We study bidding in first-price sealed-bid auctions with independent private values. We use a one-shot experiment in which a single human bidder bids against a single computerized bidder whose bids are drawn from a uniform distribution. Across our two main treatments, the value of the human bidders is fixed, while we change the upper boundary of the set of computerized bids. We find that humans' bids are higher when the boundary of the opponent is also higher. Such evidence is inconsistent with bidding based on objective functions predicting choices which are linear in the probability of winning given one's bid. This is the case of expected utility, its extensions accounting for anticipated regret, and utility of winning that is affine in value. According to such theories rescaling the probability of winning should not affect the optimal choice. The results of our experiment are difficult to be rationalized empirically by other theories as well, even though not characterized by linearity in probabilities. Probability weighting can account for the finding only for weighting function shapes lying outside of the commonly estimated range. Likewise, reference-dependent preferences can do it only under very high degrees of loss aversion. We propose a behavioral mechanism capable of rationalizing our results. We claim that when disadvantaged in the possibility of bidding, and therefore in the likelihood of winning, players react with more pessimism even in the domain of bids where probabilities should not be affected. As a consequence, they react by overbidding. We formalize such a decision making process by means of a power function distortion of the objective probability of winning, with the power depending on the symmetry of the bidding environment. Finally, we argue that such a symmetry-dependent probability distortion can be extended to account for stylized facts observed in many laboratory implementations sealed-bid auctions with multiple human bidders.