Asimple first-price auction: an item is for sale in an auction. player 1 values the item at 3 while player 2 values the item at 5. each player can bid 0, 1, or 2. if player i bids more that player j, then player i wins the item and pays his bid while player j loses and pays 0 and has a zero payoff. the winners’ payoff is his or her value less his or her bid. if the players bid the same amount, then a coin toss determines the winner, and winner gets the item and the loser does not. this is true even if both bid 0. when the bids are the same calculated the payoffs as the average of 0, the payoff of the loser, and the player’s valuation less the bid, the payoff of the winner. more precisely, when there is a ties the chance of each player winning is 50%, and you should calculate the expected payoff or the probability-weighted payoff.