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University of Oviedo Ghent University MONOTONICITY-BASED University of Oviedo Ghent University Faculty of Sciences Faculty of Bioscience Engineering MONOTONICITY-BASED CONSENSUS STATES FOR THE MONOMETRIC RATIONALISATION OF RANKING RULES WITH APPLICATION IN DECISION MAKING Ra´ulP´erez{Fern´andez Thesis submitted in fulfillment of the requirements for the degree of Doctor (Ph.D) by the University of Oviedo (Doctoral Program of Mathematics and Statistics) Doctor (Ph.D) of Applied Biological Sciences by Ghent University Academic year 2016-2017 KERMIT Supervisors: Prof. dr. Bernard De Baets Department of Mathematical Modelling, Statistics and Bioinformatics Ghent University, Belgium Prof. dr. Irene D´ıaz Department of Computer Science University of Oviedo, Spain Prof. dr. Susana Montes Department of Statistics and O.R. University of Oviedo, Spain Examination board: Prof. dr. Pedro Alonso Prof. dr. Bernard De Baets Prof. dr. Jos´eLuis Garc´ıa-Lapresta Prof. dr. Ignacio Montes Prof. dr. ir. Wim Verbeke Dean Faculty of Sciences (University of Oviedo): Prof. dr. Jos´eManuel Noriega Rector University of Oviedo: Prof. dr. Santiago Garc´ıa-Granda Dean Faculty of Bioscience Engineering (Ghent University): Prof. dr. ir. Marc Van Meirvenne Rector Ghent University: Prof. dr. Anne De Paepe Ra´ulP´erez{Fern´andez MONOTONICITY-BASED CONSENSUS STATES FOR THE MONOMETRIC RATIONALISATION OF RANKING RULES WITH APPLICATION IN DECISION MAKING Thesis submitted in fulfillment of the requirements for the degree of Doctor (Ph.D) by the University of Oviedo (Doctoral Program of Mathematics and Statistics) Doctor (Ph.D) of Applied Biological Sciences by Ghent University Academic year 2016-2017 Spanish translation of the title: Estados de consenso basados en la propiedad de monoton`ıapara la racionalizaci´onmono- m´etricade las reglas de ordenaci´oncon aplicaci´onen la toma de decisiones Dutch translation of the title: Monotoniteit-gebaseerde consensustoestanden voor de monometrische rationalisatie van rankschikkingsregels met toepassing in besluitvorming Please refer to this work as follows: Ra´ulP´erez{Fern´andez(2017). Monotonicity-based consensus states for the monometric rationalisation of ranking rules with application in decision making, PhD Thesis. University of Oviedo, Oviedo, Spain. KERMIT, Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Ghent, Belgium. This research has been partially supported by the Campus of International Excellence of the University of Oviedo, by the BOF grant with code 01DI2816 and by the project IWT-SBO- 130036: CHECKPACK - Integrated optical sensors in food packaging to simultaneously detect early-spoilage and check package integrity. ISBN The author and the supervisors give the authorisation to consult and to copy parts of this work for personal use only. Every other use is subject to the copyright laws. Permission to reproduce any material contained in this work should be obtained from the author. Agradecimientos A todos los que me acompa~naronen estos a~nosde altos y bajos, de caricias y hachazos. A todos los que injustamente olvidar´een estos m´ıserosp´arrafos.A todos los que, de una manera u otra, me han ayudado a llegar hasta aqu´ı.Gracias por ayudarme a entender que la vida consiste en ser feliz. A Samu, Sabi, Aitor, Enol, Mikel, Victor, Andr´es,Christian, Redon y resto de amigos del cachopo y la caleya. A Pelayo, cachopero, caleyero y hermano. A Julia, Kike y Jandro, eternos compa~nerosde indecisiones, escapes y sopas agripicantes. A Coru, que aparece y desaparece, que siempre est´apresente en mi coraz´on. To Marcelo and Marcela, with whom I still have lots of saideiras to share. Gracias por todas esas innumerables noches de filosof´ıay cerveza. To Michael, my partner in crime, my vriendo. To Andreia and Niels, my very best friends in Ghent. To Marc, my foal, who taught me that life tastes better mixed with ginger jenever. To Aisling, my favourite Irish-Canadian, my favourite Irish, my favourite Canadian. To David, who shared my fever, who will always be my favoured coffee partner. To (bad) Wouter, my co-organiser, who managed to create a magic potion from the sewer. To Peter, Bram, Tim, Gang, Stijn, Giacomo, Timpe, Andr´ePur´e,and rest of biomatchers, partners of victories and cheers, of defeats and tears, of fun and beers. To Youri, the only beard I feel envy for. To Guillermo, who enlivened this department even more than he can imagine. To Jos´e,el pollo loco, the spicy chili. To Christina, who suffered my face all day in front. To Dorien, who should never stop smiling. To Michiel, with whom I wandered the last paths of this PhD life. To Hassane, who was as sweet to me as his dates. To Giulio, Bac, Yuanyuan, Laura, Carlos, Shuyun and Wenwen, who meant so much in a ridiculous amount of time. To Nacho, Davide, Quique, Susana D., and rest of members of the research unit UNIMODE. To Timothy, Ivo, Cha¨ım,Marlies, Tinne, Jenna, Jim, Han, i and all members of this bw10 family, which I hold dear. Thanks for making me feel at home during these amasing years. To Micha¨el,who helped this little duck take his first steps, who called my attention to the field of social choice, who is close to being the best man I have ever met. Thanks for your help, time and friendship. A Pedro, Irene y Susana, que me iniciaron en la investigaci´on,que guiaron mis pasos, que, desgraciadamente, vieron demasiadas veces mi lado m´asgris. To Bernard, who has supported me in every single sense, who believed in me when not even I did. Thanks for getting me here. A mi padre, mi h´eroe, mi futuro ser. A mi madre, la persona con la que m´asinjusto soy, a la que no digo suficiente lo mucho que quiero. A mi hermana, que me ha sufrido m´as de 21 a~nosy, a´unas´ı,sigue a mi lado. A mi abuelo, que me mima con locura. A Juanjo, Ana, Dani y Alba, que vienen de Oriente por Navidad. A Carmen y Alberto, mi familia adoptiva. Al resto de familia repartida por el mundo: Sanse, M´alaga,Alcal´a,Par´ıs... A aquellos que, desafortunadamente, ya no est´an.A los que no hay palabras suficientes para agradecer lo mucho que debo. Gracias por ayudarme a ser quien quiero ser. Gracias por ayudarme a querer ser quien soy. A la sonrisa que alumbra mis ma~nanas. A los labios que soplan mis noches en vela. A la mano que, con mimo, mece mi vida. A mi cita con la felicidad. A mi compa~nerade viaje. A ti, siempre a ti, Laura. Gracias por estos inolvidables ocho a~nosjuntos. Happiness only real when shared. Christopher J. McCandless ii Contents Agradecimientos i Contents ix List of symbols xi 1 Prologue 1 1.1 Preface . .1 1.2 Objectives . .3 1.3 Outline of this dissertation . .3 I The state of the art 5 2 Introduction to social choice theory 7 2.1 Social choice theory . .7 2.2 Best-known ranking rules in social choice theory . .9 2.2.1 Plurality (first-past-the-post) . .9 iii 2.2.2 Borda count . 10 2.2.3 Scoring ranking rules . 11 2.2.4 Majority . 13 2.2.5 Condorcet . 16 2.2.6 Kemeny (Kemeny-Young) . 20 2.2.7 Litvak . 22 2.2.8 Bucklin . 23 2.2.9 Copeland . 25 2.2.10 Simpson (Simpson-Kramer, Minimax, Maximin) . 26 2.2.11 Tideman (ranked pairs) . 27 2.2.12 Schulze . 29 2.2.13 WVM and EWVM . 30 2.2.14 Elimination methods . 33 2.3 A ranking rule based on monotonicity . 35 II The methodology 39 3 Rationalisation of ranking rules 41 3.1 Metric rationalisation of ranking rules . 41 3.2 Monometrics . 44 3.2.1 General case . 44 3.2.2 Monometrics on L(C )......................... 50 iv 3.2.3 Monometrics on L(C )r ......................... 55 3.2.4 Monometrics on Mr(L(C )) ...................... 60 3.3 Cost calculation as an optimization problem . 68 3.3.1 Calculating the cost of changing an anonymised profile of rankings into another one . 68 3.3.2 Searching the `closest' (anonymised) profile of rankings . 69 3.3.3 The special case of the zero-one monometric . 71 3.4 Hierarchical combination of monometrics . 72 3.5 Monometric rationalisation of ranking rules . 74 4 Representations of votes 79 4.1 The scorix . 79 4.2 The votrix . 82 4.3 The votex . 85 4.4 The beatpath matrix . 95 5 Monotonicity of a representation of votes 97 5.1 Monotonicity of the scorix . 97 5.2 Recursive monotonicity of the scorix . 100 5.3 Monotonicity of the votrix . 106 5.4 Monotonicity of the votex . 108 5.5 Monotonicity of the beatpath matrix . 110 5.6 Monotonicity of the (anonymised) profile . 113 v 5.7 Acclamation . 116 6 The search for monotonicity as an optimization problem 125 6.1 Changes in the representation of votes . 126 6.1.1 Search for a monotone scorix . 126 6.1.2 Search for a monotone (quasi)votrix . 131 6.1.3 Search for a monotone (quasi)votex . 136 6.2 Changes in the profile of rankings . 143 7 Monotonicity and axiomatic social choice theory 147 7.1 Axiomatic social choice theory . 147 7.2 Independence of all proposed ranking rules . 148 7.2.1 The search for monotonicity of the scorix and the search for recursive monotonicity of the scorix . 149 7.2.2 The search for (recursive) monotonicity of the scorix and all scoring ranking rules . 149 7.2.3 The search for monotonicity of the votrix and the search for mono- tonicity of the votex . 151 7.2.4 The search for monotonicity of the votrix and the search for a Con- dorcet ranking and unanimity .
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