在賽局理論中，雙邊匹配是一項常見的議題，而且更有多種的雙邊匹配早已經被分析並描述成賽局的形式[1]。Shapley和Shubik[2]曾研究過其中一種稱作雙邊市場的雙邊匹配。他們將雙邊市場描述成一合作賽局，並命名為指派賽局以及分析了它。而Kaneko[3]則是進一步推廣了他們所提的指派賽局。論文[2][3]顯示，該指派賽局之部分核心解可以藉由線性規劃的解來得到，然而用這種方法並沒有辦法得到它所擁有的完整核心解。在本論文中，我們考慮了其中一種雙邊市場，並且也將它描述成一合作賽局，命名為馬匹交換賽局。在這賽局裡，貨品為均質馬匹，換言之，所有馬匹對於同一人而言，其價值是一樣的。而參與這賽局的玩家，是一位賣家以及多位買家。賣家在開始時擁有多匹馬匹，而買家在開始時則不擁有馬匹。除此之外，我們假設所有買家對於馬匹的需求量皆至多一匹，以及所有玩家對於馬匹所認定的價值是彼此不相同的。本篇論文最主要的貢獻為，我們找到所提馬匹交換賽局的完整核心解，並給其一明確表示法。

Two-sided matching is a popular issue in game theory. Several kinds of two-sided matching have been analyzed and modeled as games [1]. Shapley and Shubik [2] investigate a kind of two-sided matching called a two-sided market, and model it as a coalitional game called the assignment game. It is shown in [2][3] that some elements of the core of the assignment game can be obtained by solving a linear programming problem. Nonetheless, not all elements of the core of the assignment game are given. In this thesis, we consider a variant of the two-sided market and model it as a coalitional game called a horse trading game. In this game, the goods are homogeneous horses, i.e., everyone in this game values any two horses at the same price. Moreover, there are only one owner and multiple nonowners in this game. The owner has multiple horses and each nonowner has no horse. In addition, we assume that every nonowner buys at most one horse in this game, and that everyone's valuation of a horse is different from each other in this game. The main contribution in this thesis is that we give an explicit expression of the core of the horse trading game.

Abstract i
Contents ii
1 Introduction 1
2 The Horse Trading Game 3
3 The Core of the Horse Trading Game 7
4 Conclusion 13
Appendix A Proof of Lemma 5 14
Appendix B Proof of Lemma 6 18
Appendix C Proof of Lemma 7 22
Appendix D Proof of Lemma 8 25
Appendix E Proof of Lemma 9 27
Bibliography 28

[1] A. E. Roth and M. A. O. Sotomayor, Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, Cambridge, UK: Cambridge University Press, 1990.
[2] L. S. Shapley and M. Shubik, “The assignment game I: The core,” International Journal of Game Theory, vol. 1, pp. 111–130, 1971/72.
[3] M. Kaneko, “On the core and competitive equilibria of a market with indivisible goods,” Naval Research Logistics Quarterly, vol. 23, pp321–337, 1976.
[4] M. J. Osborne, An Introduction to Game Theory, New York, NY: Oxford University Press, 2009.