The Game Take-or-Play: A paradox of Rationality in Simultaneous Move Games

We show that paradoxical conclusions similar to those emerging from reasonings of backward induction can arise also in simultaneous move games with incomplete information. In a static setting, these paradoxes are particularly puzzling, because the usual attempts to avoid the backward induction solution do not work. In a dynamic setting, there is a way out of the paradoxes, which hinges on a (possibly endogenous) uncertainty over the past behaviour of the players and does not call for a long time-horizon.