Fair Division of Land Erel Segal-Halevi Department of Computer Science Ph.D. Thesis Submitted to the Senate of Bar Ilan University Ramat-Gan, Israel December 2016 This work was carried out under the supervision of Prof. Yonatan Aumann and Prof. Avinatan Hassidim Department of Computer Science, Bar-Ilan University. I would like to acknowledge the generous sponsorship of Drs. Mordecai & Monique Katz and the Katz Graduate fellowship program for support of my Doctoral degree studies and research Acknowledgements This research is a small step in a long journey, and now is a good time to acknowledge and thank all those who accompanied me along my share in this journey. This section is organized mostly in chronological order. First and foremost, thank God who has given His guidance to us through His commandments in the Bible. Specifically, the commandment to divide the land equally (Numbers 26, Ezekiel 47) was the primary inspiration to this entire research. Many thanks to my parents Dov and Dina, who raised me and taught me, since I was a child, to love both Bible and science. The education they instilled in me still drives me forward. The current academic chapter in my life started when Sarit Kraus and Ido Dagan, of the BIU computer science department, accepted me as a research assistant. Later on, they encouraged me to start a Ph.D. research. Although I changed the topic along the way, I benefited a lot from their guidance and support. I also enjoyed a lot to work with colleagues in both labs, particularly: Inon Zuckerman, Avi Rosenfeld, Ariel Roth and Osnat Drein in Sarit’s lab, and Meni Adler, Eden Erez, Ofer Bronstein, Yonatan Berant, Eyal Schnarch, Amnon Lotan, Oren Melamud, Lili Kotlerman, Chaya Liebeskind and Hila Weismann in Ido’s lab. The present research was initiated as a seminar project in Nitzotzot Shachar (sparks of dawn) — a multi-disciplinary seminar in Judaism and Science. The course leaders, Rav Shabtay Rappaport and Roly Belfer, encouraged me to pursue this research further, and connected me to Shmuel Nitzan. Shmuel Nitzan of the BIU economics department introduced me to the classic literature on fairness, and helped me a lot to enter the community of economic theory. His guidance and support are still of great help to me. I am especially grateful to Yonatan Aumann and Avinatan Hassidim, who agreed to become my Ph.D. advisors in an unusual schedule. They were very supportive of my half-baked ideas, and helped me make them into full-blown results. They guided me through new research topics in computational economics and game theory and introduced me to the community of researchers in these fields. They provided and still provide invaluable guidance and help for my first steps as a researcher. During my research I met a lot of friends in the Bar-Ilan University. I gained a lot from participating in the BIU game theory seminar and presenting there some parts of this research. I got helpful comments from the participants, including Igal Milchtaich, Ron Peretz, Amnon Schreiber, Gilad Bavli, Ziv Hellman, Eyal Baharad, Shiri Alon-Eron and Gabi Gayer. I received helpful advice from people in the math department, including Ron Adin, Tahl Nowik, Mikhail Katz, Nathan Keller, Reuven Cohen and Simi Haber. The secretaries, Hila Atia and Daphna Gad and Sylvie Baruch and Sigal Segev-Zrahia of the CS depart- ment and Michal Glazerman and Orit Nissim and Efrat Uzan of the economics department, provided all the help that I needed with administrative issues. Nachum Weil and Pinchas Weisberg and Yisrael Yedidya and Doron Kurzberg from the CS department helped me a lot with computational issues. Many researchers inspired me through their beautiful papers before I met them personally, and even more so after meeting (some of) them. I learned a lot from Haris Aziz, Noga Alon, Reuven Bar-Yehuda, Julius Barbanel, Xiaohui Bei, Silvain Bouveret, Steve Brams, Simina Brânzei, Iannis Caragiannis, Katarina Cechlarova, Marco Dall’Aglio, Ulle Endriss, Shimon Even, Dorothea Herreiner, Michael Jones, Marc Kil- gour, Christian Klamler, David Kurokawa, Jérôme Lang, Michel Lemaitre, Nicolas Maudet, Herve Moulin, Azaria Paz, Ariel D. Procaccia, Jack M. Robertson, Nisarg Shah, Walter Stromquist, Francis E. Su, Alan D. Taylor, and William A. Webb. Herve Moulin kindly hosted me in the university of Glasgow and introduced me to interesting new topics in fair division. I enjoyed working with him, as well as with various other friends from abroad, particularly Balázs Sziklai, Arseniy Akopyan, Laszlo Kozma and Emel Ozturk. I received hundreds of answers to questions in mathematics, economics, computer science, program- ming and many other issues through the Stack-Exchange websites. I did my best to mention all the people who kindly responded to my queries in the acknowledgments sections of the relevant chapters. I feel fortu- nate to live in a time where tools such as Wikipedia and Stack-Exchange make international collaboration so easy; not sure how I could have completed this research without them. My research was funded by various funds, including the Doctoral Fellowships of Excellence Program in BIU, the ISF grants 1083/13 and 1241/12 and 1394/16, the Wolfson Chair and the Mordecai and Monique Katz Graduate Fellowship Program. Last and most beloved — many thanks to my wife Galya. She gave me a lot of love, support and inspi- ration ever since we’ve met, listened and commented on all my practice presentations, provided genuine insights and ideas on my research, accepted all my culinary challenges, and baked wonderful cakes © — Erel Segal-Halevi — Hod-Hasharon, 2017-06 Contents Abstract i 1 Introduction 1 1.1 The Land Problem . .1 1.2 The Fair Division Solution . .1 1.3 Applying Fair Division to Land . .2 1.4 Related Work . .3 2 Model 6 2.1 Cake . .6 2.2 Agents . .6 2.3 Queries . .6 2.4 Allocations . .7 2.5 Fairness . .7 2.6 Geometry . .8 3 Geometric Proportional Division 10 3.1 Introduction . 10 3.1.1 Results . 11 3.1.2 Techniques . 14 3.1.3 Related work . 14 3.2 Model . 15 3.3 Impossibility Results . 16 3.3.1 Impossibility results for two, three and four walls . 17 3.3.2 Impossibility results for one wall . 18 3.3.3 Impossibility results for zero walls . 19 3.3.4 Impossibility results with fat rectangles . 22 3.4 Auctions and Covers . 24 3.4.1 Mark auction . 24 3.4.2 Eval auction . 25 3.4.3 Cover numbers . 26 3.5 Division procedures . 27 3.5.1 Four and three walls, guillotine cuts . 27 3.5.2 Two walls . 33 3.5.3 One and zero walls . 35 3.5.4 Three walls . 36 3.5.5 Four walls, guillotine cuts, identical valuations . 41 3.5.6 Compact cakes of any shape . 46 3.6 Conclusions and Future Work . 48 3.6.1 Open questions . 48 3.6.2 Different geometric constraints . 49 3.6.3 Web implementation . 49 3.7 Acknowledgments . 49 Chapter appendix ............................................... 51 3.A Staircase Lemma . 51 3.B Non-intersection of Squares in Fat Procedure . 52 3.C Existence of Best Pieces . 53 3.D Non-Rectangular Pieces . 54 4 Geometric Envy-Free Division 57 4.1 Introduction . 58 4.1.1 Results . 59 4.1.2 Related Work . 60 4.2 Model . 61 4.3 Geometric Preliminaries . 62 4.3.1 Geometric loss . 62 4.3.2 Chooser Lemma . 63 4.3.3 Cover Numbers and Cover Lemma . 64 4.3.4 Knife functions . 65 4.3.5 Geometric loss of knife functions . 66 4.4 Envy-Free Division for Two Agents . 67 4.4.1 Squares and rectangles . 67 4.4.2 Cubes and archipelagos . 68 4.4.3 Fat rectangles . 69 4.4.4 Arbitrary fat objects . 71 4.4.5 Between envy-freeness end proportionality . 72 4.5 Envy-Free Division For n agents . 73 4.5.1 The one-dimensional procedure . 73 4.5.2 Knife tuples . 74 4.5.3 Squares and fat rectangles . 77 4.5.4 Arbitrary fat objects . ..