Computation of equilibria on integer programming games Maria Margarida da Silva Carvalho Tese de Doutoramento apresentada à Faculdade de Ciências da Universidade do Porto Ciência de Computadores 2016 Computation of equilibria on integer programming games Maria Margarida da Silva Carvalho Doutoramento em Ciência de Computadores Departamento de Ciência de Computadores 2016 Orientador João Pedro Pedroso, Professor Auxiliar, Faculdade de CCiências da Universidade do Porto Coorientador Andrea Lodi, Professor Catedrático, Università di Bologna e École Polytechnique de Montréal Esta tese foi financiada por uma bolsa de doutoramento da Funda¸c~aopara a Ci^enciae a Tecnologia (refer^enciaSFRH/BD/79201/2011) no ^ambito do programa POPH, financiado pelo Fundo Social Europeu e pelo Governo Portugu^es. 5 6 Acknowledgments I would like to acknowledge the support of the Portuguese Foundation for Science and Technology (FCT) through a PhD grant number SFRH/BD/79201/2011 (POPH/FSE program). The last four years were an exciting adventure to me. I learned, I got puzzled, I got frustrated, I conquered results, I had tons of fun. This journey was accompanied and made possible due to amazing people. I have no words to Jo~aoPedro. I am grateful that he proposed me a challenging research plan. Jo~aoPedro was a tireless supervisor, always available and extremely supportive. I am deeply thankful to my co-supervisor Andrea for his guidance and for making me feel confident about myself. I enjoyed amazing scientific experiences thanks to Andrea and his department DEI, Universit`adi Bologna. I highlight, how lucky I was for having the pleasure to collaborate with Alberto Caprara and Gerhard Woeginger while I lived in Bologna. My year in Bologna was one of the happiest years I ever lived! The role of Ana Viana during my Ph.D. was not only of a collaborator. I am grateful to Ana for sharing knowledge, including me in her research projects, being tremendously friendly and advising me. I kindly thanks Mathieu Van Vyve and Claudio Telha for receiving me warmly in Louvain-la-Neuve and enjoying with me puzzling afternoons. My scientific accomplishments could not have been possible without the strength trans- mited by family and friends. I express a sincere feeling of gratitude to the friends that I made at DCC, DEI, CORE and INESC TEC. A special thanks goes to Amaya and Ana for being my family in Italy and spoiling the Ph.D. life events to me. Agrade¸coprofundamente aos meus pais, `aminha irm~ae ao Ricardo por todo o carinho e paci^enciainfinita comigo durante estes anos. Mil obrigadas a todas e todos que equi- libraram o meu dia-a-dia com o seu amor, amizade, encorajamento, tempo para ouvir problemas matem´aticose n~aomatem´aticos,em particular, Isa, Yellow Hat Sisters, Angela,^ Mariana, Mari Sol e In^es. Many more people have been important during the Ph.D., I apologize for not mentioning everybody. 7 8 Resumo O problema da mochila, o problema de emparelhamento m´aximoe o problema de dimen- sionamento de lotes s~aoexemplos cl´assicosde modelos de otimiza¸c~aocombinat´oriaque t^emsido amplamente estudados na literatura. Nos ´ultimosanos t^emsido investigadas vers~oesmais intrincadas, o que resulta numa melhor aproxima¸c~aodos problemas do mundo real e num aperfei¸coamento das t´ecnicasde solu¸c~ao. O objetivo desta tese de doutora- mento ´eestender as ferramentas algor´ıtmicasque resolvem problemas combinat´orioscom apenas um decisor para jogos, isto ´e,para problemas combinat´orioscom v´ariosdecisores. Frequentemente um processo de decis~aodepende de par^ametrosque s~aocontrolados por decisores externos. Por conseguinte, os jogos combinat´orioss~aouma linha de investiga¸c~ao fundamental, uma vez que refletem a realidade destes problemas. Focamo-nos na classifica¸c~aoda complexidade computacional e no desenho de algoritmos para determinar equil´ıbriosde jogos em programa¸c~aointeira com utilidades quadr´aticas. Num jogo em programa¸c~aointeira, o objetivo de um jogador ´eformulado usando termi- nologia de programa¸c~aomatem´atica. Cada jogador tem o intuito de maximizar a sua utilidade, uma fun¸c~aoque depende das suas vari´aveis de decis~ao(estrat´egias)e das dos restantes. Iremos concentrar-nos em jogos onde as fun¸c~oesde utilidade de cada jogador s~aoquadr´aticasnas suas vari´aveis de decis~ao. De forma a que esta tese seja auto-contida, come¸camos por fornecer as bases essenciais da teoria de complexidade computacional, da programa¸c~aomatem´aticae da teoria dos jogos. Seguir-se-´aa apresenta¸c~aodas nossas contribui¸c~oes,as quais est~aodivididas em duas partes: competi¸c~aode Stackelberg e jogos em simult^aneo. A primeira parte ´esobre competi¸c~oesde Stackelberg (tamb´emconhecidas por programa¸c~ao com dois n´ıveis), onde os jogadores jogam de forma sequencial. Estudamos um dos modelos mais simples de competi¸c~aode Stackelberg combinat´oria, o qual ´ebaseado no problema da mochila. Caracterizamos a complexidade de calcular um equil´ıbrio e desenhamos um algoritmo novo para atacar um problema de interdi¸c~aocom dois n´ıveis, o problema da mochila com restri¸c~oesde interdi¸c~ao.Recentemente, a classe de problemas de interdi¸c~aotem recebido uma grande aten¸c~aopor parte da comunidade de investiga¸c~ao. A segunda parte ´esobre jogos em simult^aneo,isto ´e,jogos em que os jogadores selecionam as suas estrat´egiasao mesmo tempo. Esta defini¸c~aod´aj´auma ideia dos obst´aculosque iremos encontrar na determina¸c~aode estrat´egiasracionais para os jogadores, uma vez que as estrat´egiasdos seus rivais ter~aode ser previstas antecipadamente. Neste contexto, 9 investigamos a estrutura de 3 jogos em particular: o jogo de coordena¸c~aoda mochila (baseado no problema da mochila), o jogo das trocas de rins (baseado no problema de emparelhamento m´aximo)e o jogo de dimensionamento de lotes (baseado no problema de dimensionamento de lotes). Em jeito de conclus~ao,depois do estudo destes tr^esjogos olhamos para a situa¸c~aomais complexo, focando a nossa aten¸c~aono caso geral de jogos em simult^aneo.Estabelecemos a rela¸c~aoentre os jogos em simult^aneoe competi¸c~oesde Stackelberg, provando que encontrar uma solu¸c~aopara um jogo em simult^aneo´epelo menos t~aodif´ıcilcomo resolver uma competi¸c~aode Stackelberg. Por fim, constru´ımosum algoritmo para aproximar um equil´ıbriopara jogos em simult^aneo. Palavras-chave: Equil´ıbriosde Nash; jogos em programa¸c~aointeira; competi¸c~oesde Stackelberg; jogos em simult^aneo. 10 Abstract The knapsack problem, the maximum matching problem and the lot-sizing problem are classical examples of combinatorial optimization models that have been broadly studied in the literature. In recent years, more intricate variants of these problems have been investigated, resulting in better approximations of real-world problems and in improve- ments in solution techniques. The goal of this Ph.D. thesis is to extend the algorithmic tools for solving these (single) decision-maker combinatorial problems to games, that is, to combinatorial problems with several decision makers. It is frequent for a decision process to depend on parameters that are controlled by external decision makers. Therefore, combinatorial games are a crucial line of research since they reflect the reality of these problems. We focus in understanding the computational complexity and in designing algorithms to find equilibria to integer programming games with quadratic utilities. In an integer programming game, a player's goal is formulated by using the mathematical programming framework. Each player aims at maximizing her utility, a function of her and other players' decision variables (strategies). We will concentrate in games with quadratic utilities on each player's decision variables. In order to make this thesis self-contained, we start by covering the essential background in computational complexity, mathematical programming and game theory. It is followed by the presentation of our contributions, which are fleshed out in two parts: Stackelberg competition and simultaneous games. The first part concerns Stackelberg competitions (also known as bilevel programming), where players play sequentially. We study the most simple to model combinatorial Stackelberg competitions, which are based on the knapsack problem. We characterize the complexity of computing equilibria and we design a novel algorithm to tackle a bilevel interdiction problem, the knapsack problem with interdiction constraints, a special class of problems which have recently received significant attention in the research community. The second part deals with simultaneous games, i.e., games in which players select their strategies at the same time. This definition already gives a hint of the obstacles involved in finding players' rational strategies, since the opponents strategies have to be predicted. In this context, we investigate the structure of three particular games: the coordination knapsack game (based on the knapsack problem), the kidney-exchange game (based on the maximum matching problem) and the lot-sizing game (based on the lot-sizing problem). 11 To conclude, after investigating these particular games, we move on to the more complex case: general simultaneous games. We establish the connection of simultaneous games with Stackelberg competitions, and prove that finding a solution to a simultaneous game is at least as hard as solving a Stackelberg competition; finally, we devise an algorithm to approximate an equilibrium for simultaneous games. Keywords: Nash equilibria; integer programming games; bilevel programming; Stackel- berg competition; simultaneous games. 12 Contents Resumo 9 Abstract 11 1 Introduction 17 1.1 Context: Mathematical Programming and Game Theory . 17 1.2 Organization and Contributions . 18 2 Background 21 p 2.1 Complexity: P, NP, Σ2 and PPAD classes . 21 2.2 Mathematical Programming . 25 2.2.1 Classical Examples . 29 2.2.1.1 Maximum Matching in a Graph . 29 2.2.1.2 The Knapsack Problem .