With the order-statistics notation in hand we can continue the discussion of FPA. Suppose in an auction with complete information there are n bidders. Suppose also that bidder 1 has the highest value x1, so he believes that in a Nash equilibrium of the auction game he will win the object.
Let us denote the values to bidder 1’s rival bidders be Y1,Y2,…,Yn-1. So, bidder 1 knows that the value to his highest rival bidder is Y1:n-1. The same idea from discussed before extends to this case. If bidders are allowed to make bids in dollars and cents only then bidder 1 bidding Y1:n-1 and his highest rival bidder bidding Y1:n-1 – 0.01 is a Nash equilibrium. (We are assuming a few things like the remaining bidders bid no more than their values, and that bidder 1’s value is at least 1 cent more than Y1:n-1. But let’s just sweep that discussion under the rug here.)
We could say that the Nash equilibrium bidding strategy involves a bidder bidding her highest rival bidder’s value when she believes that her highest rival bidder’s value is no more than her own value. In notations, we could just write b1(v1) = Y1:n-1 | Y1:n-1 ≤ x1. We do not bother about the bids from the other bidders here, as those are just dummy bids “supporting” the Nash equilibrium with no hope of winning.