Cake-cutting Algorithms

Transcription

1 Cake-cutting Algorithms Folien zur Vorlesung Sommersemester 2016 Dozent: Prof. Dr. J. Rothe J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 1 / 22

2 Preliminary Remarks Websites Websites Vorlesungswebsite: http : //ccc.cs.uni-duesseldorf.de/ rothe/cake Anmeldung nicht nur im LSF, sondern auch unter (CCC-System für alle meine Veranstaltungen) J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 2 / 22

3 This textbook connects three vibrant areas at the interface between economics and computer science: algorithmic game theory, computational social choice, and fair division. It thus offers an interdisciplinary treatment of collective decision making from an economic and computational perspective. Part I introduces to algorithmic game theory, focusing on both noncooperative and cooperative game theory. Part II introduces to computational social choice, focusing on both preference aggregation (voting) and judgment aggregation. Part III introduces to fair division, focusing on the division of both a single divisible resource ("cake-cutting") and multiple indivisible and unshareable resources ("multiagent resource allocation"). In all these parts, much weight is given to the algorithmic and complexity-theoretic aspects of problems arising in these areas, and the interconnections between the three parts are of central interest. Business / Economics isbn Preliminary Remarks Literature Buchblock 155 x 235 mm Abstand 6 mm MM: van Dijk 2570 St 1a, Farbunverbindlicher Ausdruck SPIESZDESIGN, Tel design@spiesz.de n Buchreihe: Bildquelle: Autor/Spieszdesign Bildrechte: Änderung gegenüber Vorentwurf: Kosten / _ Werbedatei Vertreter-Freigabe MM. U1 Freigabe MM. U1-U4 Freigabe Herst. DAT Freigabe Literature J. Rothe (Herausgeber): Economics and Computation: An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division. Springer-Verlag, 2015 Springer Texts in Business and Economics Jörg Rothe Editor Economics and Computation An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division Rothe Ed. Springer Texts in Business and Economics J. Rothe D. Baumeister C. Lindner I. Rothe 1 Economics and Computation Jörg Rothe Editor Economics and Computation An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division Einführung in Computational Social Choice Individuelle Strategien und kollektive Entscheidungen beim Spielen, Wählen und Teilen J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 3 / 22

4 Preliminary Remarks Literature Literature: from Prof. Michael Wooldridge, Oxford Dear Joerg, I just received a copy of Economics and Computation. It looks FANTASTIC! I already started reading some of it, and I think we will use it on a course we are giving here next year. It was tremendously kind of you to think about sending me a copy I m very grateful! Congratulations, and thanks again! Mike Professor Michael Wooldridge mailto:mjw@cs.ox.ac.uk Department of Computer Science, University of Oxford. J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 4 / 22

5 Preliminary Remarks Literature Literature (Recommended for Additional Reading) Jack Robertson and William Webb: Cake-Cutting Algorithms: Be Fair if You Can, A K Peters, 1998 Steven J. Brams and Alan D. Taylor: Fair Division: From Cake-Cutting to Dispute Resolution, Cambridge University Press, 1996 J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 5 / 22

6 The Players and the Cake The Players Claudia Doro Edith Felix Gábor Holger J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 6 / 22

7 The Players and the Cake More Players and the Cake Anna Belle Chris David Edgar 0,5 0,1 0,09 0,01 0,1 0,2 Figure: A (normalized) cake as a metaphor of a heterogeneous resource J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 7 / 22

8 What is Cake-cutting? What is Cake-cutting? Cake-cutting describes the problem of fairly dividing an infinitely divisible resource the cake among the players (as opposed to fair division of indivisible objects as in auctions). This in particular applies to areas such as economics & law (e.g., the division of a property in a divorce or inheritance case), science & technology (e.g., the division of computing time among several users sharing a single computer, or the division of bandwidth when sharing a network), and even politics (historical example: the division of Germany among the allied powers the USA, Great Britain, France, and Russia as a result of the Second World War). J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 8 / 22

9 What is Cake-cutting? Historically: the Polish School Hugo Steinhaus Stefan Banach Bronis law Knaster ( ) ( ) ( ) It may be stated incidentally that if there are two (or more) partners with different estimations, there exists a division giving to everybody more than his due part; the fact disproves the common opinion that differences in estimations make fair division difficult. Hugo Steinhaus J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 9 / 22

10 What is Cake-cutting? Crucial Assumptions Every player p i has an individual, private valuation function v i measuring p i s preferences/utilities over all pieces of cake. (Axiomatic formalization will be given later.) Private means: Every player knows only her own valuation function. The cake is a metaphor of a heterogeneous resource/good/object: Distinct players may value one and the same piece of cake differently, i.e., their individual valuation functions will in general be distinct. Pieces of cake having equal size can be valued differently by the same player: It is not all about its size! J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 10 / 22

11 Four Methods for Two Players Four Methods for Two Players Mom wants to fairly divide the cake among: Claudia and Felix J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 11 / 22

12 Four Methods for Two Players Four Methods for Two Players Method 1: Method 2: Method 3: Mom cuts the cake into two pieces that she thinks are equal and presents Claudia with one piece and Felix with the other. Mom cuts the cake into two pieces and Claudia and Felix toss a coin to decide who may choose first. Claudia cuts the cake into two pieces and Claudia may choose first. Method 4: Claudia cuts the cake into two pieces and Felix may choose first. J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 12 / 22

13 Four Methods for Two Players Four Methods for Two Players Method 1: Method 2: Method 3: Mom cuts the cake into two pieces that she thinks are equal and presents Claudia with one piece and Felix with the other. Mom cuts the cake into two pieces and Claudia and Felix toss a coin to decide who may choose first. Claudia cuts the cake into two pieces and Claudia may choose first. Method 4: Claudia cuts the cake into two pieces and Felix may choose first. J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 13 / 22

14 Four Methods for Two Players Cut-and-Choose Protocol for Two Players Given: Cake X and players p 1 and p 2 with valuation functions v 1 and v 2. Step 1: p 1 executes one cut to divide the cake into two pieces, S 1 and S 2, that he considers equal halves: v 1 (S 1 ) = v 1 (S 2 ). It holds that X = S 1 S 2. Step 2: p 2 chooses her best piece. Step 3: p 1 receives the remaining piece. Figure: Cut-and-choose protocol for two players J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 14 / 22

15 Box Representations of Valuation Functions Box Representations of Valuation Functions Belle Edgar Figure: Example of a box representation of two valuation functions In this example, the cake consists of a total of 12 boxes. The number of boxes per column specifies the valuation of this part of the cake, i.e., every column represents a distinct portion of the cake. Here, Belle values the right edge of the cake significantly less than Edgar does, and Edgar shows only little interest in its middle part. J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 15 / 22

16 Box Representations of Valuation Functions Box Representations of Valuation Functions More specifically, the left fifth of the cake is worth 4 /12 = 1 /3 to Belle and 3 /12 = 1 /4 to Edgar. The box representation is a very coarse and simplified representation of the players valuation functions: There are valuation functions where the box representation fails to adequately describe them. However, the box representation is a very handy tool to visualize examples and to demonstrate step-by-step the procedure of a protocol for a given set of valuation functions. Markings and cuts (the latter of which will be labeled by a knife) can only be done in a vertical manner. For the sake of clarity, all examples using the box representation are designed such that cuts are always made where two columns join. J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 16 / 22

17 Box Representations of Valuation Functions Cut-and-Choose Protocol: Using the Box Representation Belle Edgar Belle Edgar (a) Belle cuts, Edgar chooses (b) Edgar cuts, Belle chooses Figure: Cut-and-choose protocol: Two examples J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 17 / 22

18 Some Questions Some Questions Is the cut-and-choose protocol fair? How can it be generalized to more than two players? What does it take for a division to be fair? What other aspects need to be taken into account when evaluating the quality of a division? How can fairness be guaranteed, irrespective of the valuation functions of the players? How can we fairly divide a cake? J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 18 / 22

19 Some Questions Some Questions Is the cut-and-choose protocol fair? How can it be generalized to more than two players? What does it take for a division to be fair? What other aspects need to be taken into account when evaluating the quality of a division? How can fairness be guaranteed, irrespective of the valuation functions of the players? How can we fairly divide a cake? J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 19 / 22

20 Some Questions Doesn t Cut It Method for Three Players Claudia Doro Edith Each player wants to get one third of the cake (in her valuation). That is, they aim at a proportional division. J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 20 / 22

21 Some Questions Doesn t Cut It Method for Three Players Given: Cake X and players Claudia, Doro, and Edith with valuation functions v C, v D, and v E. Step 1: Claudia cuts the cake X into two pieces, X 1 and X 2 with X = X 1 X 2, such that v C (X 1 ) = 1 /3 and v C (X 2 ) = 2 /3. Step 2: Doro cuts the piece X 2 into two pieces, X 21 and X 22 with X 2 = X 21 X 22, such that v D (X 21 ) = ( 1 /2) v D (X 2 ) and v D (X 22 ) = ( 1 /2) v D (X 2 ). Step 3: The three players choose one piece each in the order: 1 Edith (chooses among X 1, X 21, and X 22 ); 2 Claudia (chooses among the two remaining pieces); 3 Doro (takes the last piece). J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 21 / 22

22 Some Questions Doesn t Cut It Method for Three Players: Who is Happy? 1 Edith? Certainly! 2 Claudia? As well! 3 Doro? She is not guaranteed to be happy! J. Rothe (HHU Düsseldorf) Cake-cutting Algorithms 22 / 22