We view auctions as games with the bidders as players and borrow generously from game theory. There, a standard method of predicting the players’ behavior and the outcome of a game is to identify the Nash equilibrium. When players choose actions simultaneously (like in our case players choose how much to bid) their actions are said to be in Nash equilibrium if no player wants to change her action given the actions of the other players.
Let's see that in a simple game. Suppose there are two bidders in a first-price auction and the bidders have values V1=$10 and V2=$8. In this case the seller accepts bids in dollars and pennies only. Each bidder knows the other bidder's value, but not the bid. A bidder is not allowed to submit a bid of $9.153. The bid will be taken to be $9.15. If both bidders submit the same bid, the tie is broken with equal probabilities.
The bids b1=$7.99 by bidder 1 and b2=$ 8.00 by bidder 2 are in Nash equilibrium. This is because given bidder 1 bid of $7.99 bidder 2 will not find it profitable to change her bid. Similarly, bidder 1 will not find it profitable to deviate from her bid from $7.99 given bidder 1's bid. If you think about it a bit, you will realize how natural it is that the Nash equilibrium has to be such that the winning bidder will bid the value of the object to her highest rival (in this case her only rival).
There are other Nash equilibria for this game, but they are either similar in outcome to this Nash equilibrium or involve dominated strategies like bidding more than one's value which cannot do a bidder any good. (We do not take behavioral elements like spite into account here, that would change the nature of the value of winning.) For our purpose here we do not consider such equilibria because ultimately we will be concerned with situations where a bidders will be allowed to submit “continuous” bids and, additionally, she will not know for sure that she will lose.
The bottom line is that under complete information the Nash equilibrium consists of bidders bidding the value of her highest rival conditional on winning. Since it is reasonable for the bidder with the highest value to assume that she will win it is she whose bid is equal to the highest rival bidder's value. The rest of the bidders believe that they are not winning, so their bids are essentially dummy bids that "support" the Nash equilibrium.
Before we proceed, it is useful to know a little bit about order statistics.