Existence of an equilibrium in first price auctions

We study the first price auction game with an arbitrary number of bidders when the bidders' valuations are independent from each other. In technical words, we work within the ''independent private value model''. We show that if the supports of the valuation probability distributions have the same minimum and if this minimum is not a mass point of any of these distributions, then a Nash equilibrium of the first price auction exists. We then modify the first price auction game by adding a closed interval of messages. Every bidder has to send a message with the bid he submits. These messages are used in the resolution of the ties. The winner of the auction is chosen randomly among the highest bidders with the highest value of the message among the highest bidders. In the general case, we prove the existence of a Nash equilibrium for this ''augmented'' first price auction.