There are 2 bidders in an auction that has a style as follows. There is a price clock that continuously rises starting from zero. The bidders gradually drop out of the auction and the clock stops as soon as there is only one bidder remains. That remaining bidder wins the auction and pays the current price of the clock. Suppose that there are two bidders and the valuation of each bidder is either $0$ or $V (> 0)$. The probability that the valuation is $0$ is $\frac{1}{2}$ and the probability that the valuation is $V$ is $\frac{1}{2}$. The problem asks to calculate the expected payoff of the high bidder and the expected revenue of the seller.
I have done the following but I am stuck on both parts. For expected payoff, since we are the high bidder, it means that we are going to bid $V$. It means that the other bidder is bidding either $0$ or $V$ with probability of 1/2 in each case. Our payoff is hence going to be $\frac{1}{2}(V - 0) + \frac{1}{2}(V - V)$. However, I realize that I have not taken in account the fact that we also have a probability of $\frac{1}{2}$ that we are going to bid either $V$ or $0$ - how do we consider this?
For the expected return, there are 4 scenarios this can go: $(0, 0), (0, V), (V, 0), (V, V)$. There are 3 cases out of 4 in which there is at least one bidder who is going to bid $V$, so the revenue always be $V$ no matter what. There is one case where both bidders would bid 0 so the seller would end up not selling it. Hence, the expected revenue would be $\frac{3}{4} \cdot V + \frac{1}{4} \cdot 0 = \frac{3V}{4}$. Like above, I see that I have not taken in consideration the probability of $\frac{1}{2}$ for each bidder.
Any help would be appreciated!
