Consider the case that the highest bid is strictly higher than the second highest bid. In this case, for exactly one value of j, will be non-zero. Some agent (possibly the auctioneers) can collect the polynomials and determine the winner. Since there can be only one bidder with a higher price, this reveals no extra information.
If there is a tie for the highest bid, then , and for at least two values of j, . These values of j correspond to the tied bidders.