本論文主要研究在完全資訊的有限制預算--預算僅在該次拍賣有價值拍賣結束後失去價值，首價密封拍賣機制，兩玩家兩商品的競價模型下，所得到的混合策略納許均衡結果。其中的解析過程需要用到兩個步驟，第一步使用弱劣勢策略逐步消去法消去不會被選擇的弱劣勢策略，第二步使用各種納許均衡的求解方式求出混合策略納許均衡解。模型結果可分為三個部分來解析，分別為兩人預算限制很接近，相差較遠以及相差一點五倍以上，再以兩商品的相對價值作為參數得到混合策略的納許均衡解。經濟上的現象可以發現，擁有較多預算的優勢玩家會傾向於投標在高價值商品上，而另一名較少預算的劣勢玩家則傾向於將預算投標到低價值商品上，且分配機率與商品的相對價值有關。

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.

Armstrong, Mark(2000). “Optimal Multi-Object Auctions”, The Review of Economic Studies, Volume 67, Pages 455-481
Benoît, Jean-Pierre & Krishna, Vijay (2001). “Multiple-Object Auctions with Budget Constrained Bidders”, The Review of Economic Studies, Volume 68, Pages 155-179
Bhattacharya, Sayan & Goel, Gagan & Gollapudi, Sreenivas & Munagala, Kamesh (2010). “Budget Constrained Auctions with Heterogeneous Items”, STOC '10 Proceedings of the forty-second ACM symposium on Theory of computing, Pages 379-388
Che, Yeon-Koo & Gale, Ian (1996). “Expected Revenue of All-pay Auctions and First-price Sealed-bid Auctions with Budget Constraints”, Economics Letters, Volume 50, Pages 373-379
Goeree, Jacob K. & Offerman, Theo (2002), “Efficiency in Auctions with Private and Common Values: An Experimental Study”, The American Economic Review, Volume 92, Pages 625-643
Goeree, Jacob K. & Offerman, Theo (2003). “Competitive Bidding in Auctions with Private and Common Values”, The Economic Journal, Volume 113, Pages 598-613
Hartline, Jason D. & McGrew, Robert (2005). “From Optimal Limited to Unlimited Supply Auctions”, EC '05 Proceedings of the 6th ACM Conference on Electronic Commerce, Pages 175-182
Hörner, Johannes & Jamison, Julian (2008). “Sequential Common-Value Auctions with Asymmetrically Informed Bidders”, The Review of Economic Studies, Volume 75, Pages 475-498
Rapoport, Amnon & Amaldoss, Wilfred (2000). “Mixed strategies and iterative elimination of strongly dominated strategies: an experimental investigation of states of knowledge”, Journal of Economic Behavior & Organization, Volume 42, Pages 483-521
Sönmez, Tayfun & Ünver, M. Utku (2010). “Course Bidding at Business School”, International Economic Review, Volume 51, Pages 99-123
Vickrey, William (1961). “Counterspeculation, Auctions, and Competitive Sealed Tenders”, Journal of Finance, Volume 16, Pages 8-37
Wilson, Robert (1977). “A Bidding Model of Perfect Competition”, The Review of Economic Studies, Volume 44, Pages 511-518