神戸大学・マッチング理論
（2015年夏）

マッチング理論 | Matching Theory

夏期集中講義 | Summer Intensive Lectures  

• 講義シラバス： pdf  

• 時間： 9月3日（木）、4日（金）

  • 両日とも、1限〜4限の4コマずつ（1コマは90分）の予定。

  • 1限の開始時間は8:50、4限の終了時間は16:40。

• 教室： 六甲台本館2階230教室 地図

• 講師： 安田洋祐（yosuke.yasuda<at>gmail.com）

• TA：藤田峻（sh.ki.mti27<at>gmail.com）

最終更新日: 2015年9月5日

お知らせ |  Announcement 

• [2015/9/4] --- 履修者の方向けのレポート課題の提出期限は9月23日（水・祝）とします。詳しくは以下の成績評価の項目をご覧ください。

• [2015/8/15] --- 受講希望者の方は（聴講のみで単位が不要の方も）、できるだけ早いタイミングでこちらのフォームに記入をお願いします。講義資料などをメールで送らせて頂きます。 ← 重要！

• [2015/1/6] --- 講義は日本語で行います。

講義概要・目的 |  Course Objective and Content 

• 近年、理論と実践の両面において発展が著しいマッチング理論について学びます。

• 前半は代表的なテキストであるRoth and Sotomayor (1990) を使いながら基礎をしっかり勉強し、後半で90年代以降に研究の進展が見られたより新しいトピックを解説します。

• この分野の研究スタイルに慣れ、関連論文を読みこなす力を身につけるのが目的です。

• 特別な前提知識は必要としません。ただし、抽象的な数理モデルの記述や、数学的な定理の証明などを頻繁に行いますので、この手の作業が苦手な方にはしんどい講義になるでしょう。目安として、たとえば「厚生経済学の第一基本定理の証明を（解説を読んでも）ほとんど理解できない」という方には履修をおすすめしません。

• 全八回の短い講義ですが、マッチング理論の主要成果はほぼ網羅する予定です。マッチングやマーケットデザインに関心のある方は（もし無くても、時間に余裕のある方は）ぜひご参加ください！

成績評価 | Grading Policy 

• レポートによって評価します。締切は【9月23日（水・祝）】とします。

• 1) マッチング理論に関する研究アイデア、2) 関連する既存論文（1本だけ選ぶ）に対するコメント、3) 関連するトピックを選んでそのサーベイ、のいずれかをまとめて、e-mailに添付して安田と藤田まで提出してください。

  • ページ数はA4で2〜5ページ程度を目安としてください。

  • 使用言語は日本語・英語どちらでも構いません。

講義テキスト | Textbooks  

• 教科書: Roth and Sotomayor, Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis, 1990.  Amazon (Kindle) 

  • マッチング理論のバイブル的な教科書で、80年代までの主要な研究成果をほぼ網羅しています。

  • 講義と関連する章はコピーを配布する予定です。

  • Econometric Societyの会員は（こちらからログインして）電子版を無料でダウンロードできます。

• 参考図書1: Roth, Who Gets What — And Why: The New Economics of Matchmaking and Market Design, 2015.  Amazon (Kindle)

  • 上の教科書の著者でもあるノーベル経済学者ロス教授による一般向けのマッチング・マーケットデザイン入門書。

• 参考図書2: Vulkan, Roth, and Neeman ed., Handbook of Market Design, 2013.  Amazon 目次 

  • 非常に高価なので購入はあまりおすすめしません。講義に関連する章を配布するかもしれません。

  • より安価なペーパーバック版が今夏に出版予定のようです。

• 参考図書3: Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, 1989.  Amazon 

  • 計算機科学やオペレーションズ・リサーチの視点から書かれた伝統的なマッチング理論の教科書です。

サーベイ論文・書籍 | Survey Articles 

• Abdulkadiroglu, A. and Sonmez, T. (2013), Matching Markets: Theory and Practice, in Advances in Economics and Econometrics; 10th World Congress, Edited by Daron Acemoglu, Manuel Arellano, Eddie Dekel.

• Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences (2012), Stable Allocations and The Practice of Market Design, Scientific Background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012.  Link 

• Knuth, D. (1976), Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms, English Reprint (1996).   Amazon 

• Kojima, F. (2015), Recent Developments in Matching Theory and its Practical Applications, working paper (prepared for Advances in Economics and Econometrics; 11th world congress of Econometric Society) 

• Kojima, F. and Troyan, P. (2011), Matching and Market Design: An Introduction to Selected Topics, Japanese Economic Review, 62: 82–98. 

• Niederle, M., Roth, A. and Sonmez, T. (2008), Matching, in The New Palgrave Dictionary of Economics, edited by Steven N. Durlauf and Lawrence E. Blume. 

• Roth, A. (2002), The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics, Econometrica, 70: 1341-1378. 

• Roth, A. (2003), The Origins, History, and Design of the Resident Match, Journal of the American Medical Association, 289 :909-912.

• Roth, A. (2008a), What Have We Learned from Marked Design?, Economic Journal, 118: 285-310. 

• Roth, A. (2008b), Deferred acceptance algorithms: history, theory, practice, and open questions, International Journal of Game Theory, 36: 537-569.

• Roth, A. E., & Sotomayor, M. (1992). Two-sided matching. Handbook of Game Theory with Economic Applications, Vol. 1, 485-541. 

  • メインテキストRSのダイジェスト版といった感じで便利です。多くの証明は省略されているので、RSを直接参照してください。

• Sonmez, T. and Unver, U. (2011), Matching, Allocation, and Exchange of Discrete Resources, in Handbook of Social Economics, Vol. 1A, edited by J. Benhabib, A. Bisin, and M. Jackson.

• Topkis, D. (1998), Supermodularity and Complementarity.  Amazon (Kindle) 

• Vives, X. (1999), Oligopoly Pricing: Old Ideas and New Tools.  Amazon 

• Vohra, R. (2004), Advanced Mathematical Economics.  Amazon (Kindle) 

• 小島武仁・安田洋祐, マッチング・マーケットデザイン, 経済セミナー, (647), 135-145, 2009.  増補改訂版 

• 坂井豊貴, マーケットデザイン: 最先端の実用的な経済学, ちくま新書, 2013.  Amazon 

  • マーケットデザインの日本語による包括的な入門書です。交換問題（講義では扱わない予定）と、その応用例である臓器移植マッチングについての記述が非常に詳しいです。

• 安田洋祐, マッチングの数理, 数学セミナー, 4月号, 2013. 草稿

• 安田洋祐, 学校選択問題のマッチング理論分析, 現代経済学の潮流2014（第4章, pp.95-122）, 東洋経済新報社, 2014.

• 安田洋祐（編著）, 学校選択制のデザイン：ゲーム理論アプローチ（編著）, NTT出版, 2010.  Amazon 

  • 学校選択制のマッチング理論分析に関する本格的な研究書です。日米の現実の制度的な背景や、マッチング理論の基礎および新展開などについて詳しく論じています。

マッチング・マーケットデザインに関する講義 | Lectures on Matching Market Design 

• Tayfun Sonmez | Boston College  

• Itay P. Fainmesser | Brown U. 

• Alvin E. Roth | Harvard U.

• Parag Pathak | MIT  

• Nicole Immorlica | Northwestern U. 

• Yosuke Yasuda | Osaka U. 

• Fuhito Kojima | Stanford U. 

• Muriel Niederle | Stanford U.   

• Scott Duke Kominers | U. of Chicago   

講義日程 | Lecture Schedule 

Introductory Slides 

マッチング問題と、それを解決する代表的な仕組みであるGale-Shapleyのアルゴリズムに関する入門的な解説です。

Lecture Slides 

Lecture 1〜5のスライドです。

Lecture 1  One-to-One Matching Model  9/3

• Two-Sided Market 

• Stable Matching

• Gale-Shapley "Deferred Acceptance" Algorithm

• M(W)-Optimal Matching

• Top-Top Match

• Lattice  

• Weak Pareto Optimality

• Path to Stable Matching

References 

• **Roth and Sotomayor: Ch2 

• **Gale, D. and Shapley, L. (1962)

• *Gale, D and Sotomayor, M. (1985a) 

• *Roth, A. (2008b)

• Blair, C. (1988) 

• Demange, G., Gale, D and Sotomayor, M. (1987) 

• Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences (2012) 

• Eeckhout, J. (2000) 

• Knuth, D. (1976) 

• Roth, A. and Vande Vate, J. (1990) 

You are expected to read all double-starred (**) articles and encouraged to read starred(*) articles.

Lecture 2  Incentive Issues  9/3

• Decomposition Lemma

• Core

• Blocking Lemma  

• Strategy Proofness

• Impossibility/Possibility Theorem

• Equilibrium Behavior 

References

• **Roth and Sotomayor: Ch3; Ch4 

• *Gale, D and Sotomayor, M. (1985b) 

• *Roth, A. (1982) 

• Alcalde, J. (1996) 

• Alcalde, J. and Barbera, S. (1994) 

• Dubins, L. and Freedman, D. (1981) 

• Roth, A. and Rothblum, U. (1999) 

• Sotomayor, M. (2004, 2007)

Lecture 3  One-to-Many Matching Model  9/3

• College Admissions Model

• Responsive Preferences

• Strict Core 

• Rural Hospitals Theorem 

• National Resident Matching Program (NRMP)

• Couples Problem

References

• **Roth and Sotomayor: Ch5

• *Roth, A. (1984b) 

• *Roth, A. and Peranson, E. (1999) 

• *小島・安田 (2009) 

• Bulow, J. and Levin, J. (2006) 

• Kojima, F. (2007) 

• Niederle, M. and Roth, A. (2003) 

• Roth, A. (1984a, 1985, 1986, 1991, 2002, 2003)

• Roth, A. and Sotomayor, M. (1989) 

Lecture 4  Monotone Method  9/4

• Partial Order

• Lattice and Sublattice

• Tarski's Fixed Point Theorem 

• Pre-Matching 

• T-Mapping 

• Strategic Complementarities 

References

• **Adachi, H. (2000) 

• *Vives, X. (1999): Ch2 

• *Vohra, R. (2004): Ch7 

• *安田 (2013)

• Echenique, F. (2005) 

• Echenique, F. and Oviedo, J. (2004) 

• Fleiner, T (2003) 

• Kandori, M., Kojima, F. and Yasuda, Y. (2008) 

• Sotomayor, M. (1996) 

• Tarski, A. (1955)

• Topkis, D. (1998) 

Lecture 5  Matching with Money  9/4 

• Substitutable Preferences

• Labor Market with Salaries

• Gross Substitutes Condition

• Salary-Adjustment Process 

• Assignment Game

• Budget Constraints

References

• **Roth and Sotomayor: Ch6 

• *Kelso, A. and Crawford, V. (1982) 

• *Shapley, L. and Shubik, M. (1971)

• Crawford, V. and Knoer, E. (1981)

• Demange, G. and Gale, D. (1985) 

• Mongell, S. and Roth, A. (1986)

Lecture 6  Matching with Contracts  9/4

講義の進行状況を考慮してカット。

マッチングの研究を目指す方はHatfield, J. and Milgrom, P. (2005)は必読です！

• Contracts 

• Generalized Gale-Shapley Algorithm 

• Substitutability 

• Law of Aggregate Demand 

• Strategy-Proof Matching

• Cumulative Offer Process 

References

• **Hatfield, J. and Milgrom, P. (2005) 

• *Aygun, O. and Sonmez, T. (2013) 

• *Echenique, F. (2012) 

• Kominers, S. D. (2012) 

• Ostrovsky, M. (2007) 

• Hatfield, J. and Kojima, F. (2008, 2009, 2010)

• Sonmez, T. and Switzer, T. (2014) 

• Westkamp, A. (2010) 

Lecture 7  School Choice  9/4

2013年の日本経済学会での招待講演のスライド（とほぼ同じもの）になります。

• One-sided Matching 

• Boston Mechanism 

• Top Trading Cycles Mechanism 

• Acyclicity

• Tie-Breaking 

• Ex-ante Efficiency 

• Constrained School Choice

• Preferential Treatment  

References

• **Abdulkadiroglu, A. and Sonmez, T. (2003) 

• *Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2011) 

• *Kesten, O. (2010) 

• *安田 (2014) 

• Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2015) 

• Abdulkadiroglu, A., Pathak, P. and Roth, A. (2009) 

• Balinski M and Sonmez, T. (1999) 

• Calsamiglia, C., Haeringer, G., and Klijn, F. (2010) 

• Chen, Y., and Kesten, O. (2013)

• Erdil, A. and Ergin, H. (2008) 

• Ergin, H. (2002) 

• Ergin, H. and Sonmez, T. (2006) 

• Haeringer, G., and Klijn, F. (2009) 

• Pathak, P. and Sonmez, T. (2008, 2013)

• 安田 (2010) 

Lecture 8  Recent Developments  9/4 

スタンフォード大学の小島武仁氏のスライドを（本人の許可を得て）使用させて頂きます。感謝！

（しかし、彼のウェブサイトはなぜ「ボスニア語」設定なのだろう…）

• Distributional Constraints

• Minimum Quota 

• Regional Quota 

• Affirmative Actions 

• Non Substitutable Preferences 

Other Important Topics

• Large Market 

• Network 

• Dynamic Matching 

• Decentralized Matching 

• Asymmetric Information 

• Externalities 

References

• **Kojima, F. (2015)

• *Kamada, Y. and Kojima, F. (2015) 

• *Goto, M., Iwasaki, A., Kawasaki, Y., Yasuda, Y. and Yokoo, M. (2015) 

• Abdulkadiroğlu, A. (2005) 

• Echenique, F., and Yenmez, M. B. (2012) 

• Ehlers, L., Hafalir, I. E., Yenmez, M. B., and Yildirim, M. A. (2014) 

• Goto, M., Hashimoto, N., Iwasaki, A., Kawasaki, Y., Ueda, S., Yasuda, Y. and Yokoo, M. (2014) 

• Kojima, F. (2012) 

• Kojima, F. and Pathak, P. (2009) 

• Kojima, F., Tamura, A., and Yokoo, M. (2014) 

• Hafalir, I., Yenmez, B. and Yildrim, M. (2013) 

• Monte, D. and Tumennasan, N. (2013) 

Other Possible Topic  One-Sided Matching Model  9/4 

• House Allocation Problem 

• Serial Dictatorship 

• Housing Market

• Top Trading Cycles

• YRMH-IGYT

• Kidney Exchange Problem 

References

• **Sonmez, T. and Unver, U. (2011)

• *Roth, A. and Postlewaite, A. (1977) 

• *Shapley, L. and Scarf, H. (1974) 

• Abdulkadiroglu, A. and Sonmez, T. (1999) 

• Hylland, A. and Zeckhauser, R. (1977) 

• Ma, J. (1994) 

• 坂井 (2013)

参考論文 | References  

• Abdulkadiroğlu, A. (2005). College admissions with affirmative action. International Journal of Game Theory, 33(4), 535-549. 

• Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2011), Resolving Conflicting Preferences in School Choice: the "Boston Mechanism" Reconsidered, American Economic Review, 101: 399-410.

• Abdulkadiroglu, A., Che, Y.-K. and Yasuda, Y. (2015), Expanding "Choice" in School Choice, American Economic Journal: Microeconomics, 7: 1-42.

• Abdulkadiroglu, A., Pathak, P. and Roth, A. (2009), Strategy-proofness vs. Efficiency in Matching with Indifferences: Redesigning the NYC Match, American Economic Review, 99: 1954-1978.

• Abdulkadiroglu, A. and Sonmez, T. (2003), School choice: a mechanism design approach, American Economic Review, 93: 729-747. 

• Adachi, H. (2000), On a characterization of stable matchings, Economics Letters, 68: 43–49. 

• Alcalde, J. (1996), Implementation of Stable Solutions to Marriage Problems, Journal of Economic Theory, 69: 240–254. 

• Alcalde, J. and Barbera, S. (1994), Top dominance and the possibility of strategy-proof stable solutions to matching problems, Economic Theory, 4: 417-435. 

• Aygun, O. and Sonmez, T. (2013), Matching with Contracts: Comment, American Economic Review, 103: 2050-2051.

• Balinski M and Sonmez, T. (1999), A tale of two mechanisms: student placement, Journal of Economic Theory, 84: 73–94. 

• Blair, C. (1988), The Lattice Structure of the Set of Stable Matchings with Multiple Partners, Mathematics of Operations Research, 13: 619-628. 

• Bulow, J. and Levin, J. (2006), Matching and price competition, American Economic Review, 96: 652–668. 

• Calsamiglia, C., Haeringer, G., and Klijn, F. (2010). Constrained school choice: An experimental study. The American Economic Review, 1860-1874.

• Chen, Y., and Kesten, O. (2013). From boston to chinese parallel to deferred acceptance: theory and experiments on a family of school choice mechanisms. 

• Crawford, V. and Knoer, E. (1981), Job Matching with Heterogeneous Firms and Workers, Econometrica, 49: 437-450. 

• Demange, G. and Gale, D. (1985),  The Strategy Structure of Two-Sided Matching Markets, Econometrica, 53: 873-888.  

• Demange, G., Gale, D. and Sotomayor, M. (1987), A further note on the stable matching problem, Discrete Applied Mathematics, 16: 217–222. 

• Dubins, L. and Freedman, D. (1981), Machiavelli and the Gale-Shapley algorithm, American Mathematics Monthly, 88: 485–494/ 

• Echenique, F. (2005), A Short And Constructive Proof of Tarski's Fixed-Point Theorem, International Journal of Game Theory, 33: 215-218. 

• Echenique, F. (2012), Contracts vs. Salaries in Matching, American Economic Review, 102: 594-601. 

• Echenique, F. and Oviedo, J. (2004), Core many-to-one matchings by fixed point methods, Journal of Economic Theory, 115: 358–376. 

• Echenique, F., and Yenmez, M. B. (2012). How to control controlled school choice. Available at SSRN 2189706.

• Eeckhout, J. (2000). On the uniqueness of stable marriage matchings. Economics Letters, 69(1), 1-8.

• Erdil, A. and Ergin, H. (2008), What's the Matter with Tie-Breaking? Improving Efficiency in School Choice, American Economic Review, 98: 669-689.

• Ergin, H. (2002), Efficient resource allocation on the basis of priorities, Econometrica, 70: 2489–2497. 

• Ergin, H. and Sonmez, T. (2006), Games of school choice under the Boston mechanism, Journal of Public Economics, 90: 215–237. 

• Ehlers, L., Hafalir, I. E., Yenmez, M. B., and Yildirim, M. A. (2014). School choice with controlled choice constraints: Hard bounds versus soft bounds. Journal of Economic Theory, 153, 648-683.

• Fleiner, T (2003), A fixed-point approach to stable matchings and some applications, Mathematics of Operations Research, 28: 103–126. 

• Gale, D. and Shapley, L. (1962), College Admissions and the Stability of Marriage, American Mathematical Monthly, 69: 9-15.

• Gale, D and Sotomayor, M. (1985a), Some remarks on the stable matching problem, Discrete Applied Mathematics, 11: 223-32.

• Gale, D and Sotomayor, M. (1985b), Ms Machiavelli and the stable matching problem, American Mathematical Monthly, 92: 261-8. 

• Goto, M., Hashimoto, N., Iwasaki, A., Kawasaki, Y., Ueda, S., Yasuda, Y. and Yokoo, M. (2014), Strategy-proof Matching with Regional Minimum Quotas, AAMAS2014 (Full Paper), 1225-1232.

• Goto, M., Iwasaki, A., Kawasaki, Y., Yasuda, Y. and Yokoo, M. (2015), Improving Fairness and Efficiency in Matching with Distributional Constraints: An Alternative Solution for the Japanese Medical Residency Match.. 

• Haeringer, G., and Klijn, F. (2009). Constrained school choice. Journal of Economic Theory, 144(5), 1921-1947.

• Hafalir, I., Yenmez, B. and Yildrim, M. (2013), Effective Affirmative Action in School Choice, Theoretical Economics, 8: 325-363.

• Hatfield, J. W., and Kojima, F. (2008). Matching with contracts: Comment. The American Economic Review, 98(3), 1189-1194.

• Hatfield, J. W., and Kojima, F. (2009). Group incentive compatibility for matching with contracts. Games and Economic Behavior, 67(2), 745-749.

• Hatfield, J. W., and Kojima, F. (2010). Substitutes and stability for matching with contracts. Journal of Economic Theory, 145(5), 1704-1723. 

• Hatfield, J. and Milgrom, P. (2005), Matching with Contracts, American Economic Review, 95: 913-935. 

• Hylland, A. and Zeckhauser, R. (1977), The efficient allocation of individuals to positions, Journal of Political Economy, 87: 293-314. 

• Kamada, Y. and Kojima, F. (2015), Efficient matching under distributional constraints: Theory and applications, The American Economic Review, 105(1), 67-99.

• Kandori, M., Kojima, F., and Yasuda, Y. (2008), Understanding Stable Matchings: A Non-Cooperative Approach, mimeo. 

• Kelso, A. and Crawford, V. (1982), Job Matching, Coalition Formation, and Gross Substitutes, Econometrica, 50: 1483-1504. 

• Kesten, O. (2010), School Choice with Consent, Quarterly Journal of Economics, 125: 1297-1348.

• Kojima, F. (2007). Matching and price competition: comment. The American economic review, 1027-1031.

• Kojima, F. (2012). School choice: Impossibilities for affirmative action. Games and Economic Behavior, 75(2), 685-693.

• Kojima, F. and Pathak, P. (2009), Incentives and Stability in Large Two-Sided Matching Markets, American Economic Review, 99: 608-627.

• Kojima, F., Pathak, P., and Roth, A. (2013), Stability and Incentives in Matching with Couples, Quarterly Journal of Economics, 128: 1585-1632. 

• Kojima, F., Tamura, A., and Yokoo, M. (2014). Designing Matching Mechanisms under Constraints: An Approach from Discrete Convex Analysis. 

• Kominers, S. D. (2012). On the correspondence of contracts to salaries in (many-to-many) matching. Games and Economic Behavior, 75(2), 984-989.

• Ma, J. (1994), Strategy-Proofness and the Strict Core in a Market with Indivisibilities, International Journal of Game Theory, 23: 75-83.  

• Mongell, S. and Roth, A. (1986), A Note on Job Matching with Budget Constraints, Economics Letters, 21: 135-138. 

• Monte, D. and Tumennasan, N. (2013), Matching with quorums, Economics Letters, 120: 14–17. 

• Niederle, M. and Roth, A. (2003), Relationship Between Wages and Presence of a Matching in Medical Fellowships, Journal of the American Medical Associations, 290: 1153-1154.

• Ostrovsky, M. (2007), Stability in Supply-Chain Networks, American Economic Review, 98: 897-923. 

• Pathak, P. and Sonmez, T. (2008), Leveling the Playing Field: Sincere and Sophisticated Players in the Boston Mechanism, American Economic Review, 98: 1636-1652.

• Pathak, P. and Sonmez, T. (2013), School Admissions Reform in Chicago and England: Comparing Mechanisms by their Vulnerability to Manipulation, American Economic Review, 103: 80-106. 

• Roth, A. (1982), The economics of matching: stability and incentives, Mathematics of Operations Research, 7: 617-28.

• Roth, A. (1984a), The evolution of the labor market for medical interns and residents: a case study in game theory, Journal of Political Economy, 92: 991–1016. 

• Roth, A. (1984b), Stability and Polarization of Interests in Job Matching, Econometrica, 52: 47-58.  

• Roth, A. (1985), The college admissions problem is not equivalent to the marriage problem, Journal of Economic Theory, 36: 277–288.

• Roth, A. (1986), On the Allocation of Residents to Rural Hospitals: A General Property of Two-Sided Matching Markets, Econometrica, 54: 425-427. 

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