That's ok, except that you really have a one-player game now. If the first player is locked into playing randomly forever, irrespective of what the second play

That's ok, except that you really have a one-player game now. If the first player is locked into playing randomly forever, irrespective of what the second player does, then for the remaining player any strategy is an "equilibrium" and can't be improved (or worsened). We're just talking about roulette or something. But I'm a little disappointed that someone would downvote the comment above. The "something is wrong with the theory" claim is very premature (ok, it's wrong), but the important thing is now we are fighting with the text instead of just citing it. That's a step forward, not a step back.