> the subjects who held on to their first choice the longest received 10x the reward
If the experiment was designed right, there wasn't a second offer to the same subject and it was made clear that there won't be one. Instead separate test groups were offered different amounts. Or at the very least the game was repeated.
One rational reason not to switch for $6 would be the hope/expectation that the offered incentive will increase. In the experiment, the subjects who held on to their first choice the longest received 10x the reward, while the quick switchers got only 1.2x (assuming the alternative cup was winning).