# 6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2,

2 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma Infinitely-repeated games and cooperation Folk theorems Cooperation in finitely-repeated games Social preferences Reading: Osborne, Chapters 14 and 15. 2

3 Prisoners Dilemma Networks: Lecture 15 Introduction How to sustain cooperation in the society? Recall the prisoners dilemma, which is the canonical game for understanding incentives for defecting instead of operating. Cooperate Defect Cooperate 1, 1 1, 2 Defect 2, 1 0, 0 Recall that the strategy profile (D, D) is the unique NE. In fact, D strictly dominates C and thus (D, D) is the dominant equilibrium. In society, we have many situations of this form, but we often observe some amount of cooperation. Why? 3

4 Introduction Repeated Games In many strategic situations, players interact repeatedly over time. Perhaps repetition of the same game might foster cooperation. By repeated games we refer to a situation in which the same stage game (strategic form game) is played at each date for some duration of T periods. Such games are also sometimes called supergames. Key new concept: discounting. We will imagine that future payoffs are discounted and are thus less valuable (e.g., money and the future is less valuable than money now because of positive interest rates; consumption in the future is less valuable than consumption now because of time preference). 4

5 Introduction Discounting We will model time preferences by assuming that future payoffs are discounted proportionately ( exponentially ) at some rate δ [0, 1), called the discount rate. For example, in a two-period game with stage payoffs given by u 1 and u 2, overall payoffs will be U = u 1 + δu 2. With the interest rate interpretation, we would have where r is the interest rate. δ = r, 5

6 Introduction Mathematical Model More formally, imagine that I players are playing a strategic form game G = I, (A i ) i I, (u i ) i I for T periods. At each period, the outcomes of all past periods are observed by all players. Let us start with the case in which T is finite, but we will be particularly interested in the case in which T =. Here A i denotes the set of actions at each stage, and where A = A 1 A I. u i : A R, That is, u i ( a t i, a t i) is the state payoff to player i when action profile a t = ( a t i, at i) is played. 6

7 Mathematical Model (continued) Introduction We use the notation a = {a t } T t=0 to denote the sequence of action profiles. We could also define σ = {σ t } T t=0 to be the profile of mixed strategies. The payoff to player i in the repeated game where δ [0, 1). U(a) = T δ t u i (ai t, a i) t t=0 We denote the T -period repeated game with discount factor δ by G T (δ). 7

8 Introduction Finitely-Repeated Prisoners Dilemma Recall Cooperate Defect Cooperate 1, 1 1, 2 Defect 2, 1 0, 0 What happens if this game was played T < times? We first need to decide what the equilibrium notion is. Natural choice, subgame perfect Nash equilibrium (SPE). Recall: SPE backward induction. Therefore, start in the last period, at time T. What will happen? 8

9 Introduction Finitely-Repeated Prisoners Dilemma (continued) In the last period, defect is a dominant strategy regardless of the history of the game. So the subgame starting at T has a dominant strategy equilibrium: (D, D). Then move to stage T 1. By backward induction, we know that at T, no matter what, the play will be (D, D). Then given this, the subgame starting at T 1 (again regardless of history) also has a dominant strategy equilibrium. With this argument, we have that there exists a unique SPE: (D, D) at each date. In fact, this is a special case of a more general result. 9

10 Introduction Equilibria of Finitely-Repeated Games Theorem Consider repeated game G T (δ) for T <. Suppose that the stage game G has a unique pure strategy equilibrium a. Then G T has a unique SPE. In this unique SPE, a t = a for each t = 0, 1,..., T regardless of history. Proof: The proof has exactly the same logic as the prisoners dilemma example. By backward induction, at date T, we will have that (regardless of history) a T = a. Given this, then we have a T 1 = a, and continuing inductively, a t = a for each t = 0, 1,..., T regardless of history. 10

11 Infinitely-Repeated Games Infinitely-Repeated Games Now consider the infinitely-repeated game G. The notation a = {a t } t=0 now denotes the (infinite) sequence of action profiles. The payoff to player i is then where, again, δ [0, 1). U(a) = (1 δ) δ t u i (ai t, a i) t t=0 Note: this summation is well defined because δ < 1. The term in front is introduced as a normalization, so that utility remains bounded even when δ 1. 11

12 Trigger Strategies Networks: Lecture 15 Infinitely-Repeated Games In infinitely-repeated games we can consider trigger strategies. A trigger strategy essentially threatens other players with a worse, punishment, action if they deviate from an implicitly agreed action profile. A non-forgiving trigger strategy (or grim trigger strategy) s would involve this punishment forever after a single deviation. A non-forgiving trigger strategy (for player i) takes the following form: { ai t āi if a = τ = ā for all τ < t if a τ ā for some τ < t a i Here if ā is the implicitly agreed action profile and a i is the punishment action. This strategy is non-forgiving since a single deviation from ā induces player i to switch to a i forever. 12

13 Infinitely-Repeated Games Cooperation with Trigger Strategies in the Repeated Prisoners Dilemma Recall Cooperate Defect Cooperate 1, 1 1, 2 Defect 2, 1 0, 0 Suppose both players use the following non-forgiving trigger strategy s : Play C in every period unless someone has ever played D in the past Play D forever if someone has played D in the past. We next show that the preceding strategy is an SPE if δ 1/2. 13

14 Infinitely-Repeated Games Cooperation with Trigger Strategies in the Repeated Prisoners Dilemma Step 1: cooperation is best response to cooperation. Suppose that there has so far been no D. Then given s being played by the other player, the payoffs to cooperation and defection are: Payoff from C : (1 δ)[1 + δ + δ 2 + ] = (1 δ) 1 1 δ = 1 Payoff from D : (1 δ)[ ] = 2(1 δ) Cooperation better if 2(1 δ) 1. This shows that for δ 1/2, deviation to defection is not profitable. 14

15 Infinitely-Repeated Games Cooperation with Trigger Strategies in the Repeated Prisoners Dilemma (continued) Step 2: defection is best response to defection. Suppose that there has been some D in the past, then according to s, the other player will always play D. Against this, D is a best response. This argument is true in every subgame, so s is a subgame perfect equilibrium. Note: cooperating in every period would be a best response for a player against s. But unless that player herself also plays s, her opponent would not cooperate. Thus SPE requires both players to use s. 15

16 Infinitely-Repeated Games Multiplicity of Equilibria Cooperation is an equilibrium, but so are many other strategy profiles. Multiplicity of equilibria endemic in repeated games. Note that this multiplicity only occurs at T =. In particular, for any finite T (and thus by implication for T ), prisoners dilemma has a unique SPE. Why? The set of Nash equilibria is an upper hemi-continuous correspondence in parameters. It is not necessarily lower hemi-continuous. 16

17 Infinitely-Repeated Games Repetition Can Lead to Bad Outcomes The following example shows that repeated play can lead to worse outcomes than in the one shot game: A B C A 2, 2 2, 1 0, 0 B 1, 2 1, 1 1, 0 C 0, 0 0, 1 1, 1 For the game defined above, the action A strictly dominates both B and C for both players; therefore the unique Nash equilibrium of the stage game is (A, A). If δ 1/2, this game has an SPE in which (B, B) is played in every period. It is supported by the trigger strategy: Play B in every period unless someone deviates, and play C if there is any deviation. It can be verified that for δ 1/2, (B, B) is an SPE. 17

18 Folk Theorems Folk Theorems In fact, it has long been a folk theorem that one can support cooperation in repeated prisoners dilemma, and other non-one-stage equilibrium outcomes in infinitely-repeated games with sufficiently high discount factors. These results are referred to as folk theorems since they were believe to be true before they were formally proved. Here we will see a relatively strong version of these folk theorems. 18

19 Folk Theorems Feasible Payoffs Consider stage game G = I, (A i ) i I, (u i ) i I and infinitely-repeated game G (δ). Let us introduce the Set of feasible payoffs: V = Conv{v R I there exists a A such that u(a) = v}. That is, V is the convex hull of all I - dimensional vectors that can be obtained by some action profile. Convexity here is obtained by public randomization. Note: V is not equal to {v R I there exists σ Σ such that u(σ) = v}, where Σ is the set of mixed strategy profiles in the stage game. 19

20 Minmax Payoffs Networks: Lecture 15 Folk Theorems Minmax payoff of player i: the lowest payoff that player i s opponent can hold him to: [ ] v i = min max u i (a i, a i ) a i a i [ ] = max min u i (a i, a i ). a i a i The player can never receive less than this amount. Minmax strategy profile against i: m i i = arg min a i [ ] max u i (a i, a i ) a i 20

21 Example Networks: Lecture 15 Folk Theorems Consider L R U 2, 2 1, 2 M 1, 1 2, 2 D 0, 1 0, 1 To compute v 1, let q denote the probability that player 2 chooses action L. Then player 1 s payoffs for playing different actions are given by: U 1 3q M 2 + 3q D 0 21

22 Example Networks: Lecture 15 Folk Theorems Therefore, we have v 1 = min 0 q 1 [max{1 3q, 2 + 3q, 0}] = 0, and m2 1 [ 1 3, 2 3 ]. Similarly, one can show that: v 2 = 0, and m1 2 = (1/2, 1/2, 0) is the unique minimax profile. 22

23 Minmax Payoff Lower Bounds Folk Theorems Theorem 1 Let σ be a (possibly mixed) Nash equilibrium of G and u i (σ) be the payoff to player i in equilibrium σ. Then u i (σ) v i. 2 Let σ be a (possibly mixed) Nash equilibrium of G (δ) and U i (σ) be the payoff to player i in equilibrium σ. Then U i (σ) v i. Proof: Player i can always guarantee herself v i = min a i [max ai u i (a i, a i )] in the stage [ game and also in each stage of the repeated game, since v i = max ai mina i u i (a i, a i ) ], meaning that she can always achieve at least this payoff against even the most adversarial strategies. 23

24 Folk Theorems Folk Theorems Definition A payoff vector v R I is strictly individually rational if v i > v i for all i. Theorem (Nash Folk Theorem) If (v 1,..., v I ) is feasible and strictly individually rational, then there exists some δ < 1 such that for all δ > δ, there is a Nash equilibrium of G (δ) with payoffs (v 1,, v I ). 24

25 Folk Theorems Proof Proof: Suppose for simplicity that there exists an action profile a = (a 1,, a I ) s.t. u i (a) = v [otherwise, we have to consider mixed strategies, which is a little more involved]. Let m i i these the minimax strategy of opponents of i and m i i be i s best response to m i i. Now consider the following grim trigger strategy. For player i: Play (a 1,, a I ) as long as no one deviates. If some player deviates, then play m j i thereafter. We next check if player i can gain by deviating form this strategy profile. If i plays the strategy, his payoff is v i. 25

26 Proof (continued) Networks: Lecture 15 Folk Theorems If i deviates from the strategy in some period t, then denoting v i = max a u i (a), the most that player i could get is given by: [ ] (1 δ) v i + δv i + + δ t 1 v i + δ t v i + δ t+1 v i + δ t+2 v i +. Hence, following the suggested strategy will be optimal if thus if v i 1 δ 1 δt 1 δ v i + δ t v i + δt+1 1 δ v i, v i ( 1 δ t) v i + δ t (1 δ) v i + δ t+1 v i = v i δ t [v i (1 δ)v i δv i ]. The expression in the bracket is non-negative for any v i v i δ δ max. i v i v i This completes the proof. 26

27 Folk Theorems Problems with Nash Folk Theorem The Nash folk theorem states that essentially any payoff can be obtained as a Nash Equilibrium when players are patient enough. However, the corresponding strategies involve this non-forgiving punishments, which may be very costly for the punisher to carry out (i.e., they represent non-credible threats). This implies that the strategies used may not be subgame perfect. The next example illustrates this fact. L (q) R (1 q) U 6, 6 0, 100 D 7, 1 0, 100 The unique NE in this game is (D, L). It can also be seen that the minmax payoffs are given by v 1 = 0, v 2 = 1, and the minmax strategy profile of player 2 is to play R. 27

28 Folk Theorems Problems with the Nash Folk Theorem (continued) Nash Folk Theorem says that (6,6) is possible as a Nash equilibrium payoff of the repeated game, but the strategies suggested in the proof require player 2 to play R in every period following a deviation. While this will hurt player 1, it will hurt player 2 a lot, it seems unreasonable to expect her to carry out the threat. Our next step is to get the payoff (6, 6) in the above example, or more generally, the set of feasible and strictly individually rational payoffs as subgame perfect equilibria payoffs of the repeated game. 28

29 Subgame Perfect Folk Theorem Folk Theorems The first subgame perfect folk theorem shows that any payoff above the static Nash payoffs can be sustained as a subgame perfect equilibrium of the repeated game. Theorem (Friedman) Let a NE be a static equilibrium of the stage game with payoffs e NE. For any feasible payoff v with v i > ei NE for all i I, there exists some δ < 1 such that for all δ > δ, there exists a subgame perfect equilibrium of G (δ) with payoffs v. Proof: Simply construct the non-forgiving trigger strategies with punishment by the static Nash Equilibrium. Punishments are therefore subgame perfect. For δ sufficiently close to 1, it is better for each player i to obtain v i rather than deviate get a high deviation payoff for one period, and then obtain ei NE forever thereafter. 29

30 Folk Theorems Subgame Perfect Folk Theorem (continued) Theorem (Fudenberg and Maskin) Assume that the dimension of the set V of feasible payoffs is equal to the number of players I. Then, for any v V with v i > v i for all i, there exists a discount factor δ < 1 such that for all δ δ, there is a subgame perfect equilibrium of G (δ) with payoffs v. The proof of this theorem is more difficult, but the idea is to use the assumption on the dimension of V to ensure that each player i can be singled out for punishment in the event of a deviation, and then use rewards and punishments for other players to ensure that the deviator can be held down to her minmax payoff. 30

31 Cooperation in Finitely-Repeated Games Cooperation in Finitely-Repeated Games We saw that finitely-repeated games with unique stage equilibrium do not allow corporation or any other outcome than the repetition of this unique equilibrium. But this is no longer the case when there are multiple equilibria in the stage game. Consider the following example A B C A 3, 3 0, 4 2, 0 B 4, 0 1, 1 2, 0 C 0, 2 0, 2 1, 1 The stage game has two pure Nash equilibria (B, B) and (C, C). The most cooperative outcome, (A, A), is not an equilibrium. Main result in example: in the twice repeated version of this game, we can support (A, A) in the first period. 31

32 Cooperation in Finitely-Repeated Games Cooperation in Finitely-Repeated Games (continued) Idea: use the threat of switching to (C, C) in order to support (A, A) in the first period and (B, B) in the second. Suppose, for simplicity, no discounting. If we can support (A, A) in the first period and (B, B) in the second, then each player will receive a payoff of 4. If a player deviates and plays B in the first period, then in the second period the opponent will play C, and thus her best response will be C as well, giving her -1. Thus total payoff will be 3. Therefore, deviation is not profitable. 32

33 Social Preferences How Do People Play Repeated Games? In lab experiments, there is more cooperation in prisoners dilemma games than predicted by theory. More interestingly, cooperation increases as the game is repeated, even if there is only finite rounds of repetition. Why? Most likely, in labs, people are confronted with a payoff matrix of the form: Cooperate Defect Cooperate 1, 1 1, 2 Defect 2, 1 0, 0 Entries are monetary payoffs. But we should really have people s full payoffs. These may differ because of social preferences. 33

34 Social Preferences Networks: Lecture 15 Social Preferences Types of social preferences: 1 Altruism: people receive utility from being nice to others. 2 Fairness: people receive utility from being fair to others. 3 Vindictiveness: people like to punish those deviating from fairness or other accepted norms of behavior. All of these types of social preferences seem to play some role in experimental results. 34

### Jeux finiment répétés avec signaux semi-standards

Jeux finiment répétés avec signaux semi-standards P. Contou-Carrère 1, T. Tomala 2 CEPN-LAGA, Université Paris 13 7 décembre 2010 1 Université Paris 1, Panthéon Sorbonne 2 HEC, Paris Introduction Repeated

### Economics 200C, Spring 2012 Practice Midterm Answer Notes

Economics 200C, Spring 2012 Practice Midterm Answer Notes I tried to write questions that are in the same form, the same length, and the same difficulty as the actual exam questions I failed I think that

### 6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium

6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed

### Economics 431 Homework 5 Due Wed,

Economics 43 Homework 5 Due Wed, Part I. The tipping game (this example due to Stephen Salant) Thecustomercomestoarestaurantfordinnereverydayandisservedbythe same waiter. The waiter can give either good

### Infinitely Repeated Games with Discounting Ù

Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4

### Chapter 12. Repeated Games Finitely-repeated games

Chapter 12 Repeated Games In real life, most games are played within a larger context, and actions in a given situation affect not only the present situation but also the future situations that may arise.

### 6.254 : 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

### Nash Equilibrium. Ichiro Obara. January 11, 2012 UCLA. Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31

Nash Equilibrium Ichiro Obara UCLA January 11, 2012 Obara (UCLA) Nash Equilibrium January 11, 2012 1 / 31 Best Response and Nash Equilibrium In many games, there is no obvious choice (i.e. dominant action).

### 0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text)

September 2 Exercises: Problem 2 (p. 21) Efficiency: p. 28-29: 1, 4, 5, 6 0.0.2 Pareto Efficiency (Sec. 4, Ch. 1 of text) We discuss here a notion of efficiency that is rooted in the individual preferences

### Chapter 4 : Oligopoly.

Chapter 4 : Oligopoly. Oligopoly is the term typically used to describe the situation where a few firms dominate a particular market. The defining characteristic of this type of market structure is that

### Game Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions

Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards

### 6.254 : 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

### Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games

Session 1: 1, 2, 3, 4 Session 2: 5, 6, 8, 9 Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games 1. Consider the following game in the normal form: Player 1 Player

### ECO 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

### I 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,

### Game Theory and Global Warming. Steve Schecter (North Carolina State University) Mary Lou Zeeman (Bowdoin College) Sarah Iams (Cornell University)

Game Theory and Global Warming Steve Schecter (North Carolina State University) Mary Lou Zeeman (Bowdoin College) Sarah Iams (Cornell University) 1 2 What is game theory? What is a game? There are several

### Lecture 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

### When is Reputation Bad? 1

When is Reputation Bad? 1 Jeffrey Ely Drew Fudenberg David K Levine 2 First Version: April 22, 2002 This Version: November 20, 2005 Abstract: In traditional reputation theory, the ability to build a reputation

### Game Theory: Supermodular Games 1

Game Theory: Supermodular Games 1 Christoph Schottmüller 1 License: CC Attribution ShareAlike 4.0 1 / 22 Outline 1 Introduction 2 Model 3 Revision questions and exercises 2 / 22 Motivation I several solution

### Chapter 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)

### Game 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

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### Repeated Games and Reputations

1 / 1 Repeated Games and Reputations George J Mailath UPenn June 26, 2014 9th Tinbergen Institute Conference: 70 Years of Theory of Games and Economic Behaviour The slides and associated bibliography are

### Psychology and Economics (Lecture 17)

Psychology and Economics (Lecture 17) Xavier Gabaix April 13, 2004 Vast body of experimental evidence, demonstrates that discount rates are higher in the short-run than in the long-run. Consider a final

### UCLA. Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory

UCLA Department of Economics Ph. D. Preliminary Exam Micro-Economic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.

### Introduction to Game Theory Lecture Note 3: Mixed Strategies

Introduction to Game Theory Lecture Note 3: Mixed Strategies Haifeng Huang University of California, Merced Mixed strategies and von Neumann-Morgenstern preferences So far what we have considered are pure

### 10 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

### Avoiding the Consolation Prize: The Mathematics of Game Shows

Avoiding the Consolation Prize: The Mathematics of Game Shows STUART GLUCK, Ph.D. CENTER FOR TALENTED YOUTH JOHNS HOPKINS UNIVERSITY STU@JHU.EDU CARLOS RODRIGUEZ CENTER FOR TALENTED YOUTH JOHNS HOPKINS

### 6.254 : Game Theory with Engineering Applications Lecture 10: Evolution and Learning in Games

6.254 : Game Theory with Engineering Applications Lecture 10: and Learning in Games Asu Ozdaglar MIT March 9, 2010 1 Introduction Outline Myopic and Rule of Thumb Behavior arily Stable Strategies Replicator

### What is Evolutionary Game Theory?

What is Evolutionary Game Theory? Origins: Genetics and Biology Explain strategic aspects in evolution of species due to the possibility that individual fitness may depend on population frequency Basic

### 6.207/14.15: 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

### A Simple Model of Price Dispersion *

Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion

### 9 Repeated Games. Tomorrow, and tomorrow, and tomorrow, Creeps in this petty pace from day to day To the last syllable of recorded time Shakespeare

9 Repeated Games Tomorrow, and tomorrow, and tomorrow, Creeps in this petty pace from day to day To the last syllable of recorded time Shakespeare When a game G is repeated an indefinite number of times

### I. Noncooperative Oligopoly

I. Noncooperative Oligopoly Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j s actions affect firm i s profits Example: price

### A Game Theoretical Framework for Adversarial Learning

A Game Theoretical Framework for Adversarial Learning Murat Kantarcioglu University of Texas at Dallas Richardson, TX 75083, USA muratk@utdallas Chris Clifton Purdue University West Lafayette, IN 47907,

### Investing without credible inter-period regulations:

Investing without credible inter-period regulations: A bargaining approach with application to investments in natural resources jell Bjørn Nordal ecember 999 Permanent address and affiliation: Post doctor

### Théorie 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

### Lecture 5: Mixed strategies and expected payoffs

Lecture 5: Mixed strategies and expected payoffs As we have seen for example for the Matching pennies game or the Rock-Paper-scissor game, sometimes game have no Nash equilibrium. Actually we will see

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### An Example of a Repeated Partnership Game with Discounting and with Uniformly Inefficient Equilibria

An Example of a Repeated Partnership Game with Discounting and with Uniformly Inefficient Equilibria Roy Radner; Roger Myerson; Eric Maskin The Review of Economic Studies, Vol. 53, No. 1. (Jan., 1986),

### Sequential 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

### On Stability Properties of Economic Solution Concepts

On Stability Properties of Economic Solution Concepts Richard J. Lipton Vangelis Markakis Aranyak Mehta Abstract In this note we investigate the stability of game theoretic and economic solution concepts

### CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY

CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY EXERCISES 3. Two computer firms, A and B, are planning to market network systems for office information management. Each firm can develop either a fast,

### Mathematical finance and linear programming (optimization)

Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may

### Week 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

### LECTURE 10: GAMES IN EXTENSIVE FORM

LECTURE 10: GAMES IN EXTENSIVE FORM Sequential Move Games 2 so far, we have only dealt with simultaneous games (players make the decisions at the same time, or simply without knowing what the action of

### GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

### A Full Participation Agreement On Global Emission Reduction Through Strategic Investments in R&D

Institut für Volkswirtschaftslehre A Full Participation Agreement On Global Emission Reduction Through Strategic Investments in R&D by Uwe Kratzsch, Gernot Sieg and Ulrike Stegemann Economics Department

### Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti)

Lecture 3: Growth with Overlapping Generations (Acemoglu 2009, Chapter 9, adapted from Zilibotti) Kjetil Storesletten September 10, 2013 Kjetil Storesletten () Lecture 3 September 10, 2013 1 / 44 Growth

### Competition 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

### Equilibrium Paths in Discounted Supergames

Equilibrium Paths in Discounted Supergames Kimmo Berg and Mitri Kitti Aalto University July 13, 2010 Research Questions Setup: infinitely repeated game with discounting perfect monitoring pure strategies

### ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

### Labor Economics, 14.661. Lecture 3: Education, Selection, and Signaling

Labor Economics, 14.661. Lecture 3: Education, Selection, and Signaling Daron Acemoglu MIT November 3, 2011. Daron Acemoglu (MIT) Education, Selection, and Signaling November 3, 2011. 1 / 31 Introduction

### WEAK DOMINANCE: A MYSTERY CRACKED

WEAK DOMINANCE: A MYSTERY CRACKED JOHN HILLAS AND DOV SAMET Abstract. What strategy profiles can be played when it is common knowledge that weakly dominated strategies are not played? A comparison to the

### Oligopoly 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

### Decision Making under Uncertainty

6.825 Techniques in Artificial Intelligence Decision Making under Uncertainty How to make one decision in the face of uncertainty Lecture 19 1 In the next two lectures, we ll look at the question of how

### ECON 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,

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### ECON 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

### Notes from Week 1: Algorithms for sequential prediction

CS 683 Learning, Games, and Electronic Markets Spring 2007 Notes from Week 1: Algorithms for sequential prediction Instructor: Robert Kleinberg 22-26 Jan 2007 1 Introduction In this course we will be looking

### The Market-Clearing Model

Chapter 5 The Market-Clearing Model Most of the models that we use in this book build on two common assumptions. First, we assume that there exist markets for all goods present in the economy, and that

### Lecture 1 EVOLUTIONARY GAME THEORY Toulouse School of Economics

Lecture 1 EVOLUTIONARY GAME THEORY Toulouse School of Economics Jörgen Weibull November 13, 2012 1 Economic theory and as if rationality The rationalistic paradigm in economics: Savage rationality [Leonard

### The Multiplicative Weights Update method

Chapter 2 The Multiplicative Weights Update method The Multiplicative Weights method is a simple idea which has been repeatedly discovered in fields as diverse as Machine Learning, Optimization, and Game

### Math 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

### Lecture 11: Oligopoly and Strategic Behavior

Lecture 11: Oligopoly and Strategic Behavior Few Firms in the Market: Each aware of others actions Each firm in the industry has market power Entry is Feasible, although incumbent(s) may try to deter it.

### Content. From Nash to Nash s Game Theory -- an explanation of the mathematics in the movie A Beautiful Mind and John Nash's Nobel-winning theory

From Nash to Nash s Game Theory -- an explanation of the mathematics in the movie A Beautiful Mind and John Nash's Nobel-winning theory Dr. Ng Tuen Wai Department of Mathematics, HKU Content Life of John

### 10.2 Series and Convergence

10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

### Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

### Evolutionary Game Theory

Evolutionary Game Theory James Holland Jones Department of Anthropology Stanford University December 1, 2008 1 The Evolutionarily Stable Strategy (ESS) An evolutionarily stable strategy (ESS) is a strategy

### AN INTRODUCTION TO GAME THEORY

AN INTRODUCTION TO GAME THEORY 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. MARTIN J. OSBORNE University of Toronto

### University of Oslo Department of Economics

University of Oslo Department of Economics Exam: ECON3200/4200 Microeconomics and game theory Date of exam: Tuesday, November 26, 2013 Grades are given: December 17, 2013 Duration: 14:30-17:30 The problem

### Game 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

### On Reputation in Game Theory Application on Online Settings

On Reputation in Game Theory Application on Online Settings Karl Aberer, Zoran Despotovic Distributed Information Systems Laboratory (LSIR) Institute of Core Computing Sciences (IIF) Swiss Federal Institute

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### Tiers, Preference Similarity, and the Limits on Stable Partners

Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### 1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

### Bad Reputation. March 26, 2002

Bad Reputation Jeffrey C. Ely Juuso Välimäki March 26, 2002 Abstract We study the classical reputation model in which a long-run player faces a sequence of short-run players and would like to establish

### 1 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

### 10 Dynamic Games of Incomple Information

10 Dynamic Games of Incomple Information DRAW SELTEN S HORSE In game above there are no subgames. Hence, the set of subgame perfect equilibria and the set of Nash equilibria coincide. However, the sort

### Games 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

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

### Credible Termination Threats, Reputation and Private Monitoring.

Credible Termination Threats, Reputation and Private Monitoring. Olivier Compte First Version: June 2001 This Version: April 2005 Abstract In repeated principal-agent relationships, termination threats

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### Economics of Insurance

Economics of Insurance In this last lecture, we cover most topics of Economics of Information within a single application. Through this, you will see how the differential informational assumptions allow

### Program Equilibrium. Moshe Tennenholtz Faculty of Industrial Engineering and Management Technion Israel Institute of Technology Haifa 32000, Israel

Program Equilibrium Moshe Tennenholtz Faculty of Industrial Engineering and Management Technion Israel Institute of Technology Haifa 32000, Israel This work has been carried out while the author was with

### Decentralised bilateral trading, competition for bargaining partners and the law of one price

ECONOMICS WORKING PAPERS Decentralised bilateral trading, competition for bargaining partners and the law of one price Kalyan Chatterjee Kaustav Das Paper Number 105 October 2014 2014 by Kalyan Chatterjee

### Game 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

### Equilibrium computation: Part 1

Equilibrium computation: Part 1 Nicola Gatti 1 Troels Bjerre Sorensen 2 1 Politecnico di Milano, Italy 2 Duke University, USA Nicola Gatti and Troels Bjerre Sørensen ( Politecnico di Milano, Italy, Equilibrium

### Unraveling 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

### Biology X: Introduction to (evolutionary) game theory

Biology X: Introduction to (evolutionary) game theory Andreas Witzel see Gintis, Game theory evolving, Chapters 2, 3, 10 2010-09-20 Overview Evolutionary game theory: motivation and background Classical

### Working Paper An exact non-cooperative support for the sequential Raiffa solution. Working papers // Institute of Mathematical Economics, No.

econstor www.econstor.eu Der Open-Access-Publikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Trockel,

### An Extension Model of Financially-Balanced Bonus-Malus System

An Extension Model of Financially-Balanced Bonus-Malus System Xiao Yugu 1), Meng Shengwang 1), Robert Conger 2) 1. School of Statistics, Renmin University of China 2. Towers Perrin, Chicago Contents Introduction

### Midterm Advanced Economic Theory, ECO326F1H Marcin Pęski February, 2015

Midterm Advanced Economic Theory, ECO326F1H Marcin Pęski February, 2015 There are three questions with total worth of 100 points. Read the questions carefully. You must give a supporting argument and an

### Introduction to Game Theory Lecture 7: Bayesian Games

Introduction to Game Theory Lecture 7: Bayesian Games Haifeng Huang University of California, Merced Shanghai, Summer 2011 Games of Incomplete Information: Bayesian Games In the games we have studies so

### Chapter 11: Game Theory & Competitive Strategy

Chapter 11: Game Theory & Competitive Strategy Game theory helps to understand situations in which decision-makers strategically interact. A game in the everyday sense is: A competitive activity... according

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650