We are treating an auction as a game. So, when we say we are predicting how bidders will bid using the idea of a Nash equilibrium, it is not like we are guaranteeing that this is how a bidder will bid in reality. We are making a guess, and we are as much confident about our guess as the stock market analyst is about his prediction of stock prices tomorrow. Identifying the Nash equilibrium bidding strategy is the best guess we make about how bidders would bid in an auction. It is based on this guess we will then predict the outcome (e.g., the expected price that the object will fetch).
If bidders know each other’s values predicting the Nash equilibrium strategy is easy. A bidder who knows she is going to win, bids her highest rival bidder’s value b1(v1) = Y1:n-1 | Y1:n-1 ≤ x1. The inequality Y1:n-1 ≤ x1 describes is a reminder that this is the situation where she is with winning bidder (because she has the highest value). BTW, if the equality in Y1:n-1 ≤ x1 is bothering make it strict. It won’t make a difference to what we are saying.
Under incomplete information, each bidder knows her own value but not the values of the other bidders. Instead, she believes that the values of the other bidders as random variables following certain distributions.
A bidder knows that if she does not have the highest value then she does not win in equilibrium, so what she bids matters only when she wins, i.e., she has the highest value. In that case, in equilibrium she is supposed to bid the highest rival’s value. The only thing is that she does not know what that value is. So she has to estimate that for that situation by taking expectation. So under incomplete information the Nash equilibrium bid involves this expectation:
b1(v1) = E[ Y1:n-1 | Y1:n-1 ≤ x1].