Prove that if a player in an extensive-form game has only one information set, then his set of mixed strategies equals his set of behavior strategies.
This is the exercise $6.4$ on page $246$ in Game Theory by Michael Maschler, Eilon Solan and Shmuel Zamir.
But I think the example game $A$(absent-minded driver game) in $6.3$ invalidate this claim, since Player I can't reach the payoff $10$ by a mixed strategy but it's not the case for a behavior stategy. What's wrong?
Added：
@EilonSolan Many thanks for drawing our attention to the way this exercise was phrased. We should have been more careful. Here is the new version of the exercise. I hope now it is clear.