DOCSLIB.ORG

Game Theory: Minimax, Maximin, and Iterated Removal

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Game Theory: Minimax, Maximin, and Iterated Removal Game Theory: Minimax, Maximin, and Iterated Removal Naima Hammoud March 14, 2017 Last Lecture: expected value principle Colin A B Rose A 2 -2 -3 3 B 0 0 3 -3 Rose’s expected Suppose that Rose knows Colin will play ½ A + ½ B payoff if she plays strategy A is -1/2 Rose’s Expectations for playing pure strategies E (A)=1/2 2+1/2 ( 3) = 1/2 Rose ⇥ ⇥ − − ERose(B)=1/2 0+1/2 (3) = 3/2 Rose’s expected ⇥ ⇥ payoff if she plays strategy B is 3/2 Last Lecture: expected value principle Colin A B Rose A 2 -2 -3 3 B 0 0 3 -3 Suppose that Rose knows Colin will play ½ A + ½ B Because 3/2 > -1/2 Rose’s Expectations for playing pure strategies Rose chooses to maximize her payoff ERose(A)=1/2 2+1/2 ( 3) = 1/2 by playing B. That’s ⇥ ⇥ − − of course only if ERose(B)=1/2 0+1/2 (3) = 3/2 Colin is playing ⇥ ⇥ ½ A + ½ B Last Lecture: expected value principle Colin A B Rose A 2 -2 -3 3 B 0 0 3 -3 Rule of thumb: If you know your opponent is playing a mixed strategy and will continue to play it, you should use a strategy that maximizes your expected payoff. Last Lecture • We saw that in soccer penalty-kick data collected by Ignacio Palacios- Huerta (2003) that kickers and goal-keepers seem to be playing the Nash equilibrium! But is that really the case? Goalie Left Goalie Right Kicker Left Kicker Right Nash frequency 0.42 0.58 0.38 0.62 Actual frequency 0.42 0.58 0.4 0.6 Last Lecture • We saw that in soccer penalty-kick data collected by Ignacio Palacios- Huerta (2003) that kickers and goal-keepers seem to be playing the Nash equilibrium! But is that really the case? • The player is actually trying to maximize their own gain and minimize the gain of the goal keeper • It turns out that in zero-sum games, the Nash equilibrium, maximizing your own gain, and minimizing your opponent’s gain actually coincide. Zero-sum Games zero-sum game: A zero-sum game is one in which the sum of the individual payoffs for each outcome is zero. Example: Matching pennies Colin The sum of payoffs for Heads Tails this outcome is zero, as is the sum of payoffs for every other outcome. Heads 1 -1 -1 1 Rose Tails -1 1 1 -1 Minimax, Maximin zero-sum game: A zero-sum game is one in which the sum of the individual payoffs for each outcome is zero. Minimax strategy: minimizing one’s own maximum loss Maximin strategy: maximize one’s own minimum gain Zero-sum game example Column Column player 2 player 2 2, 2 0, 0 1, 1 201 Row 4, −4 3, 3 2, −2 Row 4 32 player 1 player 1 21, −1 −2, 2 2, −23 21 −223 − − − − 4 5 4 5 Since the payoffs of the column player (shown red) are just the negative of the payoffs of the row player, we can write a matrix only showing payoffs of the row player (on the right). Once we have that, we can find the maximin & minimax. Maximin strategy for player 1: maximize their own minimum gain Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 2 − 3 C 1 22 − If player 1 plays the first strategy4 (strategy A) then their5 minimum gain is 0. Maximin strategy for player 1: maximize their own minimum gain Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 − 3 C 1 22 − − If player 1 plays4 strategy B then their minimum5 gain is -3. Maximin strategy for player 1: maximize their own minimum gain Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 − 3 C 1 22 −2 − − If player 1 plays4 strategy C then their minimum5 gain is -2. Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 − 3 C 1 22 −2 maximum loss 4 − − If player 2 plays strategy A then4 their maximum loss is 4 (their max5 loss is player 1’s max gain) Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 − 3 C 1 22 −2 maximum loss 4 −0 − If player 2 plays strategy B then4 their maximum loss is 0 (their max5 loss is player 1’s max gain) Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column player 2 A B C minimum gain A 201 0 Row player 1 B 4 32 3 2 − 3 C 1 22 −2 maximum loss 4 −0 2 − If player 2 plays strategy C then4 their maximum loss is 2 (their max5 loss is player 1’s max gain) Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column player 2 A B C minimum gain A 201 0 maximin Row player 1 B 4 32 3 2 − 3 C 1 22 −2 maximum loss 4 −0 2 − 4 minimax 5 Take the maximum of the minimum gains, i.e. the maximum of row minima (maximin), and the minimum of the maximum losses, i.e. the minimum of column maxima (minimax). If they are equal, you have a saddle point. Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column saddle point player 2 A B C minimum gain A 201 0 maximin Row player 1 B 4 32 3 2 − 3 C 1 22 −2 maximum loss 4 −0 2 − 4 minimax 5 If a saddle point exists, it should always be played. Here player 1 plays A and player 2 plays B Maximin strategy for player 1: maximize their own minimum gain Minimax strategy for player 2: minimize their own maximum loss Column saddle point player 2 A B C minimum gain A 201 0 maximin Row player 1 B 4 32 3 2 − 3 C 1 22 −2 maximum loss 4 −0 2 − 4 minimax 5 A saddle point is a Nash equilibrium More examples player 2 player 2 32100 31100 01200 01200 player 1 2 3 player 1 10210 210213 0 631227 1 maximin 631227 1 6 7 6 7 43 2 2 25minimax 43 1 2 25 None of the row minima The highlighted entry is the equals any of the column saddle point, and both maxima, so no saddle points players will play it. Dominated strategies: iterated removal Dominated strategy: There is some other strategy that does better than it. • A dominated strategy will never be played, so we can remove it from the game • We can iterate until we get to to the dominant strategy • This is called iterated removal of dominated strategies iterated removal example Column player 2 Left Center Right Up 3 0 2 1 0 0 Row player 1 Middle 1 1 1 1 5 0 Down 0 1 4 2 0 1 Column player 2 Left Center Right Up 3 0 2 1 0 0 Row player 1 Middle 1 1 1 1 5 0 Down 0 1 4 2 0 1 Column player will never play Right because it is strictly dominated by Center. The payoffs of player 2 playing Right are (0, 0, 1), which are dominated by (1, 1, 2) from playing Center. Therefore we can remove Right. Column player 2 Left Center Right Up 3 0 2 1 0 0 Row player 1 Middle 1 1 1 1 5 0 Down 0 1 4 2 0 1 Row player will never play Middle because it is strictly dominated by Up. Payoffs of Middle are (1, 1) which are dominated by (3, 2) from Up. Column player 2 Left Center Up 3 0 2 1 Row player 1 Down 0 1 4 2 The new game matrix is now smaller. Column player 2 Left Center Up 3 0 2 1 Row player 1 Down 0 1 4 2 Column player will never play Left because it is strictly dominated by Center. Payoff of (0, 1) from Left versus (1, 2) from Center. Column player 2 Center Row Up 2 1 player 1 Down 4 2 Now row player is better off playing Down than Up, because the payoff is 4 instead of 2. So (4, 2) is a unique Nash equilibrium “FAITH” — TELEVISION’S NEW HIT GAMESHOW You have observed the host to be 99.98% accurate in the last 10,000 games. If he predicted that the contestant chooses only Box #2, he rewards their faith with the million dollars. ! Do you take both boxes or only Box #2? Box 2 Box 1 $1 million $1000 or nothing 4 THE MATRIX FOR NEWCOMB’S PROBLEM HOST Predicts that you Predicts that you select both boxes select Box #2 You select both $1,000 $1,001,000 CONTESTANT You select Box #2 $0 $1,000,000 TWO ARGUMENTS Argument 1: Have faith and take Box #2 In your observations of the last 10,000 games, the host has been shown to possess 99.98% accuracy in predicting the contestants choice.
Load more

Recommended publications
  • Game Theory 2: Extensive-Form Games and Subgame Perfection
    Game Theory 2: Extensive-Form Games and Subgame Perfection 1 / 26 Dynamics in Games How should we think of strategic interactions that occur in sequence? Who moves when? And what can they do at different points in time? How do people react to different histories? 2 / 26 Modeling Games with Dynamics Players Player function I Who moves when Terminal histories I Possible paths through the game Preferences over terminal histories 3 / 26 Strategies A strategy is a complete contingent plan Player i's strategy specifies her action choice at each point at which she could be called on to make a choice 4 / 26 An Example: International Crises Two countries (A and B) are competing over a piece of land that B occupies Country A decides whether to make a demand If Country A makes a demand, B can either acquiesce or fight a war If A does not make a demand, B keeps land (game ends) A's best outcome is Demand followed by Acquiesce, worst outcome is Demand and War B's best outcome is No Demand and worst outcome is Demand and War 5 / 26 An Example: International Crises A can choose: Demand (D) or No Demand (ND) B can choose: Fight a war (W ) or Acquiesce (A) Preferences uA(D; A) = 3 > uA(ND; A) = uA(ND; W ) = 2 > uA(D; W ) = 1 uB(ND; A) = uB(ND; W ) = 3 > uB(D; A) = 2 > uB(D; W ) = 1 How can we represent this scenario as a game (in strategic form)? 6 / 26 International Crisis Game: NE Country B WA D 1; 1 3X; 2X Country A ND 2X; 3X 2; 3X I Is there something funny here? I Is there something funny here? I Specifically, (ND; W )? I Is there something funny here?
    [Show full text]
  • Lecture 4 Rationalizability & Nash Equilibrium Road
    Lecture 4 Rationalizability & Nash Equilibrium 14.12 Game Theory Muhamet Yildiz Road Map 1. Strategies – completed 2. Quiz 3. Dominance 4. Dominant-strategy equilibrium 5. Rationalizability 6. Nash Equilibrium 1 Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy). Matching pennies with perfect information 2’s Strategies: HH = Head if 1 plays Head, 1 Head if 1 plays Tail; HT = Head if 1 plays Head, Head Tail Tail if 1 plays Tail; 2 TH = Tail if 1 plays Head, 2 Head if 1 plays Tail; head tail head tail TT = Tail if 1 plays Head, Tail if 1 plays Tail. (-1,1) (1,-1) (1,-1) (-1,1) 2 Matching pennies with perfect information 2 1 HH HT TH TT Head Tail Matching pennies with Imperfect information 1 2 1 Head Tail Head Tail 2 Head (-1,1) (1,-1) head tail head tail Tail (1,-1) (-1,1) (-1,1) (1,-1) (1,-1) (-1,1) 3 A game with nature Left (5, 0) 1 Head 1/2 Right (2, 2) Nature (3, 3) 1/2 Left Tail 2 Right (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: Si = {si1,si2,…,sik} σ → A mixed strategy: i: S [0,1] s.t. σ σ σ i(si1) + i(si2) + … + i(sik) = 1. If the other players play s-i =(s1,…, si-1,si+1,…,sn), then σ the expected utility of playing i is σ σ σ i(si1)ui(si1,s-i) + i(si2)ui(si2,s-i) + … + i(sik)ui(sik,s-i).
    [Show full text]
  • Algorithms and Collusion - Background Note by the Secretariat
    Unclassified DAF/COMP(2017)4 Organisation for Economic Co-operation and Development 9 June 2017 English - Or. English DIRECTORATE FOR FINANCIAL AND ENTERPRISE AFFAIRS COMPETITION COMMITTEE Cancels & replaces the same document of 17 May 2017. Algorithms and Collusion - Background Note by the Secretariat 21-23 June 2017 This document was prepared by the OECD Secretariat to serve as a background note for Item 10 at the 127th Meeting of the Competition Committee on 21-23 June 2017. The opinions expressed and arguments employed herein do not necessarily reflect the official views of the Organisation or of the governments of its member countries. More documentation related to this discussion can be found at http://www.oecd.org/daf/competition/algorithms-and-collusion.htm Please contact Mr. Antonio Capobianco if you have any questions about this document [E-mail: [email protected]] JT03415748 This document, as well as any data and map included herein, are without prejudice to the status of or sovereignty over any territory, to the delimitation of international frontiers and boundaries and to the name of any territory, city or area. 2 │ DAF/COMP(2017)4 Algorithms and Collusion Background note by the Secretariat Abstract The combination of big data with technologically advanced tools, such as pricing algorithms, is increasingly diffused in today everyone’s life, and it is changing the competitive landscape in which many companies operate and the way in which they make commercial and strategic decisions. While the size of this phenomenon is to a large extent unknown, there are a growing number of firms using computer algorithms to improve their pricing models, customise services and predict market trends.
    [Show full text]
  • Game Theory- Prisoners Dilemma Vs Battle of the Sexes EXCERPTS
    Lesson 14. Game Theory 1 Lesson 14 Game Theory c 2010, 2011 ⃝ Roberto Serrano and Allan M. Feldman All rights reserved Version C 1. Introduction In the last lesson we discussed duopoly markets in which two firms compete to sell a product. In such markets, the firms behave strategically; each firm must think about what the other firm is doing in order to decide what it should do itself. The theory of duopoly was originally developed in the 19th century, but it led to the theory of games in the 20th century. The first major book in game theory, published in 1944, was Theory of Games and Economic Behavior,byJohnvon Neumann (1903-1957) and Oskar Morgenstern (1902-1977). We will return to the contributions of Von Neumann and Morgenstern in Lesson 19, on uncertainty and expected utility. Agroupofpeople(orteams,firms,armies,countries)areinagame if their decision problems are interdependent, in the sense that the actions that all of them take influence the outcomes for everyone. Game theory is the study of games; it can also be called interactive decision theory. Many real-life interactions can be viewed as games. Obviously football, soccer, and baseball games are games.Butsoaretheinteractionsofduopolists,thepoliticalcampaignsbetweenparties before an election, and the interactions of armed forces and countries. Even some interactions between animal or plant species in nature can be modeled as games. In fact, game theory has been used in many different fields in recent decades, including economics, political science, psychology, sociology, computer science, and biology. This brief lesson is not meant to replace a formal course in game theory; it is only an in- troduction.
    [Show full text]
  • Lecture Notes
    GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies ..................................
    [Show full text]
  • A New Approach to Stable Matching Problems
    August1 989 Report No. STAN-CS-89- 1275 A New Approach to Stable Matching Problems bY Ashok Subramanian Department of Computer Science Stanford University Stanford, California 94305 3 DISTRIBUTION /AVAILABILITY OF REPORT Unrestricted: Distribution Unlimited GANIZATION 1 1 TITLE (Include Securrty Clamfrcat!on) A New Approach to Stable Matching Problems 12 PERSONAL AUTHOR(S) Ashok Subramanian 13a TYPE OF REPORT 13b TtME COVERED 14 DATE OF REPORT (Year, Month, Day) 15 PAGE COUNT FROM TO August 1989 34 16 SUPPLEMENTARY NOTATION 17 COSATI CODES 18 SUBJECT TERMS (Contrnue on reverse If necessary and jdentrfy by block number) FIELD GROUP SUB-GROUP 19 ABSTRACT (Continue on reverse if necessary and identrfy by block number) Abstract. We show that Stable Matching problems are the same as problems about stable config- urations of X-networks. Consequences include easy proofs of old theorems, a new simple algorithm for finding a stable matching, an understanding of the difference between Stable Marriage and Stable Roommates, NTcompleteness of Three-party Stable Marriage, CC-completeness of several Stable Matching problems, and a fast parallel reduction from the Stable Marriage problem to the ’ Assignment problem. 20 DISTRIBUTION /AVAILABILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION q UNCLASSIFIED/UNLIMITED 0 SAME AS Rf’T 0 DTIC USERS 22a NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE (Include Area Code) 22c OFFICE SYMBOL Ernst Mavr DD Form 1473, JUN 86 Prevrous edrtions are obsolete SECURITY CLASSIFICATION OF TYS PAGt . 1 , S/N 0102-LF-014-6603 \ \ ““*5 - - A New Approach to Stable Matching Problems * Ashok Subramanian Department of Computer Science St anford University Stanford, CA 94305-2140 Abstract.
    [Show full text]
  • More Experiments on Game Theory
    More Experiments on Game Theory Syngjoo Choi Spring 2010 Experimental Economics (ECON3020) Game theory 2 Spring2010 1/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Playing Unpredictably In many situations there is a strategic advantage associated with being unpredictable. Experimental Economics (ECON3020) Game theory 2 Spring2010 2/28 Mixed (or Randomized) Strategy In the penalty kick, Left (L) and Right (R) are the pure strategies of the kicker and the goalkeeper. A mixed strategy refers to a probabilistic mixture over pure strategies in which no single pure strategy is played all the time. e.g., kicking left with half of the time and right with the other half of the time. A penalty kick in a soccer game is one example of games of pure con‡icit, essentially called zero-sum games in which one player’s winning is the other’sloss. Experimental Economics (ECON3020) Game theory 2 Spring2010 3/28 The use of pure strategies will result in a loss. What about each player playing heads half of the time? Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails. One player takes both coins if the pennies match; otherwise, the other takes both coins. Left Right Top 1, 1, 1 1 Bottom 1, 1, 1 1 Does there exist a Nash equilibrium involving the use of pure strategies? Experimental Economics (ECON3020) Game theory 2 Spring2010 4/28 Matching Pennies Games In a matching pennies game, each player uncovers a penny showing either heads or tails.
    [Show full text]
  • Prisoners of Reason Game Theory and Neoliberal Political Economy
    C:/ITOOLS/WMS/CUP-NEW/6549131/WORKINGFOLDER/AMADAE/9781107064034PRE.3D iii [1–28] 11.8.2015 9:57PM Prisoners of Reason Game Theory and Neoliberal Political Economy S. M. AMADAE Massachusetts Institute of Technology C:/ITOOLS/WMS/CUP-NEW/6549131/WORKINGFOLDER/AMADAE/9781107064034PRE.3D iv [1–28] 11.8.2015 9:57PM 32 Avenue of the Americas, New York, ny 10013-2473, usa Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107671195 © S. M. Amadae 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Amadae, S. M., author. Prisoners of reason : game theory and neoliberal political economy / S.M. Amadae. pages cm Includes bibliographical references and index. isbn 978-1-107-06403-4 (hbk. : alk. paper) – isbn 978-1-107-67119-5 (pbk. : alk. paper) 1. Game theory – Political aspects. 2. International relations. 3. Neoliberalism. 4. Social choice – Political aspects. 5. Political science – Philosophy. I. Title. hb144.a43 2015 320.01′5193 – dc23 2015020954 isbn 978-1-107-06403-4 Hardback isbn 978-1-107-67119-5 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
    [Show full text]
  • The Three Types of Collusion: Fixing Prices, Rivals, and Rules Robert H
    University of Baltimore Law ScholarWorks@University of Baltimore School of Law All Faculty Scholarship Faculty Scholarship 2000 The Three Types of Collusion: Fixing Prices, Rivals, and Rules Robert H. Lande University of Baltimore School of Law, [email protected] Howard P. Marvel Ohio State University, [email protected] Follow this and additional works at: http://scholarworks.law.ubalt.edu/all_fac Part of the Antitrust and Trade Regulation Commons, and the Law and Economics Commons Recommended Citation The Three Types of Collusion: Fixing Prices, Rivals, and Rules, 2000 Wis. L. Rev. 941 (2000) This Article is brought to you for free and open access by the Faculty Scholarship at ScholarWorks@University of Baltimore School of Law. It has been accepted for inclusion in All Faculty Scholarship by an authorized administrator of ScholarWorks@University of Baltimore School of Law. For more information, please contact [email protected]. ARTICLES THE THREE TYPES OF COLLUSION: FIXING PRICES, RIVALS, AND RULES ROBERTH. LANDE * & HOWARDP. MARVEL** Antitrust law has long held collusion to be paramount among the offenses that it is charged with prohibiting. The reason for this prohibition is simple----collusion typically leads to monopoly-like outcomes, including monopoly profits that are shared by the colluding parties. Most collusion cases can be classified into two established general categories.) Classic, or "Type I" collusion involves collective action to raise price directly? Firms can also collude to disadvantage rivals in a manner that causes the rivals' output to diminish or causes their behavior to become chastened. This "Type 11" collusion in turn allows the colluding firms to raise prices.3 Many important collusion cases, however, do not fit into either of these categories.
    [Show full text]
  • Kranton Duke University
    The Devil is in the Details – Implications of Samuel Bowles’ The Moral Economy for economics and policy research October 13 2017 Rachel Kranton Duke University The Moral Economy by Samuel Bowles should be required reading by all graduate students in economics. Indeed, all economists should buy a copy and read it. The book is a stunning, critical discussion of the interplay between economic incentives and preferences. It challenges basic premises of economic theory and questions policy recommendations based on these theories. The book proposes the path forward: designing policy that combines incentives and moral appeals. And, therefore, like such as book should, The Moral Economy leaves us with much work to do. The Moral Economy concerns individual choices and economic policy, particularly microeconomic policies with goals to enhance the collective good. The book takes aim at laws, policies, and business practices that are based on the classic Homo economicus model of individual choice. The book first argues in great detail that policies that follow from the Homo economicus paradigm can backfire. While most economists would now recognize that people are not purely selfish and self-interested, The Moral Economy goes one step further. Incentives can amplify the selfishness of individuals. People might act in more self-interested ways in a system based on incentives and rewards than they would in the absence of such inducements. The Moral Economy warns economists to be especially wary of incentives because social norms, like norms of trust and honesty, are critical to economic activity. The danger is not only of incentives backfiring in a single instance; monetary incentives can generally erode ethical and moral codes and social motivations people can have towards each other.
    [Show full text]
  • Collusion Constrained Equilibrium
    Theoretical Economics 13 (2018), 307–340 1555-7561/20180307 Collusion constrained equilibrium Rohan Dutta Department of Economics, McGill University David K. Levine Department of Economics, European University Institute and Department of Economics, Washington University in Saint Louis Salvatore Modica Department of Economics, Università di Palermo We study collusion within groups in noncooperative games. The primitives are the preferences of the players, their assignment to nonoverlapping groups, and the goals of the groups. Our notion of collusion is that a group coordinates the play of its members among different incentive compatible plans to best achieve its goals. Unfortunately, equilibria that meet this requirement need not exist. We instead introduce the weaker notion of collusion constrained equilibrium. This al- lows groups to put positive probability on alternatives that are suboptimal for the group in certain razor’s edge cases where the set of incentive compatible plans changes discontinuously. These collusion constrained equilibria exist and are a subset of the correlated equilibria of the underlying game. We examine four per- turbations of the underlying game. In each case,we show that equilibria in which groups choose the best alternative exist and that limits of these equilibria lead to collusion constrained equilibria. We also show that for a sufficiently broad class of perturbations, every collusion constrained equilibrium arises as such a limit. We give an application to a voter participation game that shows how collusion constraints may be socially costly. Keywords. Collusion, organization, group. JEL classification. C72, D70. 1. Introduction As the literature on collective action (for example, Olson 1965) emphasizes, groups often behave collusively while the preferences of individual group members limit the possi- Rohan Dutta: [email protected] David K.
    [Show full text]
  • Alpha-Beta Pruning
    Alpha-Beta Pruning Carl Felstiner May 9, 2019 Abstract This paper serves as an introduction to the ways computers are built to play games. We implement the basic minimax algorithm and expand on it by finding ways to reduce the portion of the game tree that must be generated to find the best move. We tested our algorithms on ordinary Tic-Tac-Toe, Hex, and 3-D Tic-Tac-Toe. With our algorithms, we were able to find the best opening move in Tic-Tac-Toe by only generating 0.34% of the nodes in the game tree. We also explored some mathematical features of Hex and provided proofs of them. 1 Introduction Building computers to play board games has been a focus for mathematicians ever since computers were invented. The first computer to beat a human opponent in chess was built in 1956, and towards the late 1960s, computers were already beating chess players of low-medium skill.[1] Now, it is generally recognized that computers can beat even the most accomplished grandmaster. Computers build a tree of different possibilities for the game and then work backwards to find the move that will give the computer the best outcome. Al- though computers can evaluate board positions very quickly, in a game like chess where there are over 10120 possible board configurations it is impossible for a computer to search through the entire tree. The challenge for the computer is then to find ways to avoid searching in certain parts of the tree. Humans also look ahead a certain number of moves when playing a game, but experienced players already know of certain theories and strategies that tell them which parts of the tree to look.
    [Show full text]