I had a short chat with one of the prospective Ph.D. students recently in which he told me about the hypergame paradox. Suppose we have a set of finite games. Finite games are games which must end in a finite number of turns. What are games? The guy said that it was just the intuitive notion of a game. I didn’t get more detail so I won’t give more. Hypergame is a game in which the only move available is to pick a finite game, then that game is played to completion, at which point hypergame is over. The question is: is hypergame a finite game? Suppose it is, then for each move pick to play hypergame. It never ends, so it isn’t finite. Suppose it isn’t. Then pick any game from the set of finite games, and that will end in a finite number of steps, and so will hypergame. Therefore it is finite. Boom. Paradox.

I was trying to figure out where the paradox comes from. At first I thought it was because hypergame was impredicative. This was probably because I’ve been thinking about Russell, his paradox, and such a lot lately. I couldn’t really formulate how this was supposed to work though. I eventually came around to thinking that there was probably a way of reducing this to the halting problem. The idea is this. Hypergame is a Turing machine that tells you whether other Turing machines (games) halt (end in a finite number of steps). As Turing taught us, such a machine cannot exist on pain of paradox. All that is needed to get this to work is a precise way of correlating games with Turing machines. Unfortunately, I didn’t get enough details for this. No idea what exactly constitutes a game, so no idea of how to line up games with Turing machines in anything more than a handwavy, metaphorical way. Alas!