I'm investigating optimal bidding in auctions, and am using logistic regression to predict the probability of winning an auction given a set of explanatory variables (e.g. the price I bid, number of competing bids etc).
One explanatory variable I want to use is the second highest price that was paid. However, by the design of the auction, I only observe the second highest price paid when I am the highest bidder (i.e. when I win the auction).
This missing data is a major issue as my dataset indicates that there is a winning bid only ~20% of the time, hence I don't know the second highest price paid 80% of the time. Yet intuitively, I don't want to drop this variable as it seems to me knowledge of the second highest bid is extremely valuable in determining my chances of being the winning bid.
Thus are there any standard methods to cope with this kind of missing data for logistic regression?