Abstract
This article mainly try to find the mixed strategy Nash equilibriums result on a complete in formation limited-use budget, which mean the budget only has value in the current auction and loses its value after the auction, first-price sealed bid auction, two players two commodities auction model. The analyzation needs two steps, first step is to use iterative elimination of weakly dominated strategies to eliminated the unchosen weakly dominated strategies, and the second step is to use various method to solve the mixed strategy Nash equilibriums. The result can be separated into three parts, two player’s budget constraint are close, further and different by 1.5 times, then uses the relative value of two commodities as a parameter to find the mixed strategy Nash equilibriums. In the economics point of view, we can find that the player who has higher budget tends to bid more often on the high value commodity and the one who has lower budget tends to bid more often on the low value commodity, and the distribution probabilities are related to the relative value of the commodities.