An item is allocated among two bidders that value the item very differently. The allocation is based on a probability that each of the bidders is allocated the item. The probability that each bidder is allocated the item is determined based on a non-linear function that is applied to bids that are received from the bidders. The item can be allocated semi-randomly subject to the probability that each bidder is allocated the item. A bidder can be required to pay its bid price only when allocated the item or each bidder can be required to pay an all-pay price regardless of which bidder is allocated the item. If the item is allocated in multiple auctions, the bidders can be ensured allocation of the item a minimum number of integer times based on the probabilities.