Consider an auction in which bidders are cash-constrained. As in the model we saw in class, there is one object for sale and n ≥ 2 bidders. Each bidder i has a valuation vi drawn from a uniform distribution in [0, 1]. The valuation of each bidder is independently distributed.
The new feature relative to what we saw in class is that bidders are cash-constrained. In partic- ular, each bidder i is subject to a budget wi: in no circumstances can a bidder with budget wi pay more than wi. If bidder i were to bid more than wi and defaults, then a (small) penalty is imposed on her.
Assume that each bidder’s budget wi is also drawn from a uniform distribution on [0, 1]. Assume that each bidder’s budgets are independently distributed (and are also independently distributed from the signals). At the beginning of the auction (i.e., before submitting bids) each bidder learns her own valuation vi and her own budget wi, but not the signals and budgets of her opponents. After observing her signal and budget, each bidder has to choose which bid to submit.
Suppose that the auction format is a second-price auction: the player who submitted the highest bid wins, but pays a price equal to the second highest bid.
- Show that, for each bidder, bidding her own budget always gives a higher payoff than bidding above her budget.
- Consider a bidder with valuation vi ≤ wi. Show that, in this case, bidding her own valuation gives her a higher payoff than any other bid. Hint: recall the arguments we used in class to show that in a standard second-price auction it is optimal for each bidder to submit a bid equal to her valuation.
- Consider a bidder with valuation vi > wi. Show that, in this case, bidding her own budget gives the bidder a higher payoff than any other bid.
- Use your answers to parts (b) and (c) to conclude that each bidder submits a bid equal b(x, w) = min{x, w}.