Bidding among friends and enemies

We consider an auction setting in which potential buyers, even if they fail to obtain the good, care about the price paid by the winner. We study the impact of these price-externalities on the first-price auction and the second-price auction in a symmetric information framework. First, we consider situations in which bidders care about the price paid independently from the identity of the winner. We prove that the first-price auction is not affected by this kind of price-externalities while the second-price auction is. In broader specifications, we observe though that the first-price auction can be affected by the presence of such price-externalities. In any case, in comparison with the first-price auction, the second-price auction exacerbates the effects of price-externalities whatever their types. Therefore, there is no revenue equivalence between the two auction formats.