Dear A Reader,

I'm considering this more. You're right--I had trouble finding a game theoretic model that would match up perfectly with the very simple auction model I'd want.

I do understand the definition of the strong and weak bidders. But the key here is that in all the models I've seen are based on multiple auctions in each of which bidders can bid up to their private value. So the private value is equal to the maximum bid. In this case bidder with a private value drawn from a lower distribution is the same as the bidder with a lower maximum bid.

If you want to define it more formally, consider an auction with two bidders for an object of type T where T is a number from 1 to 10, with private values PV(1) and PV(2).

PV(1) = T if T<4 or T=4 and PV(1) = 4 if T>4

PV(2) = T if T<9 or T=9 and PV(2)=9 if T>9

This is the equivalent of having two bidders with maximum bids of 4 and 9, bidding on an object with a type (or "desirability") of T. So now we've defined the strong and weak bidders here more formally in terms of a distribution of values.

Am I still cheating here? Yeah. I know I am. For one thing the paper I cite looks at a two bidder case, not a multiple bidder case. I think you should be able to derive an equivalence between the case of two bidders with independent private values and the very simple model I'm using with only one possible bid (the maximum) and a model with multiple bidders with values drawn from a range of distributions. In both cases the decision on whether to bid will be independent of the value that other bidders assign to each object. I would need to think about how to show this more rigorously.

My approach here may be backwards: there may be a way to show this much more simply with a common value model. That might even be trivial, but I haven't seen anything that would apply for that case.

Mark