DOCSLIB.ORG

Game Theory a Playful Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Game Theory a Playful Introduction STUDENT MATHEMATICAL LIBRARY Volume 80 Game Theory A Playful Introduction Matt DeVos Deborah A. Kent Game Theory A Playful Introduction https://doi.org/10.1090//stml/080 STUDENT MATHEMATICAL LIBRARY Volume 80 Game Theory A Playful Introduction Matt DeVos Deborah A. Kent Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 91-01, 91A46, 91A06, 91B06. For additional information and updates on this book, visit www.ams.org/bookpages/stml-80 Library of Congress Cataloging-in-Publication Data Names: DeVos, Matthew Jared, 1974- j Kent, Deborah A., 1978- Title: Game theory : a playful introduction / Matthew DeVos, Deborah A. Kent. Description: Providence, Rhode Island : American Mathematical Society, [2016] j Series: Student mathematical library ; volume 80 j Includes bibliographical references and index. Identifiers: LCCN 2016035452 j ISBN 9781470422103 (alk. paper) Subjects: LCSH: Game theory–Textbooks. j Combinatorial analysis–Textbooks. j AMS: Game theory, economics, social and behavioral sciences – Instructional exposition (textbooks, tutorial papers, etc.). msc j Game theory, economics, social and behavioral sciences – Game theory – Combinatorial games. msc j Game theory, economics, social and behavioral sciences – Game theory – n-person games, n > 2. msc j Game the- ory, economics, social and behavioral sciences – Mathematical economics – Decision theory. msc Classification: LCC QA269 .D45 2016 j DDC 519.3–dc23 LC record available at https://lccn.loc.gov/2016035452 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions r of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copy- right ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. © 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 21 20 19 18 17 16 Dedicated to my family, MD. For Mom, in memory with love, DK. Contents Preface xi Chapter 1. Combinatorial Games 1 §1.1. Game Trees 3 §1.2. Zermelo’s Theorem 9 §1.3. Strategy 14 Exercises 19 Chapter 2. Normal-Play Games 25 §2.1. Positions and Their Types 27 §2.2. Sums of Positions 30 §2.3. Equivalence 36 Exercises 41 Chapter 3. Impartial Games 45 §3.1. Nim 46 §3.2. The Sprague-Grundy Theorem 52 §3.3. Applying the MEX Principle 54 Exercises 59 vii viii Contents Chapter 4. Hackenbush and Partizan Games 63 §4.1. Hackenbush 64 §4.2. Dyadic Numbers and Positions 71 §4.3. The Simplicity Principle 77 Exercises 83 Chapter 5. Zero-Sum Matrix Games 89 §5.1. Dominance 91 §5.2. Mixed Strategies 95 §5.3. Von Neumann Solutions 100 Exercises 104 Chapter 6. Von Neumann’s Minimax Theorem 111 §6.1. Equating the Opponent’s Results 113 §6.2. Two-Dimensional Games 118 §6.3. Proof of the Minimax Theorem 123 Exercises 128 Chapter 7. General Games 133 §7.1. Utility 135 §7.2. Matrix Games 139 §7.3. Game Trees 145 §7.4. Trees vs. Matrices 150 Exercises 155 Chapter 8. Nash Equilibrium and Applications 161 §8.1. Nash Equilibrium 162 §8.2. Evolutionary Biology 169 §8.3. Cournot Duopoly 176 Exercises 182 Chapter 9. Nash’s Equilibrium Theorem 187 §9.1. Sperner’s Lemma 189 §9.2. Brouwer’s Fixed Point Theorem 192 Contents ix §9.3. Strategy Spaces 198 §9.4. Nash Flow and the Proof 202 Exercises 208 Chapter 10. Cooperation 213 §10.1. The Negotiation Set 214 §10.2. Nash Arbitration 221 §10.3. Repeated Games and the Folk Theorem 228 Exercises 238 Chapter 11. 푛-Player Games 245 §11.1. Matrix Games 247 §11.2. Coalitions 251 §11.3. Shapley Value 260 Exercises 270 Chapter 12. Preferences and Society 275 §12.1. Fair Division 277 §12.2. Stable Marriages 285 §12.3. Arrow’s Impossibility Theorem 290 Exercises 298 Appendix A. On Games and Numbers 301 Appendix B. Linear Programming 309 Basic Theory 310 A Connection to Game Theory 313 LP Duality 317 Appendix C. Nash Equilibrium in High Dimensions 323 Game Boards 331 Bibliography 335 Index of Games 339 Index 341 Preface The story of this book began in 2002 when Matt, then a postdoc at Prince- ton University, was given the opportunity to teach an undergraduate class in game theory. Thanks largely to the 2001 release of a Hollywood movie on the life of the famous Princeton mathematician and (classical) game theorist John Nash, this course attracted a large and highly diverse audience. Princeton’s mathematics department featured not only Nash, but also John Conway, the father of modern combinatorial game theory. So it seemed only natural to blend the two sides of game theory, combi- natorial and classical, into one (rather ambitious) class. The varied back- grounds of the students and the lack of a suitable textbook made for an extremely challenging teaching assignment (that sometimes went awry). However, the simple fun of playing games, the rich mathematical beauty of game theory, and its significant real-world connections still made for an amazing class. Deborah adopted a variant of this material a few years later and fur- ther developed it for a general undergraduate audience. Over the ensu- ing years, Deborah and Matt have both taught numerous incarnations of this course at various universities. Through exchange and collaboration, the material has undergone a thorough evolution, and this textbook rep- resents the culmination of our process. We hope it will provide an intro- ductory course in mathematical game theory that you will find inviting, entertaining, mathematically interesting, and meaningful. xi xii Preface Combinatorial game theory is the study of games like Chess and Checkers in which two opponents alternate moves, each trying to win the game. This part of game theory focuses on deterministic games with full information and is thus highly amenable to recursive analysis. Com- binatorial game theory traces its roots to Charles Bouton’s theory of the game Nim and a classification theorem attributed independently to Roland Sprague and Patrick Grundy. The 1982 publication of the classic Winning Ways for Your Mathematical Plays by Elwyn Berlekamp, John Conway, and Richard Guy laid a modern foundation for the subject— now a thriving branch of combinatorics. In contrast, classical game the- ory is an aspect of applied mathematics frequently taught in departments of economics. Classical game theory is the study of strategic decision- making in situations with two or more players, each of whom may af- fect the outcome. John von Neumann and Oskar Morgenstern are com- monly credited with the foundation of classical game theory in their groundbreaking work Theory of Games and Economic Behavior published in 1944. This treatise established a broad mathematical framework for reasoning about rational decision-making in a wide variety of contexts and it launched a new branch of academic study. Although there have been many significant developments in this theory, John Nash merits mention for his mathematical contributions, most notably the Nash Equilibrium Theorem. Traditionally, the classical and combinatorial sides of game theory are separated in the classroom. A strong theme of strategic thinking nonetheless connects them and we have found the combination to re- sult in a rich and engaging class. The great fun we have had teaching this broad mathematical tour through game theory undergirded our de- cision to write this book. From the very beginning of this project, our goal has been to give an honest introduction to the mathematics of game theory (both combinatorial and classical) that is accessible to an early un- dergraduate student. Over the years, we have developed an approach to teaching combinatorial game theory that avoids some of the set-theoretic complexities found in advanced treatments yet still holds true to the sub- ject. As a result, we achieve the two cornerstones of the Sprague-Grundy Theorem and the Simplicity Principle in an efficient and student-friendly Preface xiii manner. The classical game theory portion of the book contains numer- ous carefully sculpted and easy-to-follow proofs to establish the theoret- ical core of the subject (including the Minimax Theorem, Nash arbitra- tion, Shapley Value, and Arrow’s Paradox). Most significantly, Chapter 9 is entirely devoted to an extremely gentle proof of Nash’s Equilibrium Theorem. For the sake of concreteness, the chapter focuses on 2 × 2 matrices, but each argument generalizes and Appendix C contains full details. Sperner’s Lemma appears in this chapter as the first step of our proof and we offer an intuitive exposition of this lemma by treating itas a game of solitaire. More broadly, Sperner’s Lemma provides a touch- stone through other chapters.
Load more

Recommended publications
  • Is Shapley Cost Sharing Optimal?∗ (For the Special Issue in Honor of Lloyd Shapley)
    Is Shapley Cost Sharing Optimal?∗ (For the special issue in honor of Lloyd Shapley) Shahar Dobzinskiy Aranyak Mehtaz Tim Roughgardenx Mukund Sundararajan{ August 29, 2016 Abstract A general approach to the design of budget-balanced cost-sharing mechanisms is to use the Shapley value, applied to the given cost function, to define payments from the players to the mechanism. Is the corresponding Shapley value mechanism \optimal" in some sense? We consider the objective of minimizing worst-case inefficiency subject to a revenue constraint, and prove results in three different regimes. 1. For the public excludable good problem, the Shapley value mechanism minimizes the worst-case efficiency loss over all truthful, deterministic, and budget-balanced mechanisms that satisfy equal treatment. This result follows from a characteriza- tion of the Shapley value mechanism as the unique one that satisfies two additional technical conditions. 2. For the same problem, even over a much more general mechanism design space that allows for randomization and approximate budget-balance and does not im- pose equal treatment, the worst-case efficiency loss of the Shapley value mecha- nism is within a constant factor of the minimum possible. ∗The results in Sections 3{5 appeared, in preliminary form, in the extended abstract [8]. The results in Section 6 appeared in the PhD thesis of the fourth author [27]. yDepartment of Applied Math and Computer Science, the Weizmann Institute of Science, Rohovot 76100, Israel. This work was done while the author was visiting Stanford University, and supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities, and by grants from the Israel Science Foundation and the USA-Israel Bi-national Science Foundation.
    [Show full text]
  • On Variations of Nim and Chomp Arxiv:1705.06774V1 [Math.CO] 18
    On Variations of Nim and Chomp June Ahn Benjamin Chen Richard Chen Ezra Erives Jeremy Fleming Michael Gerovitch Tejas Gopalakrishna Tanya Khovanova Neil Malur Nastia Polina Poonam Sahoo Abstract We study two variations of Nim and Chomp which we call Mono- tonic Nim and Diet Chomp. In Monotonic Nim the moves are the same as in Nim, but the positions are non-decreasing numbers as in Chomp. Diet-Chomp is a variation of Chomp, where the total number of squares removed is limited. 1 Introduction We study finite impartial games with two players where the same moves are available to both players. Players alternate moves. In a normal play, the person who does not have a move loses. In a misère play, the person who makes the last move loses. A P-position is a position from which the previous player wins, assuming perfect play. We can observe that all terminal positions are P-positions. An N-position is a position from which the next player wins given perfect play. When we play we want to end our move with a P-position and want to see arXiv:1705.06774v1 [math.CO] 18 May 2017 an N-position before our move. Every impartial game is equivalent to a Nim heap of a certain size. Thus, every game can be assigned a non-negative integer, called a nimber, nim- value, or a Grundy number. The game of Nim is played on several heaps of tokens. A move consists of taking some tokens from one of the heaps. The game of Chomp is played on a rectangular m by n chocolate bar with grid lines dividing the bar into mn squares.
    [Show full text]
  • Stable Allocations and the Practice of Market Design
    15 OCTOBER 2012 Scientific Background on the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012 STABLE ALLOCATIONS AND THE PRACTICE OF MARKET DESIGN compiled by the Economic Sciences Prize Committee of the Royal Swedish Academy of Sciences THE ROYAL SWEDISH ACADEMY OF SCIENCES has as its aim to promote the sciences and strengthen their influence in society. BOX 50005 (LILLA FRESCATIVÄGEN 4 A), SE-104 05 STOCKHOLM, SWEDEN TEL +46 8 673 95 00, FAX +46 8 15 56 70, [email protected] HTTP://KVA.SE 1 Introduction Economists study how societies allocate resources. Some allocation problems are solved by the price system: high wages attract workers into a particu- lar occupation, and high energy prices induce consumers to conserve energy. In many instances, however, using the price system would encounter legal and ethical objections. Consider, for instance, the allocation of public-school places to children, or the allocation of human organs to patients who need transplants. Furthermore, there are many markets where the price system operates but the traditional assumption of perfect competition is not even approximately satis…ed. In particular, many goods are indivisible and het- erogeneous, whereby the market for each type of good becomes very thin. How these thin markets allocate resources depends on the institutions that govern transactions. This year’s prizewinning work encompasses a theoretical framework for analyzing resource allocation, as well as empirical studies and actual redesign of real-world institutions such as labor-market clearinghouses and school ad- missions procedures. The foundations for the theoretical framework were laid in 1962, when David Gale and Lloyd Shapley published a mathematical inquiry into a certain class of allocation problems.
    [Show full text]
  • Stable Matching: Theory, Evidence, and Practical Design
    THE PRIZE IN ECONOMIC SCIENCES 2012 INFORMATION FOR THE PUBLIC Stable matching: Theory, evidence, and practical design This year’s Prize to Lloyd Shapley and Alvin Roth extends from abstract theory developed in the 1960s, over empirical work in the 1980s, to ongoing efforts to fnd practical solutions to real-world prob- lems. Examples include the assignment of new doctors to hospitals, students to schools, and human organs for transplant to recipients. Lloyd Shapley made the early theoretical contributions, which were unexpectedly adopted two decades later when Alvin Roth investigated the market for U.S. doctors. His fndings generated further analytical developments, as well as practical design of market institutions. Traditional economic analysis studies markets where prices adjust so that supply equals demand. Both theory and practice show that markets function well in many cases. But in some situations, the standard market mechanism encounters problems, and there are cases where prices cannot be used at all to allocate resources. For example, many schools and universities are prevented from charging tuition fees and, in the case of human organs for transplants, monetary payments are ruled out on ethical grounds. Yet, in these – and many other – cases, an allocation has to be made. How do such processes actually work, and when is the outcome efcient? Matching theory The Gale-Shapley algorithm Analysis of allocation mechanisms relies on a rather abstract idea. If rational people – who know their best interests and behave accordingly – simply engage in unrestricted mutual trade, then the outcome should be efcient. If it is not, some individuals would devise new trades that made them better of.
    [Show full text]
  • An Introduction to Conway's Games and Numbers
    AN INTRODUCTION TO CONWAY’S GAMES AND NUMBERS DIERK SCHLEICHER AND MICHAEL STOLL 1. Combinatorial Game Theory Combinatorial Game Theory is a fascinating and rich theory, based on a simple and intuitive recursive definition of games, which yields a very rich algebraic struc- ture: games can be added and subtracted in a very natural way, forming an abelian GROUP (§ 2). There is a distinguished sub-GROUP of games called numbers which can also be multiplied and which form a FIELD (§ 3): this field contains both the real numbers (§ 3.2) and the ordinal numbers (§ 4) (in fact, Conway’s definition gen- eralizes both Dedekind sections and von Neumann’s definition of ordinal numbers). All Conway numbers can be interpreted as games which can actually be played in a natural way; in some sense, if a game is identified as a number, then it is under- stood well enough so that it would be boring to actually play it (§ 5). Conway’s theory is deeply satisfying from a theoretical point of view, and at the same time it has useful applications to specific games such as Go [Go]. There is a beautiful microcosmos of numbers and games which are infinitesimally close to zero (§ 6), and the theory contains the classical and complete Sprague-Grundy theory on impartial games (§ 7). The theory was founded by John H. Conway in the 1970’s. Classical references are the wonderful books On Numbers and Games [ONAG] by Conway, and Win- ning Ways by Berlekamp, Conway and Guy [WW]; they have recently appeared in their second editions.
    [Show full text]
  • Impartial Games
    Combinatorial Games MSRI Publications Volume 29, 1995 Impartial Games RICHARD K. GUY In memory of Jack Kenyon, 1935-08-26 to 1994-09-19 Abstract. We give examples and some general results about impartial games, those in which both players are allowed the same moves at any given time. 1. Introduction We continue our introduction to combinatorial games with a survey of im- partial games. Most of this material can also be found in WW [Berlekamp et al. 1982], particularly pp. 81{116, and in ONAG [Conway 1976], particu- larly pp. 112{130. An elementary introduction is given in [Guy 1989]; see also [Fraenkel 1996], pp. ??{?? in this volume. An impartial game is one in which the set of Left options is the same as the set of Right options. We've noticed in the preceding article the impartial games = 0=0; 0 0 = 1= and 0; 0; = 2: {|} ∗ { | } ∗ ∗ { ∗| ∗} ∗ that were born on days 0, 1, and 2, respectively, so it should come as no surprise that on day n the game n = 0; 1; 2;:::; (n 1) 0; 1; 2;:::; (n 1) ∗ {∗ ∗ ∗ ∗ − |∗ ∗ ∗ ∗ − } is born. In fact any game of the type a; b; c;::: a; b; c;::: {∗ ∗ ∗ |∗ ∗ ∗ } has value m,wherem =mex a;b;c;::: , the least nonnegative integer not in ∗ { } the set a;b;c;::: . To see this, notice that any option, a say, for which a>m, { } ∗ This is a slightly revised reprint of the article of the same name that appeared in Combi- natorial Games, Proceedings of Symposia in Applied Mathematics, Vol. 43, 1991. Permission for use courtesy of the American Mathematical Society.
    [Show full text]
  • Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi
    HEXAFLEXAGONS, PROBABILITY PARADOXES, AND THE TOWER OF HANOI For 25 of his 90 years, Martin Gard- ner wrote “Mathematical Games and Recreations,” a monthly column for Scientific American magazine. These columns have inspired hundreds of thousands of readers to delve more deeply into the large world of math- ematics. He has also made signifi- cant contributions to magic, philos- ophy, debunking pseudoscience, and children’s literature. He has produced more than 60 books, including many best sellers, most of which are still in print. His Annotated Alice has sold more than a million copies. He continues to write a regular column for the Skeptical Inquirer magazine. (The photograph is of the author at the time of the first edition.) THE NEW MARTIN GARDNER MATHEMATICAL LIBRARY Editorial Board Donald J. Albers, Menlo College Gerald L. Alexanderson, Santa Clara University John H. Conway, F.R. S., Princeton University Richard K. Guy, University of Calgary Harold R. Jacobs Donald E. Knuth, Stanford University Peter L. Renz From 1957 through 1986 Martin Gardner wrote the “Mathematical Games” columns for Scientific American that are the basis for these books. Scientific American editor Dennis Flanagan noted that this column contributed substantially to the success of the magazine. The exchanges between Martin Gardner and his readers gave life to these columns and books. These exchanges have continued and the impact of the columns and books has grown. These new editions give Martin Gardner the chance to bring readers up to date on newer twists on old puzzles and games, on new explanations and proofs, and on links to recent developments and discoveries.
    [Show full text]
  • COMPSCI 575/MATH 513 Combinatorics and Graph Theory
    COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #34: Partisan Games (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 9 December 2016 Partisan Games • Conway's Game Theory • Hackenbush and Domineering • Four Types of Games and an Order • Some Games are Numbers • Values of Numbers • Single-Stalk Hackenbush • Some Domineering Examples Conway’s Game Theory • J. H. Conway introduced his combinatorial game theory in his 1976 book On Numbers and Games or ONAG. Researchers in the area are sometimes called onagers. • Another resource is the book Winning Ways by Berlekamp, Conway, and Guy. Conway’s Game Theory • Games, like everything else in the theory, are defined recursively. A game consists of a set of left options, each a game, and a set of right options, each a game. • The base of the recursion is the zero game, with no options for either player. • Last time we saw non-partisan games, where each player had the same options from each position. Today we look at partisan games. Hackenbush • Hackenbush is a game where the position is a diagram with red and blue edges, connected in at least one place to the “ground”. • A move is to delete an edge, a blue one for Left and a red one for Right. • Edges disconnected from the ground disappear. As usual, a player who cannot move loses. Hackenbush • From this first position, Right is going to win, because Left cannot prevent him from killing both the ground supports. It doesn’t matter who moves first.
    [Show full text]
  • ~Umbers the BOO K O F Umbers
    TH E BOOK OF ~umbers THE BOO K o F umbers John H. Conway • Richard K. Guy c COPERNICUS AN IMPRINT OF SPRINGER-VERLAG © 1996 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a re­ trieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Published in the United States by Copernicus, an imprint of Springer-Verlag New York, Inc. Copernicus Springer-Verlag New York, Inc. 175 Fifth Avenue New York, NY lOOlO Library of Congress Cataloging in Publication Data Conway, John Horton. The book of numbers / John Horton Conway, Richard K. Guy. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-4612-8488-8 e-ISBN-13: 978-1-4612-4072-3 DOl: 10.l007/978-1-4612-4072-3 1. Number theory-Popular works. I. Guy, Richard K. II. Title. QA241.C6897 1995 512'.7-dc20 95-32588 Manufactured in the United States of America. Printed on acid-free paper. 9 8 765 4 Preface he Book ofNumbers seems an obvious choice for our title, since T its undoubted success can be followed by Deuteronomy,Joshua, and so on; indeed the only risk is that there may be a demand for the earlier books in the series. More seriously, our aim is to bring to the inquisitive reader without particular mathematical background an ex­ planation of the multitudinous ways in which the word "number" is used.
    [Show full text]
  • ES.268 Dynamic Programming with Impartial Games, Course Notes 3
    ES.268 , Lecture 3 , Feb 16, 2010 http://erikdemaine.org/papers/AlgGameTheory_GONC3 Playing Games with Algorithms: { most games are hard to play well: { Chess is EXPTIME-complete: { n × n board, arbitrary position { need exponential (cn) time to find a winning move (if there is one) { also: as hard as all games (problems) that need exponential time { Checkers is EXPTIME-complete: ) Chess & Checkers are the \same" computationally: solving one solves the other (PSPACE-complete if draw after poly. moves) { Shogi (Japanese chess) is EXPTIME-complete { Japanese Go is EXPTIME-complete { U. S. Go might be harder { Othello is PSPACE-complete: { conjecture requires exponential time, but not sure (implied by P 6= NP) { can solve some games fast: in \polynomial time" (mostly 1D) Kayles: [Dudeney 1908] (n bowling pins) { move = hit one or two adjacent pins { last player to move wins (normal play) Let's play! 1 First-player win: SYMMETRY STRATEGY { move to split into two equal halves (1 pin if odd, 2 if even) { whatever opponent does, do same in other half (Kn + Kn = 0 ::: just like Nim) Impartial game, so Sprague-Grundy Theory says Kayles ≡ Nim somehow { followers(Kn) = fKi + Kn−i−1;Ki + Kn−i−2 j i = 0; 1; :::;n − 2g ) nimber(Kn) = mexfnimber(Ki + Kn−i−1); nimber(Ki + Kn−i−2) j i = 0; 1; :::;n − 2g { nimber(x + y) = nimber(x) ⊕ nimber(y) ) nimber(Kn) = mexfnimber(Ki) ⊕ nimber(Kn−i−1); nimber(Ki) ⊕ nimber(Kn−i−2) j i = 0; 1; :::n − 2g RECURRENCE! | write what you want in terms of smaller things Howe do w compute it? nimber(K0) = 0 (BASE CASE) nimber(K1) = mexfnimber(K0) ⊕ nimber(K0)g 0 ⊕ 0 = 0 = 1 nimber(K2) = mexfnimber(K0) ⊕ nimber(K1); 0 ⊕ 1 = 1 nimber(K0) ⊕ nimber(K0)g 0 ⊕ 0 = 0 = 2 so e.g.
    [Show full text]
  • Minmax Shapley Value and Biform Contests Subgame Perfect Nash Equilibrium
    1 סדרת מאמרים לדיון Working paper series No. 14 – 2011 Minmax Shapley Value and Biform Contests Subgame Perfect Nash Equilibrium Mordechai E. Schwarz 142 מערכות מידע חשבונאיות ממוחשבות 1 Minmax Shapley Value and Biform Contests Subgame Perfect Nash Equilibrium Mordechai E. Schwarz1 Abstract A biform contest is a two-stage game. In the first stage, agents exert resources to build up their position towards the second stage. The second stage is either a bargaining game or a cooperative game, but if no cooperative solution is achieved it turns to a rent-seeking contest. It is shown that with a passive regulator, the second stage is a cooperative game and if all contestants are risk-averse, the biform contest non-cooperatively implements the minmax Shapley value, as defined in the paper, in subgame perfect equilibrium. Keywords: biform game, biform contest, rent-seeking, externalities, minmax approach, minmax coalitional worth, minmax Shapley value. JEL classification: C7, D7 1 Department of Economics and Management, The Open University of Israel. Email: [email protected]. I am truly indebted to J. Atsu Amegashie, Shmuel Nitzan and Aviad Heifetz for very helpful comments on earlier versions of this work. Of course, the responsibility for every error is exclusively mine. 2 1. Introduction Cooperation among rent-seekers is usually modeled as a two-stage game. There are at least three main categories of these models. One category which I call wolf-pack games studied a two-stage group rent-seeking without a specific sharing rule of the prize among the winning group members. Contestants cooperate within group rent-seeking against rival groups in the first stage, and then the individual members of the winning group engage in another rent- seeking contest for the prize in the second stage2.
    [Show full text]
  • Comparing the Fairness of Two Popular Solution Concepts of Coalition Games: Shapley Value and Nucleolus1
    1 COMPARING THE FAIRNESS OF TWO POPULAR SOLUTION CONCEPTS OF COALITION GAMES: SHAPLEY VALUE AND NUCLEOLUS1 AUTHORS REDACTED TO PRESERVE ANONYMITY ABSTRACT An analytics model is developed to compare two popular solution concepts of coalition games, and to determine which one is fairer and more appropriate to use in real life applications of coalition games. Coalition games have found wide application in economics, finance, politics, and computing. When applications are developed using coalition game theory, designers have the option of choosing from several solutions concepts and the effectiveness of the solution depends on the accuracy of the underlying solution concepts. The algorithm presented sheds light on the effectiveness and fairness of the solution concepts and, in particular, is used to compare the fairness between Shapley Value and Nucleolus solution concepts. When imputations for coalitions are expressed in a barycentric triangle, they create a core, and larger core areas leave room for more competition among the coalitions which can create instability. Smaller core areas are desirable because they leave little room for further collaboration and come to a quick resolution. An example supports the idea of smaller core comparison, and imputations have been calculated using Shapley Value and Nucleolus expressed in a barycentric triangle for comparison. INTRODUCTION Game theory has its foundations in applied mathematics. The mathematical models and built-in consistency of game theory make it a suitable framework and basis for modeling and designing of automated bargaining and decision-making software systems in interactive negotiation. For example, a game theory framework can serve as the most efficient bidding rules for Web Services, e-commerce auction website, or tamper-proof automated negotiations for purchasing communication bandwidth.
    [Show full text]