0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text)
|
|
- Kathlyn Tate
- 7 years ago
- Views:
Transcription
1 September 2 Exercises: Problem 2 (p. 21) Efficiency: p : 1, 4, 5, Pareto Efficiency (Sec. 4, Ch. 1 of text) We discuss here a notion of efficiency that is rooted in the individual preferences of group members over choices that may be made for the group. The word "efficiency" has other meanings in economics, such as "cost minimization" or "optimization." In the above example, for instance, it is efficient for the government to assign the task of reducing emissions to the firm that can do so at the lowest cost. Example 4 Suppose there are 3 people (1,2,3) and 4 possible choices of a restaurant as a choice for dinner for the group of three people: I (Italian), C (Chinese), S (seafood), and T (Thai). The three people rank the 4 restaurants as follows: 1 I C S T 2 I S=T C Here, " " indicates strictly prefers while "=" indicates that the individual is indifferent between the two alternatives. Q: What are the efficient choices of restaurants? The point of answering this question is to illustrate what economists typically mean by "efficient." Definition. Suppose a group of people can choose from a set of alternatives. Alternative is efficient if it is not possible to switch to some other alternative and in the process make some people in the group strictly better off and without making anyone strictly worse off. An alternative with this property is often referred to as Pareto efficient or Pareto optimal, after the 19th century Italian economist Vilfredo Pareto, who came up with the idea. I personally prefer this terminology, which is more precise than simply "efficient". As noted above, "efficiency" has other meanings in economics. If an alternative exists such that is ranked at least as good as by every person, and at least one person strictly prefers to,then is said to Pareto dominate. 1. Is I efficient? Yes, because switching to either of C, S, or T would make person 1 (and person 2) strictly worse off. 2. Is C efficient? Looking at person 1 s preferences, we can see that the only possible improvement from his perspective is switching from C to I. Switching to I from C, however, makes person 3 strictly worse off. It is therefore not possible to switch from C to some other alternative and make some people strictly better off without hurting some other person. Yes, C is efficient. 3. Is S efficient? Looking at person 1 s preferences, we can see that the only possible moves that would not hurt him are I and C. Moving from S to I would make persons 1 8
2 and 2 strictly better off and it would not hurt person 3. No, S is not efficient. 4. Is T efficient? No, because every person is made strictly better off by switching from T to I. Incidentally, it is also also true that switching from T to S shows that T is not efficient. We conclude that I and C are efficient but that S and T are not. What does this mean? First, it would be really dumb for the group to go out together to either the seafood (S) or the Thai (T) restaurants. Second, efficiency does not help the group to choose between I and C. You might be tempted to say, "But 1 and 2 rate I as their best choice, and 3 rates I as second best; 2 in fact ranks C as his worst choice!" But this argument presumes either (i) a method by which the choice is made such as majority rule, or (ii) the assumption that each individual s well-being counts equally. As to (ii), it bears emphasizing that this notion of efficiency respects the preferences of each individual. There is no weighing of one person s interests against anothers. Efficiency rules out the stupid options but leaves the problem of weighing one person s interests against another s in selecting among the efficient choices. Notice also that I and C are each rated as strictly better than every other choice by at least one individual. This is a sufficient condition for an alternative to be efficient in the above sense: if some individual ranks some alternative as strictly better than every other alternative, then it is clearly impossible to switch to some other alternative without hurting that individual. Consequently, it must be efficient. Example 5 Can an alternative be efficient even if every individual has some other alternative that he thinks is strictly better? Yes, though not in the above example. Consider 1 C I S T 2 S=T I C This changes the above table by lowering I in the rankings of persons 1 and 2, switching it with the next best alternative(s) for those individuals. Notice that I remains efficient: looking at person 1 s ranking, the only possible improvement would be to switch from I to C, but that would make person 2 worse off. Since 3 ranks C as strictly best, it is also efficient. Consider S. Looking at 2 s ranking, we see that a switch to T is the only possible way to not hurt him. But this would strictly hurt both persons 1 and 2, and so S is also efficient. Finally, what about T? A switch from T to S doesn t hurt person 2, but it makes persons 1 and 3 strictly better off. Therefore, T is not efficient in this second table. This makes an important point: if some individual ranks an alternative as best but also equivalent to some other alternative,then is not necessarily efficient. 9
3 Efficiency with Numerical Representation of Preferences (Utility) Example 6 We will typically assume that each individual assigns a number to each alternative so that his ranking of two alternatives simply reflects the relative size of the two numbers. This assumes that each agent has a utility function over the set of alternatives. Recall the first example from above: 1 I C S T 2 I S=T C There are many utility functions that represent each of these person s preferences. Here s one set: person utility values 1 1 ( ) =5 1 ( ) =4 1 ( ) =3 1 ( )=2 2 2 ( ) = ( ) =0 2 ( ) = ( ) = ( ) = 10 3 ( ) = 5 3 ( ) = 10 3 ( )= 50 Q: Where did these numbers come from? A: I just made them up. The ordering of the utility numbers for each individual is the same as the ordering of restaurants in the above table. This is the point of representing preferences with a utility function. Q: Are there other possible representations of these same preferences? A: Of course. Q: What do the numbers mean? A: In general, nothing. We don t necessarily interpret utility as measuring something. No units are specified. In particular, the fact that 2 ( ) = 200 and 1 ( ) =5does not mean that person 2 likes Italian food 40 times as much as person 1! Utility numbers do not necessarily measure anything, though in some examples they do. Q: What s the point of numerically representing preferences with utility functions? A: There isn t much point to it in this example. More generally, however, utility representations can allow you to use the mathematics of functions to analyze problems. Utility therefore facilitates the use of mathematics to analyze economic problems. Assume we have some utility representation of each individual s preferences. An alternative is Pareto efficient if it solves the maximization problem ( ) max Here, denotes the number of individuals in the group and ( ) denotes the utility function of individual. An alternative can, however, be efficient even it fails to maximize this sum. This is a second sufficient condition for Pareto efficiency (the first, discussed above, is that some person rank the alternative as strictly better than any other alternative). 10
4 Proof. Suppose ( ) ( ) for all (1) We prove the result by contradiction. Assume that is not Pareto efficient. This means that there exist some such that Pareto dominates,i.e., ( ) ( ) for each person, with the inequality strict for at least one. Therefore, ( ) ( ) which contradicts (1). Exercise. There are 4 people (1, 2, 3, and4) and 3 alternatives (,, and ). The people rank the alternatives as follows: 1 a b c 2 b a c 3 a=c b 4 c=b a What are the Pareto efficient alternatives? Chapter 1, Section 5 Exercises: Nash equilibrium and dominant strategy equilibrium: p. 44: 1, 3, 4 Chapter 1, Section 5 of Campbell s book is an introduction to noncooperative game theory. "Noncooperative" here means that each individual is assumed to care only about his own well-being, i.e., everyone is selfish. A game requires the specification of the players, the actions available to each player, and the order in which actions are taken. We might consider this to be the rules of the game. Games are interesting when some outcome affects the well-being of the different agents (i.e., what any player receives depends upon the actions of all players). solution concept: A story or theory of what happens when the game is played a notion of equilibrium. Solution concepts typically depend upon what the players know and different senses in which an action can be interpreted as in the best interest of a player (e.g., does the optimality of an agent s strategy depend upon some specification of strategies for the other agents?) game theory: precise language of incentives. Today we ll discuss two solution concepts: dominant strategy equilibrium and Nash equilibrium Dominant and Dominated Strategies Example 7 Prisoner s Dilemma 11
5 The choice C is a dominant strategy for each of the two players. The prisoner s dilemma is a classic example because it illustrates the conflict between the best outcome for a group (here, 2 2) as compared to the outcome that results when every person pursues his own self-interest ( 5 5). It has been used to model arms races between nations. Definition of a dominant strategy: 0 is a dominant strategy iff for and, ( 0 ) ( ) A player can have more than one dominant strategy with this definition, though any two of his dominant strategies must be payoff-equivalent in the sense of providing him with exactly the same utility for all choices of his opponents strategies: if 0 and 00 are both dominant strategies, then ( 0 )= ( 00 ) for all. strictly dominate, (weakly) dominate, (weakly) dominated, strictly dominant, strictly dominated. Recall the VCG mechanism that was discussed last week. Honest reporting is a dominant strategy for each firm, i.e., the honest report maximizes a firm s profit or payoff given the reports of the other firms, regardless of the specification of those reports by the other firms Nash Equilibrium An -tuple ( 1 ) of pure strategies is a pure strategy Nash equilibrium if, for each player, ( ) ( 0 ) for all other pure strategies 0 of player. This idea is due to John Nash (1951). Notice that a dominant strategy equilibrium is necessarily also a Nash equilibrium. Example 8 MeetinginNY: Example 9 Battle of the Sexes
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma
More informationECON 40050 Game Theory Exam 1 - Answer Key. 4) All exams must be turned in by 1:45 pm. No extensions will be granted.
1 ECON 40050 Game Theory Exam 1 - Answer Key Instructions: 1) You may use a pen or pencil, a hand-held nonprogrammable calculator, and a ruler. No other materials may be at or near your desk. Books, coats,
More informationI d Rather Stay Stupid: The Advantage of Having Low Utility
I d Rather Stay Stupid: The Advantage of Having Low Utility Lior Seeman Department of Computer Science Cornell University lseeman@cs.cornell.edu Abstract Motivated by cost of computation in game theory,
More information6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games
6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses
More informationChapter 7. Sealed-bid Auctions
Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationLecture 28 Economics 181 International Trade
Lecture 28 Economics 181 International Trade I. Introduction to Strategic Trade Policy If much of world trade is in differentiated products (ie manufactures) characterized by increasing returns to scale,
More information10 Evolutionarily Stable Strategies
10 Evolutionarily Stable Strategies There is but a step between the sublime and the ridiculous. Leo Tolstoy In 1973 the biologist John Maynard Smith and the mathematician G. R. Price wrote an article in
More informationECON 202: Principles of Microeconomics. Chapter 13 Oligopoly
ECON 202: Principles of Microeconomics Chapter 13 Oligopoly Oligopoly 1. Oligopoly and Barriers to Entry. 2. Using Game Theory to Analyze Oligopoly. 3. Sequential Games and Business Strategy. 4. The Five
More informationECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY 100-3 90-99 21 80-89 14 70-79 4 0-69 11
The distribution of grades was as follows. ECO 199 B GAMES OF STRATEGY Spring Term 2004 PROBLEM SET 4 B DRAFT ANSWER KEY Range Numbers 100-3 90-99 21 80-89 14 70-79 4 0-69 11 Question 1: 30 points Games
More informationLecture V: Mixed Strategies
Lecture V: Mixed Strategies Markus M. Möbius February 26, 2008 Osborne, chapter 4 Gibbons, sections 1.3-1.3.A 1 The Advantage of Mixed Strategies Consider the following Rock-Paper-Scissors game: Note that
More informationAN ANALYSIS OF A WAR-LIKE CARD GAME. Introduction
AN ANALYSIS OF A WAR-LIKE CARD GAME BORIS ALEXEEV AND JACOB TSIMERMAN Abstract. In his book Mathematical Mind-Benders, Peter Winkler poses the following open problem, originally due to the first author:
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 Principal-Agent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationOligopoly: Firms in Less Competitive Markets
Chapter 13 Oligopoly: Firms in Less Competitive Markets Prepared by: Fernando & Yvonn Quijano 2008 Prentice Hall Business Publishing Economics R. Glenn Hubbard, Anthony Patrick O Brien, 2e. Competing with
More informationGames Manipulators Play
Games Manipulators Play Umberto Grandi Department of Mathematics University of Padova 23 January 2014 [Joint work with Edith Elkind, Francesca Rossi and Arkadii Slinko] Gibbard-Satterthwaite Theorem All
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationCapital Structure. Itay Goldstein. Wharton School, University of Pennsylvania
Capital Structure Itay Goldstein Wharton School, University of Pennsylvania 1 Debt and Equity There are two main types of financing: debt and equity. Consider a two-period world with dates 0 and 1. At
More informationExtreme cases. In between cases
CHAPTER 16 OLIGOPOLY FOUR TYPES OF MARKET STRUCTURE Extreme cases PERFECTLY COMPETITION Many firms No barriers to entry Identical products MONOPOLY One firm Huge barriers to entry Unique product In between
More informationComputational Learning Theory Spring Semester, 2003/4. Lecture 1: March 2
Computational Learning Theory Spring Semester, 2003/4 Lecture 1: March 2 Lecturer: Yishay Mansour Scribe: Gur Yaari, Idan Szpektor 1.1 Introduction Several fields in computer science and economics are
More informationChapter 14. Oligopoly
Chapter 14. Oligopoly Instructor: JINKOOK LEE Department of Economics / Texas A&M University ECON 202 504 Principles of Microeconomics Oligopoly Market Oligopoly: A market structure in which a small number
More informationGames of Incomplete Information
Games of Incomplete Information Jonathan Levin February 00 Introduction We now start to explore models of incomplete information. Informally, a game of incomplete information is a game where the players
More informationSummary of Doctoral Dissertation: Voluntary Participation Games in Public Good Mechanisms: Coalitional Deviations and Efficiency
Summary of Doctoral Dissertation: Voluntary Participation Games in Public Good Mechanisms: Coalitional Deviations and Efficiency Ryusuke Shinohara 1. Motivation The purpose of this dissertation is to examine
More informationExample 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).
Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.
More informationSequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium
Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses
More informationOptimal Auctions Continued
Lecture 6 Optimal Auctions Continued 1 Recap Last week, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] residual valuation of t 0 for
More informationBiform Games: Additional Online Material
Biform Games: Additional Online Material Adam Brandenburger Harborne Stuart July 2006 These appendices supplement Brandenburger and Stuart [1, 2006], [2, 2006] ( Biform Games ). Appendix C uses the efficiency
More informationOligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry s output.
Topic 8 Chapter 13 Oligopoly and Monopolistic Competition Econ 203 Topic 8 page 1 Oligopoly: How do firms behave when there are only a few competitors? These firms produce all or most of their industry
More informationZero-knowledge games. Christmas Lectures 2008
Security is very important on the internet. You often need to prove to another person that you know something but without letting them know what the information actually is (because they could just copy
More informationThéorie de la décision et théorie des jeux Stefano Moretti
héorie de la décision et théorie des jeux Stefano Moretti UMR 7243 CNRS Laboratoire d'analyse et Modélisation de Systèmes pour l'aide à la décision (LAMSADE) Université Paris-Dauphine email: Stefano.MOREI@dauphine.fr
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,
More informationPascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.
Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive
More informationMinimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example
Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationThe Basics of Game Theory
Sloan School of Management 15.010/15.011 Massachusetts Institute of Technology RECITATION NOTES #7 The Basics of Game Theory Friday - November 5, 2004 OUTLINE OF TODAY S RECITATION 1. Game theory definitions:
More informationCompetition and Regulation. Lecture 2: Background on imperfect competition
Competition and Regulation Lecture 2: Background on imperfect competition Monopoly A monopolist maximizes its profits, choosing simultaneously quantity and prices, taking the Demand as a contraint; The
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationPrice competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]
ECON9 (Spring 0) & 350 (Tutorial ) Chapter Monopolistic Competition and Oligopoly (Part ) Price competition with homogenous products: The Bertrand duopoly model [Simultaneous move price setting duopoly]
More information6.1 What is a Game? 156 CHAPTER 6. GAMES
From the book Networks, Crowds, and Markets: Reasoning about a Highly Connected World. By David Easley and Jon Kleinberg. Cambridge University Press, 2010. Complete preprint on-line at http://www.cs.cornell.edu/home/kleinber/networks-book/
More informationThe fundamental question in economics is 2. Consumer Preferences
A Theory of Consumer Behavior Preliminaries 1. Introduction The fundamental question in economics is 2. Consumer Preferences Given limited resources, how are goods and service allocated? 1 3. Indifference
More informationGame Theory and Nash Equilibrium
Game Theory and Nash Equilibrium by Jenny Duffy A project submitted to the Department of Mathematical Sciences in conformity with the requirements for Math 4301 (Honours Seminar) Lakehead University Thunder
More informationGames of Incomplete Information
Games of Incomplete Information Yan Chen November 16, 2005 Games of Incomplete Information Auction Experiments Auction Theory Administrative stuff Games of Incomplete Information Several basic concepts:
More informationGame Theory in Wireless Networks: A Tutorial
1 Game heory in Wireless Networks: A utorial Mark Felegyhazi, Jean-Pierre Hubaux EPFL Switzerland email: {mark.felegyhazi, jean-pierre.hubaux}@epfl.ch EPFL echnical report: LCA-REPOR-2006-002, submitted
More informationWeek 7 - Game Theory and Industrial Organisation
Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationINCIDENCE-BETWEENNESS GEOMETRY
INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationchapter >> Consumer and Producer Surplus Section 3: Consumer Surplus, Producer Surplus, and the Gains from Trade
chapter 6 >> Consumer and Producer Surplus Section 3: Consumer Surplus, Producer Surplus, and the Gains from Trade One of the nine core principles of economics we introduced in Chapter 1 is that markets
More informationeach college c i C has a capacity q i - the maximum number of students it will admit
n colleges in a set C, m applicants in a set A, where m is much larger than n. each college c i C has a capacity q i - the maximum number of students it will admit each college c i has a strict order i
More informationGame Theory 1. Introduction
Game Theory 1. Introduction Dmitry Potapov CERN What is Game Theory? Game theory is about interactions among agents that are self-interested I ll use agent and player synonymously Self-interested: Each
More information6.254 : Game Theory with Engineering Applications Lecture 1: Introduction
6.254 : Game Theory with Engineering Applications Lecture 1: Introduction Asu Ozdaglar MIT February 2, 2010 1 Introduction Optimization Theory: Optimize a single objective over a decision variable x R
More informationUnderstanding Options: Calls and Puts
2 Understanding Options: Calls and Puts Important: in their simplest forms, options trades sound like, and are, very high risk investments. If reading about options makes you think they are too risky for
More informationAn Introduction to Sponsored Search Advertising
An Introduction to Sponsored Search Advertising Susan Athey Market Design Prepared in collaboration with Jonathan Levin (Stanford) Sponsored Search Auctions Google revenue in 2008: $21,795,550,000. Hal
More informationWorking Paper Series ISSN 1424-0459. Working Paper No. 238. Stability for Coalition Formation Games. Salvador Barberà and Anke Gerber
Institute for Empirical Research in Economics University of Zurich Working Paper Series ISSN 1424-0459 Working Paper No. 238 A Note on the Impossibility of a Satisfactory Concept of Stability for Coalition
More informationChristmas Gift Exchange Games
Christmas Gift Exchange Games Arpita Ghosh 1 and Mohammad Mahdian 1 Yahoo! Research Santa Clara, CA, USA. Email: arpita@yahoo-inc.com, mahdian@alum.mit.edu Abstract. The Christmas gift exchange is a popular
More informationMicroeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012. (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3
Microeconomic Theory Jamison / Kohlberg / Avery Problem Set 4 Solutions Spring 2012 1. Subgame Perfect Equilibrium and Dominance (a) LEFT CENTER RIGHT TOP 8, 5 0, 0 6, 3 BOTTOM 0, 0 7, 6 6, 3 Highlighting
More informationWhen other firms see these potential profits they will enter the industry, causing a downward shift in the demand for a given firm s product.
Characteristics of Monopolistic Competition large number of firms differentiated products (ie. substitutes) freedom of entry and exit Examples Upholstered furniture: firms; HHI* = 395 Jewelry and Silverware:
More informationHow to Solve Strategic Games? Dominant Strategies
How to Solve Strategic Games? There are three main concepts to solve strategic games: 1. Dominant Strategies & Dominant Strategy Equilibrium 2. Dominated Strategies & Iterative Elimination of Dominated
More informationNash and game theory
Nash and game theory Antonio Cabrales 1 I am asked to give my view on the contribution of John Nash to the development of game theory. Since I have received most of my early influence through textbooks,
More informationCommon Knowledge: Formalizing the Social Applications
Common Knowledge: Formalizing the Social Applications 1 Today and Thursday we ll take a step in the direction of formalizing the social puzzles, such as omission commission. 2 First, a reminder of the
More informationOligopoly and Strategic Pricing
R.E.Marks 1998 Oligopoly 1 R.E.Marks 1998 Oligopoly Oligopoly and Strategic Pricing In this section we consider how firms compete when there are few sellers an oligopolistic market (from the Greek). Small
More informationChapter 16 Oligopoly. 16.1 What Is Oligopoly? 1) Describe the characteristics of an oligopoly.
Chapter 16 Oligopoly 16.1 What Is Oligopoly? 1) Describe the characteristics of an oligopoly. Answer: There are a small number of firms that act interdependently. They are tempted to form a cartel and
More informationStrengthening International Courts and the Early Settlement of Disputes
Strengthening International Courts and the Early Settlement of Disputes Michael Gilligan, Leslie Johns, and B. Peter Rosendorff November 18, 2008 Technical Appendix Definitions σ(ŝ) {π [0, 1] s(π) = ŝ}
More informationSimon Fraser University Spring 2015. Econ 302 D200 Final Exam Solution Instructor: Songzi Du Tuesday April 21, 2015, 12 3 PM
Simon Fraser University Spring 2015 Econ 302 D200 Final Exam Solution Instructor: Songzi Du Tuesday April 21, 2015, 12 3 PM The brief solutions suggested here may not have the complete explanations necessary
More informationTwo-Sided Matching Theory
Two-Sided Matching Theory M. Utku Ünver Boston College Introduction to the Theory of Two-Sided Matching To see which results are robust, we will look at some increasingly general models. Even before we
More informationCournot s model of oligopoly
Cournot s model of oligopoly Single good produced by n firms Cost to firm i of producing q i units: C i (q i ), where C i is nonnegative and increasing If firms total output is Q then market price is P(Q),
More informationKant s deontological ethics
Michael Lacewing Kant s deontological ethics DEONTOLOGY Deontologists believe that morality is a matter of duty. We have moral duties to do things which it is right to do and moral duties not to do things
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationOligopoly markets: The price or quantity decisions by one rm has to directly in uence pro ts by other rms if rms are competing for customers.
15 Game Theory Varian: Chapters 8-9. The key novelty compared to the competitive (Walrasian) equilibrium analysis is that game theoretic analysis allows for the possibility that utility/pro t/payo s depend
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos Hervés-Beloso, Emma Moreno- García and
More informationStupid Divisibility Tricks
Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013
More informationEconomics Instructor Miller Oligopoly Practice Problems
Economics Instructor Miller Oligopoly Practice Problems 1. An oligopolistic industry is characterized by all of the following except A) existence of entry barriers. B) the possibility of reaping long run
More informationInternet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords
Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords by Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz (EOS) presented by Scott Brinker
More informationGame Mining: How to Make Money from those about to Play a Game
Game Mining: How to Make Money from those about to Play a Game David H. Wolpert NASA Ames Research Center MailStop 269-1 Moffett Field, CA 94035-1000 david.h.wolpert@nasa.gov James W. Bono Department of
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More informationClock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter 4 described a mathematical system
CHAPTER Number Theory FIGURE FIGURE FIGURE Plus hours Plus hours Plus hours + = + = + = FIGURE. Clock Arithmetic and Modular Systems Clock Arithmetic The introduction to Chapter described a mathematical
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationVideo Poker in South Carolina: A Mathematical Study
Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game
More informationMTH6120 Further Topics in Mathematical Finance Lesson 2
MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Non-constant interest rates....................... 15 1.3 Arbitrage and Black-Scholes Theory....................... 16 1.3.1 Informal
More information6.207/14.15: Networks Lectures 19-21: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning
6.207/14.15: Networks Lectures 19-21: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning Daron Acemoglu and Asu Ozdaglar MIT November 23, 25 and 30, 2009 1 Introduction
More informationHow To Solve The Stable Roommates Problem
THE ROOMMATES PROBLEM DISCUSSED NATHAN SCHULZ Abstract. The stable roommates problem as originally posed by Gale and Shapley [1] in 1962 involves a single set of even cardinality 2n, each member of which
More informationECON101 STUDY GUIDE 7 CHAPTER 14
ECON101 STUDY GUIDE 7 CHAPTER 14 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) An oligopoly firm is similar to a monopolistically competitive
More informationNext Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due
Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 6 Sept 25 2007 Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Today: the price-discriminating
More information1. Write the number of the left-hand item next to the item on the right that corresponds to it.
1. Write the number of the left-hand item next to the item on the right that corresponds to it. 1. Stanford prison experiment 2. Friendster 3. neuron 4. router 5. tipping 6. small worlds 7. job-hunting
More informationLecture Note 7: Revealed Preference and Consumer Welfare
Lecture Note 7: Revealed Preference and Consumer Welfare David Autor, Massachusetts Institute of Technology 14.03/14.003 Microeconomic Theory and Public Policy, Fall 2010 1 1 Revealed Preference and Consumer
More informationInvalidity in Predicate Logic
Invalidity in Predicate Logic So far we ve got a method for establishing that a predicate logic argument is valid: do a derivation. But we ve got no method for establishing invalidity. In propositional
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationOnline Appendix to Stochastic Imitative Game Dynamics with Committed Agents
Online Appendix to Stochastic Imitative Game Dynamics with Committed Agents William H. Sandholm January 6, 22 O.. Imitative protocols, mean dynamics, and equilibrium selection In this section, we consider
More informationPareto Efficiency and Approximate Pareto Efficiency in Routing and Load Balancing Games
Pareto Efficiency and Approximate Pareto Efficiency in Routing and Load Balancing Games Yonatan Aumann Yair Dombb Abstract We analyze the Pareto efficiency, or inefficiency, of solutions to routing games
More informationGame Theory and Poker
Game Theory and Poker Jason Swanson April, 2005 Abstract An extremely simplified version of poker is completely solved from a game theoretic standpoint. The actual properties of the optimal solution are
More informationSolutions for Practice problems on proofs
Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some
More informationA public good is often defined to be a good that is both nonrivalrous and nonexcludable in consumption.
Theory of Public Goods A public good is often defined to be a good that is both nonrivalrous and nonexcludable in consumption. The nonrivalrous property holds when use of a unit of the good by one consumer
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
More informationNational Responses to Transnational Terrorism: Intelligence and Counterterrorism Provision
National Responses to Transnational Terrorism: Intelligence and Counterterrorism Provision Thomas Jensen October 10, 2013 Abstract Intelligence about transnational terrorism is generally gathered by national
More information8 Modeling network traffic using game theory
8 Modeling network traffic using game theory Network represented as a weighted graph; each edge has a designated travel time that may depend on the amount of traffic it contains (some edges sensitive to
More information??? Signaling Games???
??? Signaling Games??? In incomplete information games, one player knows more information than the other player. So far, we have focused on the case where the type of the more informed player was known
More informationi/io as compared to 4/10 for players 2 and 3. A "correct" way to play Certainly many interesting games are not solvable by this definition.
496 MATHEMATICS: D. GALE PROC. N. A. S. A THEORY OF N-PERSON GAMES WITH PERFECT INFORMA TION* By DAVID GALE BROWN UNIVBRSITY Communicated by J. von Neumann, April 17, 1953 1. Introduction.-The theory of
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More information1 Nonzero sum games and Nash equilibria
princeton univ. F 14 cos 521: Advanced Algorithm Design Lecture 19: Equilibria and algorithms Lecturer: Sanjeev Arora Scribe: Economic and game-theoretic reasoning specifically, how agents respond to economic
More informationThe Chinese Remainder Theorem
The Chinese Remainder Theorem Evan Chen evanchen@mit.edu February 3, 2015 The Chinese Remainder Theorem is a theorem only in that it is useful and requires proof. When you ask a capable 15-year-old why
More information