Non-explanatory equilibria: An extremely simple game with (mostly) unattainable fixed points

Equilibrium analysis pervades mathematical social science. This paper calls into question the explanatory significance of equilibrium by offering an extremely simple game, most of whose equilibria are unattainable in principle from any of its initial conditions. Moreover, the number of computation steps required to reach those (few) equilibria that are attainable is shown to grow exponentially with the number of players—making long-run equilibrium a poor predictor of the game's observed state. The paper also poses a number of combinatorially challenging problems raised by the game.