We study bidding behavior in sealed bid first price auctions, and obtain tight lower bounds on winning bid distributions for arbitrary symmetric distributions of values allowing for any information structure. The information structure and strategy profile attaining this bound have bidders uncertain about their value, uncertain about whether they will win, indifferent to all higher bids in the support of the bid distribution, and without binding constraints to bid lower. The bidding bounds imply tight lower bounds on revenue and upper bounds on bidder surplus.

The only upper bound on bids on winning bids is the distribution of highest values. Our results extend if we restrict attention to information structures where bidders know their own values. In this case, we give a non-trivial upper bound on bidder surplus and lower bound on revenue.