<Edited on September 24, following comments of Jason Hartline>
A gambler has to choose at every stage t whether to take the current prize A_t that is offered to him and go home, or whether to forgo the prize in the hope that he will get an even higher prize in the next stage. The prize A_t is chosen according to a known distribution F_t, and the process continues for T stages.
Clearly, at stage T the gambler will take the prize that is offered to him. One can then calculate by backward induction the optimal strategy of the gambler. This strategy will be a threshold strategy: for every t there exists a threshold q_t; if the prize offered at stage t is higher than q_t, the gambler will take the prize and go home, otherwise he will wait and hope for a higher prize in the future.
To calculate this strategy one needs to find the T thresholds. Apparently there is a simple strategy that is not optimal yet it is 1/2-optimal: it ensures an expected payoff which is at least 1/2 of the optimal payoff. This strategy is a threshold strategy: there is a threshold q_* such that the gambler accepts the prize if and only it is higher than q_* (regardless of the stage). In particular, it may well happen that the gambler will not accept the prize at stage T, if the game reaches that stage. The proof uses the Prophet Inequality, and was shown to us by Jason Hartline in a remarkably interesting presentation.
Jason went on and generalized this basic observation to a mechanism design setup. A single item is offered for sale on an auction. The private values of the n bidders are independent. By Myerson (1981) we know the structure of the optimal mechanism: for each player there is a function that transforms his bid into a virtual price. Then one runs a second price auction with reserve prices on the virtual prices, where the reserve price is bidder dependent. In practice the assumptions of Myerson’s theorem do not hold: private values need not be independent and their distributions need not be common knowledge. Moreover, setting bidder-dependent reserve prices will never work. It turns out that a second price auction with bidder-dependent reserve prices is a 1/2-optimal mechanism for the seller in a wide range of applications. If one insists on a common reserve price, then a second price auction is a 1/4-optimal mechanism for the seller, provided one carefully chooses the common reserve price.
Jason and co. have more results on simple mechanisms that approximate the optimal mechanism in various setups, but I am afraid that my description of these results will not be accurate, and that I will get tons of complaints on my sloppiness. So please do read his and his co-authors’ work. The original is always better than copies.