6.6 Secret price cuts
|
|
- Sydney Simpson
- 5 years ago
- Views:
Transcription
1 Joe Chen Secret price cuts As stated earlier, afirm weights two opposite incentives when it ponders price cutting: future losses and current gains. The highest level of collusion (monopoly price) is sustainable with the severest level of punishment (eternal reversion to pricing at marginal cost). When price choices are perfectly observable, it makes sense to resort to extreme punishments because such punishments never occur in equilibrium and therefore are costless to firms (they are just threats). In some situations, maximal punishments need not be optimal. Suppose firms initially coordinate on the monopoly-price perfect equilibrium, and some firm deviates by undercutting the price in the first period. By the trigger strategy with maximal punishment, firms charge the marginal cost forever after the deviation. But the firms, who expect no profits from period 2 on, have every incentive to renegotiate to avoid the punishment phase and reach another equilibrium anew. The possibility of renegotiating undermines the strength of punishments and, therefore, adds incentive to undercutting. This opens the door for the discussion of renegotiation equilibrium in supergames. We will not get into that here. Instead, we consider a situation where the demand is stochastic and firms cannot tell a secret price undercutting from a slump in the demand. Under such an uncertainty, mistakes are unavoidable and maximal punishments need not be optimal. Under imperfect information, the fully collusive outcome is not sustainable. The collusive outcome could be sustained only if firms kept on colluding (charging the monopoly price) even when making small profits, because even under collusion small profits can occur as a result of low demand. However, a firm that is confident that its rivals will continue cooperating even when their profits are low has every incentive to undercut secretly. Thus, fully collusive outcome is inconsistent with preventing (deterring) secret price undercutting.
2 Joe Chen Setup price game Porter (1983) and Green and Porter (1984) propose a supergame model that formulizes the issue of secret price cutting. In their model, Green and Porter assume quantity competition. Here, we will go through a version of the model where firms compete in prices. The essence is the same. Consider a market of two firms choosing prices in every period. The goods are perfect substitutes and are produced at constant marginal cost c. Consumers all buy from the low price firm, and the demand is split in halves if both firms charge the same price. In each period, there are two possible states of nature. With probability α, there is no demand (the low-demand state ); with probability 1 α, the demand is D(p) > 0 (the high-demandstate ) Denote p m arg max p (p c)d(p), andπ m =(p m c)d(p m ). Notethatafirm that does not sell at time t is not able to observe whether the absence of demand is due to the realization of the low-demand state or to rival s secret price cut. Let s look for an equilibrium with the following trigger strategies: At the beginning, both firms charge p m ; Both firms continue to charging p m until one or both of them makes zero profit and the game goes to the punishment phase. Call the phase that both firms charging p m the collusive phase; In the punishment phase, both firms charge the marginal cost c for exactly T periods (T can, a priori, be finite or infinite), and revert to the collusive phase. Note that at least one firm makes zero profit, whenever it happens, is common knowledge, even though both firms does not observe its rival s profits. Note also that the punishment phase is unavoidable in equilibrium; in other words, the punishment phase is on the equilibrium path.
3 Joe Chen Trigger strategy equilibrium To look for an equilibrium given the above trigger strategies is to look for a T such that the expected present discounted value of profits of each firm is maximal subject to the constraint that the associated strategies form an equilibrium. Let V + denote the expected present discounted value of a firm s profit fromt on, given that at t 1, the game is in the collusive phase. Let V denote the payoff of a firm s profit from t on, given that at t, the game starts the punishment phase. Then: V + =(1 α)( Πm 2 + δv + )+αδv V = δ T V +. Or, And, the incentive constraint is: V + = V = (1 α)π m 2[1 (1 α)δ αδ T +1 ] (*) (1 α)δ T Π m 2[1 (1 α)δ αδ T +1 ]. V + (1 α)(π m + δv )+αδv ; or (using the definition of V + ), δ(v + V ) Π m /2. (**) Note that, on the one hand, V must be sufficiently lower than V + to prevent undercutting. This implies T has to be long enough. On the other hand, because punishments are costly and occur with positive probability, T should be chosen as small as possible given that the constraint is satisfied. Substituting the equations of (*) to equation (**) yields: 2(1 α)δ +(2α 1)δ T +1 1.
4 Joe Chen 78 When the game starts in the collusive phase, the highest profit for the firms is obtained by solving the following problem: max T V + = (1 α)π m 2 1 (1 α)δ αδ T +1 s.t. 2(1 α)δ +(2α 1)δ T Observe first, V + is decreasing in T (again, this implies we need to find T as small as possible). Following conclusions can be drawn: For all T, 2(1 α)δ +(2α 1)δ T +1 < 1, ifα [1/2, 1]. Thisisbecausewhenα =1/2, 2(1 α)δ +(2α 1)δ T +1 = δ<1, and2(1 α)δ +(2α 1)δ T +1 is decreasing in α. When α is high, no equilibrium with the above trigger strategies is sustainable; Assume 2(1 α)δ 1 (1 α)δ 1/2, the above trigger strategy equilibrium is guaranteed using maximal punishments (T = ). Note that this generalizes the result for the deterministic demand case which corresponds to α =0. To maximize V + subject to the incentive constraint, given α and δ, itsuffices to choose the smallest T (finite) that satisfied the incentive constraint. As an example, if α =1/4, the incentive constraint requires 3δ δ T This is possible only when δ 2/3. Ifδ =0.7, thesmallestt is around Now we have price wars that are involuntary in that they are triggered, not by a price cut, but by an unobservable slump in demand. Note also that price wars are triggered by recessions, contrary to the Rotemberg-Saloner model.
5 Joe Chen Price rigidities Inthesupergameframework,weassumethatfirms always choose prices simultaneously (the synchronicity assumption). Two features of this setup are important: First, a firm s current profit isnotaffected by its rival s previous price choices when it chooses its own price (the game is a repeated game, not a fullly fledged dynamic one); second, the only reason a firm conditions its pricing behavior on previous price choices is that the other firms do so. Hence, the firms strategies are bootstrap strategies in that: The achievement of collusion stems from a subtle self-fulfilling expectation. There are no real business strategies, such as trying to regain losing market shares. In reality, past price choices affect current profits in many scenarios. This raises the discussions that price reactions are not bootstrap reactions but are real attempts to regain market shares. Past price choices may affect current profits through: Price rigidity: the existence of menu cost; On the demand side, consumers may face costs of product learning, or switching costs, or both; On the supply side, past prices affect current workload, if orders take time to be filled. In this subsection, we introduce the concept of Markov perfect equilibrium (MPE) through the examination of a model of asynchronous pricing. Let s consider two firms producing perfect substitutes. At odd (respectively, even) periods, firm 1 (respectively 2) chooses its price. At any period t, apricep it chosen by firm i lasts for two periods: p it = p i,t+1. In period t +2, firm i chooses a new price, which again will be locked in for another two periods. The assumption that firm 1 (respectively, firm 2) chooses a price at odd (respectively, even) periods and the assumtpion that the prices are lock-in for two periods, are not important; the key is the lock-in of prices for some periods. We look for an equilibrium in which the firms price choices are simple in that they depend only on the payoff-relevant information. More precisely, at date 2k +1, firm 2
6 Joe Chen 80 is still committed to the price p 2k itchoseatperiod2k. Notethatthispriceaffects firm 1 s profit atperiod2k +1and therefore, it is payoff-relevant. Consider a strategy of the form: p 1,2k+1 = M 1 (p 2,2k ).Thatis,firm 1 s strategy is conditioned on as little information as is consistent with profit maximizing. Similarly for firm 2, p 2,2k+2 = M 2 (p 1,2k+1 ). M i ( ) is called a Markov reaction function. A Markov perfect equilibrium is a perfect equilibrium in which firms use Markov strategies. Forpricep 2,2k at time 2k +1, firm 1 s reaction must maximize its objective function given that the firms will react according to M 1 ( ) and M 2 ( ) in the future. Mathematically, at period 2k +1,denotep 2,2k as p 2,andp 1,2k+1 as p 1, firm 1 s profit fromperiod2k +1on is: V 1 (p 2 )=max p 1 Π 1 (p 1,p 2 )+δπ 1 (p 1,M 2 (p 1 )) + δ 2 Π 1 (M 1 (M 2 (p 1 )),M 2 (p 1 )) + In equilibrium, p 1 = M 1 (p 2 ) must maximize the profit expression for all p 2.Firm2behaves similarly The kinked-demand story revisited Let D(p) =1 p, andfirms are producing at marginal cost c =0. The price grid is discrete: p h = h/6, whereh =0, 1,...,6. Note that p 0 =0is the competitive price, and p 3 =1/2 is the monopoly price. Consider a symmetric reaction function M 1 ( ) =M 2 ( ) =M( ) as follows: p Π(p) =36p(1 p) M(p) p 6 =1 0 p 3 p 5 =5/6 5 p 3 p 4 =2/3 8 p 3 p 3 =1/2 9 p 3 p 2 =1/3 8 p 1 p 1 =1/6 5 p 1 p 3 with prob. α with prob. 1 α p 0 =0 0 p 3 According to M( ), starting from p 3,ifafirm raises its price, its rival does not follow suit. If a firm undercuts to p 2, its rival reacts with a price war. At p 1,thefirms engage in a
7 Joe Chen 81 war of attrition. Bothfirms want the price to go back to p 3 ; however, each of them wants the other firm to move first, because the relenting firm loses market share in the short run. The outcome is a typical mixed strategy behavior in which firms raise price with positive probability. The pair of strategies (M( ),M( )) forms an MPE when the discount factor is close enough to one. To check, we need to verify that no firm would deviate from M( ): No price cutting at p 3.Atp 3, undercutting to p 2 results in: 8+δ 0+δ 2 0+δ 3 V (p 3 ); pricing at p 3 results in: V (p 3 )=4.5(1 + δ + δ ) =4.5/(1 δ). So, as long as δ is close enough to one, undercutting is not profitable; At p 2, continue the price war. Charging p 1 results in: 5+δ W (p 1 ),wherew (p 1 ) is the payoff when: a firm chose p 1 in the previous period, it is now the other firm s term to choose price, and firms use (M( ),M( )). Notethatatp 1, firms are indifferent between staying p 1 and raising the price to p 3. Hence, 2.5 +δw(p 1 )=δv (p 3 ) or, W (p 1 )=V (p 3 ) 2.5/δ. So,thepayoff of charging p 1 is: 2.5+δV (p 3 ).Thisislarger than the pay off of charging p 3 which is δv (p 3 ). At p 1, firmsplayamixedstrategywiththeprobabilityofplayingp 1 defined as: or, δ 4.5 =2.5+δ {α[2.5+δv (p 1 )] + (1 α)[5 + δv (p 3 )]}; 1 {z δ {z } } continuing p 1 raising price to p 3 Note that V (p 1 )=δv (p 3 ). α = 5+δ 5δ +9δ 2. Let p 3 be the focal price, this equilibrium is the same as the static kinked-demand-curve story, but now the reactions are real and fully rational.
8 Joe Chen Edgeworth cycle There also exist equilibria in which the price never settles. following strategy: For instance, consider the p Π(p) =36p(1 p) M(p) p 6 =1 0 p 4 p 5 =5/6 5 p 4 p 4 =2/3 8 p 3 p 3 =1/2 9 p 2 p 2 =1/3 8 p 1 p 1 =1/6 5 p 0 p 0 =0 0 p 0 p 5 with prob. β with prob. 1 β
9 Joe Chen Some final thoughts Despite multiple equilibria, it can be shown that in every MPE, profits are always bounded away from the competitive profit (whichis0). Both supergames and Markov games suggest some collusion is always possible as long as the discount factor is close enough to one. In Markov games, unlike supergames, the current profit is determined by pervious actions. When firms compete in prices, the response to a rival s previous action would be to regain its losing market share. Based on this reasoning, suppose one introduces demand fluctuation into the model, one would expect price adjustments more sluggish during booms than during recessions (more Green-Porter alike). Some sort of collusion is possible because of the fear of retaliation (triggering a price war). However, the motives for retaliation are very different with the two different setups. In supergames, the price war is a purely self-fulling phenomenon. A firm charges a low price because it expects the other firms to do so (bootstrapping). In Markov games with price rigidities, the reaction of one firm to a price cut by another firm is motivated by its desire to regain a market share that has been and continues to be eroded by its rivals aggressive pricing strategy. In an intertemporal setup, it is also possible that firms can sustain collusion through nonphysical factors such as reputation. We stop here.
February 23, An Application in Industrial Organization
An Application in Industrial Organization February 23, 2015 One form of collusive behavior among firms is to restrict output in order to keep the price of the product high. This is a goal of the OPEC oil
More informationAnswer Key: Problem Set 4
Answer Key: Problem Set 4 Econ 409 018 Fall A reminder: An equilibrium is characterized by a set of strategies. As emphasized in the class, a strategy is a complete contingency plan (for every hypothetical
More informationAn Alternating Move Price-Setting Duopoly Model with Stochastic Costs. Andrew Eckert Department of Economics University of Alberta Edmonton Alberta
An Alternating Move Price-Setting Duopoly Model with Stochastic Costs Andrew Eckert Department of Economics University of Alberta Edmonton Alberta June 7, 2004 Abstract This paper examines an alternating
More informationGame Theory. Wolfgang Frimmel. Repeated Games
Game Theory Wolfgang Frimmel Repeated Games 1 / 41 Recap: SPNE The solution concept for dynamic games with complete information is the subgame perfect Nash Equilibrium (SPNE) Selten (1965): A strategy
More informationLecture 9: Basic Oligopoly Models
Lecture 9: Basic Oligopoly Models Managerial Economics November 16, 2012 Prof. Dr. Sebastian Rausch Centre for Energy Policy and Economics Department of Management, Technology and Economics ETH Zürich
More informationDuopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two-stage prisoner s dilemma
Recap Last class (September 20, 2016) Duopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two-stage prisoner s dilemma Today (October 13, 2016) Finitely
More informationInfinitely Repeated Games
February 10 Infinitely Repeated Games Recall the following theorem Theorem 72 If a game has a unique Nash equilibrium, then its finite repetition has a unique SPNE. Our intuition, however, is that long-term
More informationExercises Solutions: Oligopoly
Exercises Solutions: Oligopoly Exercise - Quantity competition 1 Take firm 1 s perspective Total revenue is R(q 1 = (4 q 1 q q 1 and, hence, marginal revenue is MR 1 (q 1 = 4 q 1 q Marginal cost is MC
More informationFinitely repeated simultaneous move game.
Finitely repeated simultaneous move game. Consider a normal form game (simultaneous move game) Γ N which is played repeatedly for a finite (T )number of times. The normal form game which is played repeatedly
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 3 1. Consider the following strategic
More informationM.Phil. Game theory: Problem set II. These problems are designed for discussions in the classes of Week 8 of Michaelmas term. 1
M.Phil. Game theory: Problem set II These problems are designed for discussions in the classes of Week 8 of Michaelmas term.. Private Provision of Public Good. Consider the following public good game:
More informationOptimal selling rules for repeated transactions.
Optimal selling rules for repeated transactions. Ilan Kremer and Andrzej Skrzypacz March 21, 2002 1 Introduction In many papers considering the sale of many objects in a sequence of auctions the seller
More information13.1 Infinitely Repeated Cournot Oligopoly
Chapter 13 Application: Implicit Cartels This chapter discusses many important subgame-perfect equilibrium strategies in optimal cartel, using the linear Cournot oligopoly as the stage game. For game theory
More informationRepeated Games. Econ 400. University of Notre Dame. Econ 400 (ND) Repeated Games 1 / 48
Repeated Games Econ 400 University of Notre Dame Econ 400 (ND) Repeated Games 1 / 48 Relationships and Long-Lived Institutions Business (and personal) relationships: Being caught cheating leads to punishment
More informationA monopoly is an industry consisting a single. A duopoly is an industry consisting of two. An oligopoly is an industry consisting of a few
27 Oligopoly Oligopoly A monopoly is an industry consisting a single firm. A duopoly is an industry consisting of two firms. An oligopoly is an industry consisting of a few firms. Particularly, l each
More informationEcon 101A Final exam Mo 18 May, 2009.
Econ 101A Final exam Mo 18 May, 2009. Do not turn the page until instructed to. Do not forget to write Problems 1 and 2 in the first Blue Book and Problems 3 and 4 in the second Blue Book. 1 Econ 101A
More informationPRISONER S DILEMMA. Example from P-R p. 455; also 476-7, Price-setting (Bertrand) duopoly Demand functions
ECO 300 Fall 2005 November 22 OLIGOPOLY PART 2 PRISONER S DILEMMA Example from P-R p. 455; also 476-7, 481-2 Price-setting (Bertrand) duopoly Demand functions X = 12 2 P + P, X = 12 2 P + P 1 1 2 2 2 1
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationThe Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies:
Problem Set 4 1. (a). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: B L R U 1, 1 2, 5 A D 2, 0 0, 0 Sketch a graph of the players payoffs.
More informationNotes for Section: Week 4
Economics 160 Professor Steven Tadelis Stanford University Spring Quarter, 2004 Notes for Section: Week 4 Notes prepared by Paul Riskind (pnr@stanford.edu). spot errors or have questions about these notes.
More informationPuerto Rico, US, Dec 2013: 5-year sentence for pricefixing
Dynaic oligopoly theory Collusion price coordination Illegal in ost countries - Explicit collusion not feasible - Legal exeptions Recent EU cases - Banking approx..7 billion Euros in fines (03) - Cathodic
More informationEC 202. Lecture notes 14 Oligopoly I. George Symeonidis
EC 202 Lecture notes 14 Oligopoly I George Symeonidis Oligopoly When only a small number of firms compete in the same market, each firm has some market power. Moreover, their interactions cannot be ignored.
More informationMA300.2 Game Theory 2005, LSE
MA300.2 Game Theory 2005, LSE Answers to Problem Set 2 [1] (a) This is standard (we have even done it in class). The one-shot Cournot outputs can be computed to be A/3, while the payoff to each firm can
More informationElements of Economic Analysis II Lecture XI: Oligopoly: Cournot and Bertrand Competition
Elements of Economic Analysis II Lecture XI: Oligopoly: Cournot and Bertrand Competition Kai Hao Yang /2/207 In this lecture, we will apply the concepts in game theory to study oligopoly. In short, unlike
More informationSTOCHASTIC REPUTATION DYNAMICS UNDER DUOPOLY COMPETITION
STOCHASTIC REPUTATION DYNAMICS UNDER DUOPOLY COMPETITION BINGCHAO HUANGFU Abstract This paper studies a dynamic duopoly model of reputation-building in which reputations are treated as capital stocks that
More informationMicroeconomic Theory II Preliminary Examination Solutions Exam date: August 7, 2017
Microeconomic Theory II Preliminary Examination Solutions Exam date: August 7, 017 1. Sheila moves first and chooses either H or L. Bruce receives a signal, h or l, about Sheila s behavior. The distribution
More informationEconomics and Computation
Economics and Computation ECON 425/563 and CPSC 455/555 Professor Dirk Bergemann and Professor Joan Feigenbaum Reputation Systems In case of any questions and/or remarks on these lecture notes, please
More informationCorporate Control. Itay Goldstein. Wharton School, University of Pennsylvania
Corporate Control Itay Goldstein Wharton School, University of Pennsylvania 1 Managerial Discipline and Takeovers Managers often don t maximize the value of the firm; either because they are not capable
More informationReputation and Signaling in Asset Sales: Internet Appendix
Reputation and Signaling in Asset Sales: Internet Appendix Barney Hartman-Glaser September 1, 2016 Appendix D. Non-Markov Perfect Equilibrium In this appendix, I consider the game when there is no honest-type
More informationProblem 3 Solutions. l 3 r, 1
. Economic Applications of Game Theory Fall 00 TA: Youngjin Hwang Problem 3 Solutions. (a) There are three subgames: [A] the subgame starting from Player s decision node after Player s choice of P; [B]
More informationRepeated Games. September 3, Definitions: Discounting, Individual Rationality. Finitely Repeated Games. Infinitely Repeated Games
Repeated Games Frédéric KOESSLER September 3, 2007 1/ Definitions: Discounting, Individual Rationality Finitely Repeated Games Infinitely Repeated Games Automaton Representation of Strategies The One-Shot
More informationDiscounted Stochastic Games with Voluntary Transfers
Discounted Stochastic Games with Voluntary Transfers Sebastian Kranz University of Cologne Slides Discounted Stochastic Games Natural generalization of infinitely repeated games n players infinitely many
More informationPrice cutting and business stealing in imperfect cartels Online Appendix
Price cutting and business stealing in imperfect cartels Online Appendix B. Douglas Bernheim Erik Madsen December 2016 C.1 Proofs omitted from the main text Proof of Proposition 4. We explicitly construct
More informationCompeting Mechanisms with Limited Commitment
Competing Mechanisms with Limited Commitment Suehyun Kwon CESIFO WORKING PAPER NO. 6280 CATEGORY 12: EMPIRICAL AND THEORETICAL METHODS DECEMBER 2016 An electronic version of the paper may be downloaded
More informationPAULI MURTO, ANDREY ZHUKOV
GAME THEORY SOLUTION SET 1 WINTER 018 PAULI MURTO, ANDREY ZHUKOV Introduction For suggested solution to problem 4, last year s suggested solutions by Tsz-Ning Wong were used who I think used suggested
More informationIntroduction to Game Theory
Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 1. Dynamic games of complete and perfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas
More informationMicroeconomics II. CIDE, MsC Economics. List of Problems
Microeconomics II CIDE, MsC Economics List of Problems 1. There are three people, Amy (A), Bart (B) and Chris (C): A and B have hats. These three people are arranged in a room so that B can see everything
More informationExercise Chapter 10
Exercise 10.8.1 Where the isoprofit curves touch the gradients of the profits of Alice and Bob point in the opposite directions. Thus, increasing one agent s profit will necessarily decrease the other
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 2 1. Consider a zero-sum game, where
More informationSUPPLEMENT TO WHEN DOES PREDATION DOMINATE COLLUSION? (Econometrica, Vol. 85, No. 2, March 2017, )
Econometrica Supplementary Material SUPPLEMENT TO WHEN DOES PREDATION DOMINATE COLLUSION? (Econometrica, Vol. 85, No., March 017, 555 584) BY THOMAS WISEMAN S1. PROOF FROM SECTION 4.4 PROOF OF CLAIM 1:
More informationNot 0,4 2,1. i. Show there is a perfect Bayesian equilibrium where player A chooses to play, player A chooses L, and player B chooses L.
Econ 400, Final Exam Name: There are three questions taken from the material covered so far in the course. ll questions are equally weighted. If you have a question, please raise your hand and I will come
More informationRelative Performance and Stability of Collusive Behavior
Relative Performance and Stability of Collusive Behavior Toshihiro Matsumura Institute of Social Science, the University of Tokyo and Noriaki Matsushima Graduate School of Business Administration, Kobe
More informationMicroeconomic Theory II Preliminary Examination Solutions
Microeconomic Theory II Preliminary Examination Solutions 1. (45 points) Consider the following normal form game played by Bruce and Sheila: L Sheila R T 1, 0 3, 3 Bruce M 1, x 0, 0 B 0, 0 4, 1 (a) Suppose
More informationIntroduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4)
Introduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4) Outline: Modeling by means of games Normal form games Dominant strategies; dominated strategies,
More informationRegret Minimization and Security Strategies
Chapter 5 Regret Minimization and Security Strategies Until now we implicitly adopted a view that a Nash equilibrium is a desirable outcome of a strategic game. In this chapter we consider two alternative
More informationRepeated Games with Perfect Monitoring
Repeated Games with Perfect Monitoring Mihai Manea MIT Repeated Games normal-form stage game G = (N, A, u) players simultaneously play game G at time t = 0, 1,... at each date t, players observe all past
More informationMicroeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 2017
Microeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 07. (40 points) Consider a Cournot duopoly. The market price is given by q q, where q and q are the quantities of output produced
More informationChapter 11: Dynamic Games and First and Second Movers
Chapter : Dynamic Games and First and Second Movers Learning Objectives Students should learn to:. Extend the reaction function ideas developed in the Cournot duopoly model to a model of sequential behavior
More informationEcon 323 Microeconomic Theory. Chapter 10, Question 1
Econ 323 Microeconomic Theory Practice Exam 2 with Solutions Chapter 10, Question 1 Which of the following is not a condition for perfect competition? Firms a. take prices as given b. sell a standardized
More informationSolution to Assignment 3
Solution to Assignment 3 0/03 Semester I MA6 Game Theory Tutor: Xiang Sun October 5, 0. Question 5, in Tutorial set 5;. Question, in Tutorial set 6; 3. Question, in Tutorial set 7. Solution for Question
More informationEcon 101A Final Exam We May 9, 2012.
Econ 101A Final Exam We May 9, 2012. You have 3 hours to answer the questions in the final exam. We will collect the exams at 2.30 sharp. Show your work, and good luck! Problem 1. Utility Maximization.
More informationEconomics 171: Final Exam
Question 1: Basic Concepts (20 points) Economics 171: Final Exam 1. Is it true that every strategy is either strictly dominated or is a dominant strategy? Explain. (5) No, some strategies are neither dominated
More informationGame Theory Fall 2003
Game Theory Fall 2003 Problem Set 5 [1] Consider an infinitely repeated game with a finite number of actions for each player and a common discount factor δ. Prove that if δ is close enough to zero then
More informationAnswers to Microeconomics Prelim of August 24, In practice, firms often price their products by marking up a fixed percentage over (average)
Answers to Microeconomics Prelim of August 24, 2016 1. In practice, firms often price their products by marking up a fixed percentage over (average) cost. To investigate the consequences of markup pricing,
More informationDoes structure dominate regulation? The case of an input monopolist 1
Does structure dominate regulation? The case of an input monopolist 1 Stephen P. King Department of Economics The University of Melbourne October 9, 2000 1 I would like to thank seminar participants at
More informationFinite Memory and Imperfect Monitoring
Federal Reserve Bank of Minneapolis Research Department Finite Memory and Imperfect Monitoring Harold L. Cole and Narayana Kocherlakota Working Paper 604 September 2000 Cole: U.C.L.A. and Federal Reserve
More informationIn the Name of God. Sharif University of Technology. Graduate School of Management and Economics
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics (for MBA students) 44111 (1393-94 1 st term) - Group 2 Dr. S. Farshad Fatemi Game Theory Game:
More informationDUOPOLY. MICROECONOMICS Principles and Analysis Frank Cowell. July 2017 Frank Cowell: Duopoly. Almost essential Monopoly
Prerequisites Almost essential Monopoly Useful, but optional Game Theory: Strategy and Equilibrium DUOPOLY MICROECONOMICS Principles and Analysis Frank Cowell 1 Overview Duopoly Background How the basic
More informationFinite Memory and Imperfect Monitoring
Federal Reserve Bank of Minneapolis Research Department Staff Report 287 March 2001 Finite Memory and Imperfect Monitoring Harold L. Cole University of California, Los Angeles and Federal Reserve Bank
More informationAppendix: Common Currencies vs. Monetary Independence
Appendix: Common Currencies vs. Monetary Independence A The infinite horizon model This section defines the equilibrium of the infinity horizon model described in Section III of the paper and characterizes
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 2012
Game Theory Lecture Notes By Y. Narahari Department of Computer Science and Automation Indian Institute of Science Bangalore, India October 22 COOPERATIVE GAME THEORY Correlated Strategies and Correlated
More informationDUOPOLY MODELS. Dr. Sumon Bhaumik (http://www.sumonbhaumik.net) December 29, 2008
DUOPOLY MODELS Dr. Sumon Bhaumik (http://www.sumonbhaumik.net) December 29, 2008 Contents 1. Collusion in Duopoly 2. Cournot Competition 3. Cournot Competition when One Firm is Subsidized 4. Stackelberg
More informationAsymmetric collusion with growing demand
Asymmetric collusion with growing demand António Brandão CEF.UP and Faculdade de Economia. Universidade do Porto. Joana Pinho Facultad de Económicas. Universidad de Vigo. Hélder Vasconcelos CEF.UP and
More informationEcon 323 Microeconomic Theory. Practice Exam 2 with Solutions
Econ 323 Microeconomic Theory Practice Exam 2 with Solutions Chapter 10, Question 1 Which of the following is not a condition for perfect competition? Firms a. take prices as given b. sell a standardized
More informationOutline for Dynamic Games of Complete Information
Outline for Dynamic Games of Complete Information I. Examples of dynamic games of complete info: A. equential version of attle of the exes. equential version of Matching Pennies II. Definition of subgame-perfect
More informationAS/ECON 2350 S2 N Answers to Mid term Exam July time : 1 hour. Do all 4 questions. All count equally.
AS/ECON 2350 S2 N Answers to Mid term Exam July 2017 time : 1 hour Do all 4 questions. All count equally. Q1. Monopoly is inefficient because the monopoly s owner makes high profits, and the monopoly s
More informationECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves
University of Illinois Spring 01 ECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves Due: Reading: Thursday, April 11 at beginning of class
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Bargaining We will now apply the concept of SPNE to bargaining A bit of background Bargaining is hugely interesting but complicated to model It turns out that the
More informationECO410H: Practice Questions 2 SOLUTIONS
ECO410H: Practice Questions SOLUTIONS 1. (a) The unique Nash equilibrium strategy profile is s = (M, M). (b) The unique Nash equilibrium strategy profile is s = (R4, C3). (c) The two Nash equilibria are
More informationIMPERFECT COMPETITION AND TRADE POLICY
IMPERFECT COMPETITION AND TRADE POLICY Once there is imperfect competition in trade models, what happens if trade policies are introduced? A literature has grown up around this, often described as strategic
More informationEcon 302 Assignment 3 Solution. a 2bQ c = 0, which is the monopolist s optimal quantity; the associated price is. P (Q) = a b
Econ 302 Assignment 3 Solution. (a) The monopolist solves: The first order condition is max Π(Q) = Q(a bq) cq. Q a Q c = 0, or equivalently, Q = a c, which is the monopolist s optimal quantity; the associated
More informationEconomics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Repeated games OR 8 and 9, and FT 5
Economics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Repeated games OR 8 and 9, and FT 5 The basic idea prisoner s dilemma The prisoner s dilemma game with one-shot payoffs 2 2 0
More informationName. Answers Discussion Final Exam, Econ 171, March, 2012
Name Answers Discussion Final Exam, Econ 171, March, 2012 1) Consider the following strategic form game in which Player 1 chooses the row and Player 2 chooses the column. Both players know that this is
More informationIntroduction to Game Theory Lecture Note 5: Repeated Games
Introduction to Game Theory Lecture Note 5: Repeated Games Haifeng Huang University of California, Merced Repeated games Repeated games: given a simultaneous-move game G, a repeated game of G is an extensive
More informationLecture 6 Dynamic games with imperfect information
Lecture 6 Dynamic games with imperfect information Backward Induction in dynamic games of imperfect information We start at the end of the trees first find the Nash equilibrium (NE) of the last subgame
More informationECON/MGMT 115. Industrial Organization
ECON/MGMT 115 Industrial Organization 1. Cournot Model, reprised 2. Bertrand Model of Oligopoly 3. Cournot & Bertrand First Hour Reviewing the Cournot Duopoloy Equilibria Cournot vs. competitive markets
More informationThe value of switching costs
The value of switching costs Gary Biglaiser University of North Carolina Jacques Crémer Toulouse School of Economics CNRS & IDEI Gergely Dobos 1 Switching costs play an important role in current economic
More informationTopics in Contract Theory Lecture 1
Leonardo Felli 7 January, 2002 Topics in Contract Theory Lecture 1 Contract Theory has become only recently a subfield of Economics. As the name suggest the main object of the analysis is a contract. Therefore
More informationUC Berkeley Haas School of Business Game Theory (EMBA 296 & EWMBA 211) Summer 2016
UC Berkeley Haas School of Business Game Theory (EMBA 296 & EWMBA 211) Summer 2016 More on strategic games and extensive games with perfect information Block 2 Jun 11, 2017 Auctions results Histogram of
More informationEconometrica Supplementary Material
Econometrica Supplementary Material PUBLIC VS. PRIVATE OFFERS: THE TWO-TYPE CASE TO SUPPLEMENT PUBLIC VS. PRIVATE OFFERS IN THE MARKET FOR LEMONS (Econometrica, Vol. 77, No. 1, January 2009, 29 69) BY
More informationA Decentralized Learning Equilibrium
Paper to be presented at the DRUID Society Conference 2014, CBS, Copenhagen, June 16-18 A Decentralized Learning Equilibrium Andreas Blume University of Arizona Economics ablume@email.arizona.edu April
More informationGraduate Macro Theory II: Two Period Consumption-Saving Models
Graduate Macro Theory II: Two Period Consumption-Saving Models Eric Sims University of Notre Dame Spring 207 Introduction This note works through some simple two-period consumption-saving problems. In
More informationEconomics 431 Infinitely repeated games
Economics 431 Infinitely repeated games Letuscomparetheprofit incentives to defect from the cartel in the short run (when the firm is the only defector) versus the long run (when the game is repeated)
More informationI. Introduction and definitions
Economics 335 March 7, 1999 Notes 7: Noncooperative Oligopoly Models I. Introduction and definitions A. Definition A noncooperative oligopoly is a market where a small number of firms act independently,
More informationWhen one firm considers changing its price or output level, it must make assumptions about the reactions of its rivals.
Chapter 3 Oligopoly Oligopoly is an industry where there are relatively few sellers. The product may be standardized (steel) or differentiated (automobiles). The firms have a high degree of interdependence.
More informationSimon Fraser University Spring 2014
Simon Fraser University Spring 2014 Econ 302 D200 Final Exam Solution This brief solution guide does not have the explanations necessary for full marks. NE = Nash equilibrium, SPE = subgame perfect equilibrium,
More informationRelational Incentive Contracts
Relational Incentive Contracts Jonathan Levin May 2006 These notes consider Levin s (2003) paper on relational incentive contracts, which studies how self-enforcing contracts can provide incentives in
More informationPrisoner s dilemma with T = 1
REPEATED GAMES Overview Context: players (e.g., firms) interact with each other on an ongoing basis Concepts: repeated games, grim strategies Economic principle: repetition helps enforcing otherwise unenforceable
More informationPh.D. MICROECONOMICS CORE EXAM August 2018
Ph.D. MICROECONOMICS CORE EXAM August 2018 This exam is designed to test your broad knowledge of microeconomics. There are three sections: one required and two choice sections. You must complete both problems
More informationSolutions to Homework 3
Solutions to Homework 3 AEC 504 - Summer 2007 Fundamentals of Economics c 2007 Alexander Barinov 1 Price Discrimination Consider a firm with MC = AC = 2, which serves two markets with demand functions
More informationGame Theory with Applications to Finance and Marketing, I
Game Theory with Applications to Finance and Marketing, I Homework 1, due in recitation on 10/18/2018. 1. Consider the following strategic game: player 1/player 2 L R U 1,1 0,0 D 0,0 3,2 Any NE can be
More informationHomework 2: Dynamic Moral Hazard
Homework 2: Dynamic Moral Hazard Question 0 (Normal learning model) Suppose that z t = θ + ɛ t, where θ N(m 0, 1/h 0 ) and ɛ t N(0, 1/h ɛ ) are IID. Show that θ z 1 N ( hɛ z 1 h 0 + h ɛ + h 0m 0 h 0 +
More informationYao s Minimax Principle
Complexity of algorithms The complexity of an algorithm is usually measured with respect to the size of the input, where size may for example refer to the length of a binary word describing the input,
More informationINVESTMENT DYNAMICS IN ELECTRICITY MARKETS Alfredo Garcia, University of Virginia joint work with Ennio Stacchetti, New York University May 2007
INVESTMENT DYNAMICS IN ELECTRICITY MARKETS Alfredo Garcia, University of Virginia joint work with Ennio Stacchetti, New York University May 2007 1 MOTIVATION We study resource adequacy as an endogenous
More informationd. Find a competitive equilibrium for this economy. Is the allocation Pareto efficient? Are there any other competitive equilibrium allocations?
Answers to Microeconomics Prelim of August 7, 0. Consider an individual faced with two job choices: she can either accept a position with a fixed annual salary of x > 0 which requires L x units of labor
More information1 Two Period Exchange Economy
University of British Columbia Department of Economics, Macroeconomics (Econ 502) Prof. Amartya Lahiri Handout # 2 1 Two Period Exchange Economy We shall start our exploration of dynamic economies with
More informationCUR 412: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 2015
CUR 41: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 015 Instructions: Please write your name in English. This exam is closed-book. Total time: 10 minutes. There are 4 questions,
More informationEcon 8602, Fall 2017 Homework 2
Econ 8602, Fall 2017 Homework 2 Due Tues Oct 3. Question 1 Consider the following model of entry. There are two firms. There are two entry scenarios in each period. With probability only one firm is able
More informationGroup-lending with sequential financing, contingent renewal and social capital. Prabal Roy Chowdhury
Group-lending with sequential financing, contingent renewal and social capital Prabal Roy Chowdhury Introduction: The focus of this paper is dynamic aspects of micro-lending, namely sequential lending
More informationMA200.2 Game Theory II, LSE
MA200.2 Game Theory II, LSE Answers to Problem Set [] In part (i), proceed as follows. Suppose that we are doing 2 s best response to. Let p be probability that player plays U. Now if player 2 chooses
More information