denoted S1 and S2. Below are full descriptions of three experiments. These descriptions are also given to the subjects.
Experiment 1: S2 leaves the room. S1 faces a disk divided into two sides and has to chose one side. His choice is recorded (say, marked on the back of the disk). Then the disk is randomly rotated. Then S2 returns and has to chose a side. If both subjects chose the same side, they get one dollar each. Otherwise, they are cast into the lake of fire.
Experiment 2: Same as Experiment 1, except that the disk is not rotated.
Experiment 3: The Experiment is called `Driving in Illinois’. The rest of the game is as in Experiment 2.
A game theorists who encounters one of this experiments might write down a payoff matrix
Whaddya know ? Same matrix for the three games ! Yet we don’t have to actually run these experiments in the the laboratory to know that the outcomes will be completely different (Never worry though — people did run these games in the laboratory).
What’s going on here is that the payoff matrix has a symmetry: it stays the same if both players change actions. From the matrix alone, without labeling the actions, there is no way for the players to distinguish between the actions, so much that, at least in the first experiment, I even hesitate to say that the players have two actions — it seems that whatever choice they make, the players pick the left side and right side with probaility 1/2. In the third experiment there is clear labeling. The second experiment is somewhere in between, and I would say that it is closer to the first than to the third. Without the context, just giving name to actions doesn’t kill the symmetry.
Now there are formal frameworks to handle symmetries in games, and the literature has gone in several directions. You can ask how to modify standard solution concepts to respect the symmetry of the game — after all, if the game stays the same when we change the names of the actions then the solution should also stay the same. You can ask how symmetry can be broken and how the players will play after symmetry is broken. Read Christoph’ papers to see how these questions lead to evolutionary consideration and Morse sequence.
Let me make one point, which is not the main issue of the study of symmetries in game, but I think is nicely highlighted by this study. The rotating disk trick is clever because it is difficult to find a way to unlabel actions in the laboratory: the very description of the experiment seems to force some labeling over the actions. Even if you only give the payoff matrix to the players you essentially label one of the row player’s actions as Top and another as Bottom. One may claim that rotating disks exist only in the laboratory and in `real life’ there are no such a thing as unlabeled actions. One would be right, I think, with a caveat: In real life there is also no such a thing as labeled actions. There are no such things as actions and players either. It is ridiculously rare that a real life strategic situation can be mapped into the mathematical model that game theorists call a game, because it’s never clear who are the players, what are their actions and what is the payoff matrix. But that’s cool: models are not supposed to be imitation of reality anyway, and neither do experiments.
But hey, I wrote this post to celebrate symmetries, not to open a pandora box.