Bidding among friends and enemies with symmetric information

We consider an auction setting, in a symmetric information framework, in which bidders, even if they fail to obtain the good, care about the price paid by the winner. We prove that the outcome of the first-price auction is not affected by identity-independent price externalities, while the outcome of the second-price auction is. In contrast, identity-dependent price externalities affect the outcome of both auction formats. In any case, the second-price auction exacerbates the effects of price externalities.