In the above game, \(d\) is weakly dominated by \(u\) for Ann. If Bob knows that Ann is rational (in the sense that she will not choose a weakly dominated strategy), then he can rule out option \(d\). In the smaller game, action \(r\) is now strictly dominated by \(l\) for Bob. If Ann knows that Bob is rational and that Bob knows that she is rational (and so, rules out option \(d\)), then she can rule out option \(r\). Assuming that the above reasoning is transparent to both Ann and Bob, it is common knowledge that Ann will play \(u\) and Bob will play \(l\). But now, what is the reason for Bob to rule out the possibility that Ann will play \(d\)? He knows that Ann knows that he is going to play \(l\) and both \(u\) and \(d\) are best responses to \(l\). The problem is that assuming that the players’ beliefs are cautious conflicts with the logic of iterated removal of weakly dominated strategies. This issue is nicely described in a well-known microeconomics textbook: