I am trying to understand an example from the article "Standard Auctions with Financially Constrained Bidders" Che & Gale (1998) - Review of Economic Studies.
Two buyers each value an object at $\frac{1}{2}$ but each has a budget, $w$, drawn uniformly (and independently) from $[0,1]$. In a second-price auction, it is weakly dominant for an active bidder to bid the smaller of his budget and $\frac{1}{2}$, which yields expected revenue of 0.292. In a first-price auction, it is equilibrium behavior for a bidder to bid $w$ if $w \in (0, \frac{1}{4}]$ and to bid $\frac{1}{2}-\frac{1}{16}w$ if $w>\frac{1}{4}$. This strategy yields expected revenue of 0.385.
I understand the calculation behind the second-price case. We have three cases
1) $\text{Prob}\left(\text{Both bidders}>\frac{1}{2}\right)$ = $\frac{1}{4}$ and Expected Revenue = $\frac{1}{2}$
2) $\text{Prob}\left(\text{Both bidders}<\frac{1}{2}\right)$ = $\frac{1}{4}$ and Expected Revenue = $\frac{1}{6}$
3) $\text{Prob}\left(\text{One bidder}>\frac{1}{2},\text{ One bidder}<\frac{1}{2}\right)$ = $\frac{1}{2}$ and Expected Revenue = $\frac{1}{4}$
Total expected revenue = $\frac{1}{4}\cdot\frac{1}{2}+\frac{1}{4}\cdot\frac{1}{6}+\frac{1}{2}\cdot\frac{1}{4}\approx0.292$
However, I can't figure out how Che & Gale get the equilibrium bidding strategies and arrive at 0.385 in expected revenue for the first-price auction. Does anyone have a suggestion?
The revenue ranking that Che and Gale have in the paper is correct - see their Theoretical Economics article in (2006?) for a more general result. The characterization of the first-price auction equilibrium, however, is not. I'm not sure about the particular example you mention but they assume the equilibrium of the FPA is given by $\min\{ b(v),w\}$ where $b(\cdot)$ is the unconstrained bid and $w$ is the budget. However, even with their assumptions, the differential equation that $b(\cdot)$ must solve may have only a solution that is not always increasing -- so it is not really a solution... Few years ago, Kotowski from Yale solved (correctly this time) for the eq. of the FPA with budget constraints:
http://mfile.narotama.ac.id/files/Jurnal/MIT%202012-2013%20%28PDF%29/First-Price%20Auctions%20with%20Budget%20Constraints.pdf