Microeconomics of Banking: Lecture 5

Size: px
Start display at page:

Download "Microeconomics of Banking: Lecture 5"

Transcription

1 Microeconomics of Banking: Lecture 5 Prof. Ronaldo CARPIO Oct. 23, 2015

2 Administrative Stuff Homework 2 is due next week. Due to the change in material covered, I have decided to change the grading system for the course. We will have an open-book midterm and final exam. Homework: 15 %, Midterm: 35 %, Final: 50 % The midterm exam will be on Nov. 6, and will cover the general equilibrium model of asset pricing and game theory. There will be a true-false section, and 3 problems similar to the ones on the homeworks.

3 Review of Last Week A game is a model of a strategic situation, in which there are many decision-makers that can affect each other. We formulate a strategic game as having three components: The players For each player, a set of actions For each player, preferences over all possible outcomes. An outcome is determined by the actions chosen by all players.

4 Review of Last Week Game theory is the analysis of strategic situations. We want to have some way of predicting the outcome (i.e. the choices of all players) of a situation. Complete prediction is difficult, so we can try an easier task: find a steady state. A Nash equilibrium (NE) is a steady state, under the assumption that all players choose their actions unilaterally (i.e. acting alone). In a NE, no player has an incentive to deviate (i.e. change his action). Note that this doesn t say anything about how players learn to find NE or which NE (if there are many) is chosen.

5 Prisoner s Dilemma Player 1 Player 2 Q F Q 2,2 0,3 F 3,0 1,1 Each player has 2 actions: Q and F. Each cell shows the payoffs to the players if the corresponding action is chosen. (F, F ) is the unique Nash equilibrium.

6 Best Response Functions Suppose that the players other than Player i play the action list a i. Let B i (a i ) be the set of Player i s best (i.e. payoff - maximizing) actions, given that the other players play a i. (There may be more than one). B i is called the best response function of Player i. B i is a set-valued function, that is, it may give a result with more than one element. Every member of B i (a i ) is a best response of Player i to a i.

7 Using Best Response Functions to find Nash Eq. Proposition: The action profile a is a Nash equilibrium if and only if every player s action is a best response to the other players actions: a i B i (a i) for every player i (1) If the best-response function is single-valued: Let bi (ai ) be the single member of B i(a i ), i.e. B i (a i ) = {b i(ai )}. Then condition 1 is equivalent to: a i = b i (a i) for every player i (2) If the best-response function is single-valued and there are 2 players, condition 1 is equivalent to: a 1 = b 1 (a 2) a 2 = b 2 (a 1)

8 Prisoner s Dilemma Q F Q 2,2 0,3 F 3,0 1,1 B i (Q) = {F } for i = 1, 2 B i (F ) = {F } for i = 1, 2

9 BoS Bach Stravinsky Bach 2, 1 0, 0 Stravinsky 0, 0 1, 2 B i (Bach) = {Bach} for i = 1, 2 B i (Stravinsky) = {Stravinsky} for i = 1, 2

10 Matching Pennies Head Tail Head 1,-1-1,1 Tail -1,1 1,-1 B 1 (Head) = {Head} B 2 (Head) = {Tail} B 1 (Tail) = {Tail} B 2 (Tail) = {Head}

11 L M R T 1,1 1,0 0,1 B 1,0 0,1 1,0 B 1 (L) = {T, B} B 1 (M) = {T } B 1 (R) = {B} B 2 (T ) = {L, R} B 2 (B) = {M}

12 Finding Nash equilibrium with Best-Response functions We can use this to find Nash equilibria when the action space is continuous. Step 1: Calculate the best-response functions. Step 2: Find an action profile a that satisfies: a i B i (a i) for every player i Or, if every player s best-response function is single-valued, find a solution of the n equations (n is the number of players): a i = b i (a i) for every player i

13 Example: synergistic relationship (37.2 in book) Two individuals. Each decides how much effort to devote to relationship. Amount of effort a i is a non-negative real number (so the action space is infinite) Payoff to Player i: u i (a i ) = a i (c + a j a i ), where c > 0 is a constant.

14 Finding the Nash Equilibrium Construct players best-response functions: Player i s payoff function: u i (a i ) = a i (c + a j a i ) Given a j, this becomes a quadratic: u i (a i ) = a i c + a i a j a 2 i Best response to a j is when this quadratic is maximized. Take the derivative and set to 0. c + a j 2a i = 0 a i = c + a j 2 So, best response functions are: b1 (a 2 ) = c+a2 2 b 2 (a 1 ) = c+a1 2

15 Finding the Nash Equilibrium The pair (a 1, a 2 ) is a Nash equilibrium if a 1 = b 1 (a 2 ) and a 2 = b 2 (a 1 ). Solving the two equations a 1 = c + a 2 2 a 2 = c + a 1 2 gives a unique solution (c, c). Therefore, this game has a unique Nash equilibrium: a 1 = c, a 2 = c.

16 Finding the Nash Equilibrium The intersection of b 1 (a 2 ) = c+a2 and b 2 2 (a 1 ) = c+a1 is the Nash 2 equilibrium. Note that using calculus to find the best response requires that the payoffs are concave.

17 Direct Proof of Nash Equilibrium Sometimes, the only way to find the set of NE is to classify all possible outcomes into cases, and prove that each case is a NE or not. Consider the game we saw last week: guess 2 3 Assume there are 3 players. of the average. Players: 3 people. Action set: player i chooses a number xi [0, 100]. Preferences: The k players whose xi is closest to 2 3 (x 1 + x 2 + x 3 )/3 gets a payoff of 1/k. Everyone else gets a payoff of 0.

18 Direct Proof of Nash Equilibrium Case 1: x 1 = x 2 = x 3 = 0. All players get a payoff of 1/ (x 1 + x 2 + x 3 )/3 = 0 Suppose player i deviates, by choosing x i = y > 0. The new average becomes 2 y 3 3. Player i s distance to the average is y 2 y 3 3 = y (1 2 9 ). The distance of the other players to the average is 2 y, which is 3 3 smaller. Player i s payoff goes from 1/3 to 0, so he has no incentive to deviate. Therefore, this case is a Nash equilibrium.

19 Direct Proof of Nash Equilibrium Case 2: x 1 = x 2 = x 3 = x > 0. All players get a payoff of 1/ (x 1 + x 2 + x 3 )/3 = 2 3 x Suppose player i switches from x to x/2. The average goes down by x 9 to 5x 9. Player i becomes closest and gets a payoff of 1, and therefore has an incentive to deviate. This case is not a Nash equilibrium.

20 Direct Proof of Nash Equilibrium Case 3: Any other combination of x 1, x 2, x 3. At least one player is not one of the closest, and gets a payoff of 0. This player can always increase his payoff by changing his number to something closer to the average. Therefore, this case is not a Nash equilibrium.

21 Extensive Form Games (Chapter 5) So far, we ve been using strategic form (or normal form) games. All players are assumed to move simultaneously. This cannot capture a sequential situation, where one player moves, then another... Or, if one player can get information on the moves of the other players, before making his own move. We will introduce a way of specifying a game that allows this.

22 Example: An Entry Game Suppose we have a situation where there is an incumbent and a challenger. For example, an industry might have an established dominant firm. A challenger firm is deciding whether it wants to enter this industry and compete with the incumbent. If the challenger enters, the incumbent chooses whether to engage in intense (and possibly costly) competition, or to accept the challenger s entry.

23 Entry Game There are two players: the incumbent and the challenger. The challenger moves first, has two actions: In and Out. If the challenger chooses In, the incumbent chooses Fight or Acquiesce. Challenger s preference over outcomes: (In, Acquiesce) > (Out) > (In, Fight) Incumbent s preference over outcomes: (Out) > (In, Acquiesce) > (In, Fight) We can represent these preferences with the payoff functions (challenger is u 1 ): u 1 (In, Acquiesce) = 2, u 1 (Out) = 1, u 1 (In, Fight) = 0 u 2 (Out) = 2, u 2 (In, Acquiesce) = 1, u 2 (In, Fight) = 0

24 Game Tree We can represent this game with a tree diagram. The root node of the tree is the first move in the game (here, by the challenger). Each action at a node corresponds to a branch in the tree. Outcomes are leaf nodes (i.e. there are no more branches). The first number at each outcome is the payoff to the first player (the challenger).

25 Formal Specification of an Extensive Game Formally, we need to specify all possible sequences of actions, and all possible outcomes. A history is the sequence of actions played from the beginning, up to some point in the game. In the tree, a history is a path from the root to some node in the tree. In the entry game, all possible histories are: (i.e. at the beginning, no actions played yet), (In), (Out), (In, Acquiesce), (In, Fight). A terminal history is a sequence of actions that specifies an outcome, which is what players have preferences over. In the tree, a terminal history is a path from the root to a leaf node (a node with no branches). In the entry game, the terminal histories are: (Out), (In, Acquiesce), (In, Fight). A player function specifies whose turn it is to move, at every non-terminal history (every non-leaf node in the tree).

26 Formal Specification of an Extensive Game An extensive game is specified by four components: A set of players A set of terminal histories, with the property that no terminal history can be a subsequence of some other terminal history A player function that assigns a player to every non-terminal history For each player, preferences over the set of terminal histories The sequence of moves and the set of actions at each node are implicitly determined by these components. In practice, we will use trees to specify extensive games.

27 Solutions to Entry Game How can we find the solution to this game? First approach: Each player will imagine what will happen in future nodes, and use that to determine his choice in current nodes. Suppose we re at the node just after the challenger plays In. At this point, the payoff-maximizing choice for the incumbent is Acquiesce, which gives a payoff pair (2,1). So, at the beginning, the challenger might assume playing In gives a payoff pair of (2,1), which gives a higher payoff than Out. This approach is called backwards induction: imagining what will happen at the end, and using that to determine what to do in earlier situations.

28 Backwards Induction At each move, for each action, a player deduces the actions that all players will rationally take in the future. This gives the outcome that will occur (assuming everyone behaves rationally), and therefore gives the payoff to each current action. However, in some cases, backwards induction doesn t give a clear prediction about what will happen. In this version of the Entry Game, both Acquiesce, Fight give the same payoff to the incumbent. Unclear what to believe at the beginning of the game. Also, games with infinitely long histories (e.g. an infinitely repeating game).

29 Strategies in Extensive Form Games Another approach is to formulate this as a strategic game, then use the Nash equilibrium solution concept. We need to expand the action sets of the players to take into account the different actions at each node. For each player i, we will specify the action chosen at all of i s nodes, i.e. every history after which it s i s turn to move Definition: A strategy of player i in an extensive game with perfect information is a function that assigns to each history h after which it is i s turn to move, an action in A(h) (the actions available after h).

30 In this game, Player 1 only moves at the start (i.e. after the empty history ). The actions available are C, D, so Player 1 has two strategies: C, D. Player 2 moves after the history C and also after D. After C, available actions are E, F. After D, available actions are G, H. Player 2 has four strategies: In this case, it s simple enough to write them together. We can refer to these strategies as EG, EH, FG, FH. The first action corresponds to the first history C.

31 Strategies in Extensive Form Games We can think of a strategy as an action plan or contingency plan: If Player 1 chooses action X, do Y. However, a strategy must specify an action for all histories, even if they do not occur due to previous choices in the strategy. In this example, a strategy for Player 1 must specify an action for the history (C, E), even if it specifies D at the beginning. Think of this as allowing for the possibility of mistakes in execution.

32 Strategy Profiles & Outcomes As before, a strategy profile is a list of the strategies of all players. Given a strategy profile s, the terminal history that results by executing the actions specified by s is denoted O(s), the outcome of s. For example, in this game, the outcome of the strategy pair (DG, E) is the terminal history D. The outcome of (CH, E) is the terminal history (C, E, H).

33 Nash Equilibrium Definition The strategy profile s in an extensive game with perfect information is a Nash equilibrium if, for every player i and strategy r i of player i, the outcome O(s ) is at least as good as the outcome O(r i, s i ) generated by any other strategy profile (r i, s i ) in which player i chooses r i: u i (O(s )) u i (O(r i, s i)) for every strategy r i of player i We can construct the strategic form of an extensive game by listing all strategies of all players and finding the outcome.

34 Strategic Form of Entry Game The strategic form of the Entry Game is: There are two Nash equilibria: (In, Acquiesce) and (Out, Fight). The first NE is the same as the one found with backwards induction. In the second NE, the incumbent chooses Fight. However, if In is taken as given, this is not rational. This is called an incredible threat. If the incumbent could commit to Fight at the beginning of the game, it would be credible.

35 Subgames The concept of Nash equilibrium ignores the sequential structure of an extensive game. It treats strategies as choices made once and for all at the beginning of the game. However, the equilibria of this method may contain incredible threats. We ll define a notion of equilibrium that excludes incredible situations. Suppose Γ is an extensive form game with perfect information. The subgame following a non-terminal history h, Γ(h), is the game beginning at the point just after h. A proper subgame is a subgame that is not Γ itself.

36 Subgames This game has two proper subgames:

37 Subgame Perfect Equilibria A subgame perfect equilibrium is a strategy profile s in which each subgame s strategy profile is also a Nash equilibrium. Each player s strategy must be optimal for all subgames that have him moving at the beginning, not just the entire game. (Out, Fight) is a NE, but is not a subgame perfect equilibrium because in the subgame following In, the strategy Fight is not optimal for the incumbent.

38 Subgame Perfect Equilibria Every subgame perfect equilibrium is also a Nash equilibrium, but not vice versa. A subgame perfect equilibrium induces a Nash equilibrium in every subgame. In games with finite histories, subgame perfect equilibria are consistent with backwards induction.

39 Administrative Stuff Homework 2 is due next week. Due to the change in material covered, I have decided to change the grading system for the course. We will have an open-book midterm and final exam. Homework: 15 %, Midterm: 35 %, Final: 50 % The midterm exam will be on Nov. 6, and will cover the general equilibrium model of asset pricing and game theory. There will be a true-false section, and 3 problems similar to the ones on the homeworks.

Lecture 6 Dynamic games with imperfect information

Lecture 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 information

CUR 412: Game Theory and its Applications, Lecture 9

CUR 412: Game Theory and its Applications, Lecture 9 CUR 412: Game Theory and its Applications, Lecture 9 Prof. Ronaldo CARPIO May 22, 2015 Announcements HW #3 is due next week. Ch. 6.1: Ultimatum Game This is a simple game that can model a very simplified

More information

CUR 412: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 2015

CUR 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 information

CUR 412: Game Theory and its Applications, Lecture 12

CUR 412: Game Theory and its Applications, Lecture 12 CUR 412: Game Theory and its Applications, Lecture 12 Prof. Ronaldo CARPIO May 24, 2016 Announcements Homework #4 is due next week. Review of Last Lecture In extensive games with imperfect information,

More information

Introduction to Multi-Agent Programming

Introduction to Multi-Agent Programming Introduction to Multi-Agent Programming 10. Game Theory Strategic Reasoning and Acting Alexander Kleiner and Bernhard Nebel Strategic Game A strategic game G consists of a finite set N (the set of players)

More information

Introduction 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) 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 information

Problem 3 Solutions. l 3 r, 1

Problem 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 information

G5212: Game Theory. Mark Dean. Spring 2017

G5212: Game Theory. Mark Dean. Spring 2017 G5212: Game Theory Mark Dean Spring 2017 Modelling Dynamics Up until now, our games have lacked any sort of dynamic aspect We have assumed that all players make decisions at the same time Or at least no

More information

Duopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two-stage prisoner s dilemma

Duopoly 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 information

Econ 711 Homework 1 Solutions

Econ 711 Homework 1 Solutions Econ 711 Homework 1 s January 4, 014 1. 1 Symmetric, not complete, not transitive. Not a game tree. Asymmetric, not complete, transitive. Game tree. 1 Asymmetric, not complete, transitive. Not a game tree.

More information

1 Solutions to Homework 3

1 Solutions to Homework 3 1 Solutions to Homework 3 1.1 163.1 (Nash equilibria of extensive games) 1. 164. (Subgames) Karl R E B H B H B H B H B H B H There are 6 proper subgames, beginning at every node where or chooses an action.

More information

CMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies

CMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies CMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies Mohammad T. Hajiaghayi University of Maryland Behavioral Strategies In imperfect-information extensive-form games, we can define

More information

Answers to Problem Set 4

Answers to Problem Set 4 Answers to Problem Set 4 Economics 703 Spring 016 1. a) The monopolist facing no threat of entry will pick the first cost function. To see this, calculate profits with each one. With the first cost function,

More information

GAME THEORY: DYNAMIC. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Dynamic Game Theory

GAME THEORY: DYNAMIC. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Dynamic Game Theory Prerequisites Almost essential Game Theory: Strategy and Equilibrium GAME THEORY: DYNAMIC MICROECONOMICS Principles and Analysis Frank Cowell April 2018 1 Overview Game Theory: Dynamic Mapping the temporal

More information

UC 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 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 information

Economics 171: Final Exam

Economics 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 information

CUR 412: Game Theory and its Applications, Lecture 11

CUR 412: Game Theory and its Applications, Lecture 11 CUR 412: Game Theory and its Applications, Lecture 11 Prof. Ronaldo CARPIO May 17, 2016 Announcements Homework #4 will be posted on the web site later today, due in two weeks. Review of Last Week An extensive

More information

CUR 412: Game Theory and its Applications, Lecture 4

CUR 412: Game Theory and its Applications, Lecture 4 CUR 412: Game Theory and its Applications, Lecture 4 Prof. Ronaldo CARPIO March 27, 2015 Homework #1 Homework #1 will be due at the end of class today. Please check the website later today for the solutions

More information

Finitely repeated simultaneous move game.

Finitely 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 information

CHAPTER 15 Sequential rationality 1-1

CHAPTER 15 Sequential rationality 1-1 . CHAPTER 15 Sequential rationality 1-1 Sequential irrationality Industry has incumbent. Potential entrant chooses to go in or stay out. If in, incumbent chooses to accommodate (both get modest profits)

More information

CUR 412: Game Theory and its Applications, Lecture 4

CUR 412: Game Theory and its Applications, Lecture 4 CUR 412: Game Theory and its Applications, Lecture 4 Prof. Ronaldo CARPIO March 22, 2015 Homework #1 Homework #1 will be due at the end of class today. Please check the website later today for the solutions

More information

In reality; some cases of prisoner s dilemma end in cooperation. Game Theory Dr. F. Fatemi Page 219

In reality; some cases of prisoner s dilemma end in cooperation. Game Theory Dr. F. Fatemi Page 219 Repeated Games Basic lesson of prisoner s dilemma: In one-shot interaction, individual s have incentive to behave opportunistically Leads to socially inefficient outcomes In reality; some cases of prisoner

More information

Notes for Section: Week 4

Notes 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 information

Economics 109 Practice Problems 1, Vincent Crawford, Spring 2002

Economics 109 Practice Problems 1, Vincent Crawford, Spring 2002 Economics 109 Practice Problems 1, Vincent Crawford, Spring 2002 P1. Consider the following game. There are two piles of matches and two players. The game starts with Player 1 and thereafter the players

More information

Game Theory. Wolfgang Frimmel. Repeated Games

Game 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 information

Prisoner s dilemma with T = 1

Prisoner 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 information

Game Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati.

Game Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati. Game Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati. Module No. # 06 Illustrations of Extensive Games and Nash Equilibrium

More information

Game Theory. Important Instructions

Game Theory. Important Instructions Prof. Dr. Anke Gerber Game Theory 2. Exam Summer Term 2012 Important Instructions 1. There are 90 points on this 90 minutes exam. 2. You are not allowed to use any material (books, lecture notes etc.).

More information

CHAPTER 14: REPEATED PRISONER S DILEMMA

CHAPTER 14: REPEATED PRISONER S DILEMMA CHAPTER 4: REPEATED PRISONER S DILEMMA In this chapter, we consider infinitely repeated play of the Prisoner s Dilemma game. We denote the possible actions for P i by C i for cooperating with the other

More information

Advanced Micro 1 Lecture 14: Dynamic Games Equilibrium Concepts

Advanced Micro 1 Lecture 14: Dynamic Games Equilibrium Concepts Advanced Micro 1 Lecture 14: Dynamic Games quilibrium Concepts Nicolas Schutz Nicolas Schutz Dynamic Games: quilibrium Concepts 1 / 79 Plan 1 Nash equilibrium and the normal form 2 Subgame-perfect equilibrium

More information

The Ohio State University Department of Economics Second Midterm Examination Answers

The Ohio State University Department of Economics Second Midterm Examination Answers Econ 5001 Spring 2018 Prof. James Peck The Ohio State University Department of Economics Second Midterm Examination Answers Note: There were 4 versions of the test: A, B, C, and D, based on player 1 s

More information

Stochastic Games and Bayesian Games

Stochastic Games and Bayesian Games Stochastic Games and Bayesian Games CPSC 532L Lecture 10 Stochastic Games and Bayesian Games CPSC 532L Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games Stochastic Games

More information

Exercises Solutions: Game Theory

Exercises Solutions: Game Theory Exercises Solutions: Game Theory Exercise. (U, R).. (U, L) and (D, R). 3. (D, R). 4. (U, L) and (D, R). 5. First, eliminate R as it is strictly dominated by M for player. Second, eliminate M as it is strictly

More information

1 x i c i if x 1 +x 2 > 0 u i (x 1,x 2 ) = 0 if x 1 +x 2 = 0

1 x i c i if x 1 +x 2 > 0 u i (x 1,x 2 ) = 0 if x 1 +x 2 = 0 Game Theory - Midterm Examination, Date: ctober 14, 017 Total marks: 30 Duration: 10:00 AM to 1:00 PM Note: Answer all questions clearly using pen. Please avoid unnecessary discussions. In all questions,

More information

Not 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.

Not 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 information

Introduction to Game Theory Lecture Note 5: Repeated Games

Introduction 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 information

Microeconomics II. CIDE, MsC Economics. List of Problems

Microeconomics 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 information

An introduction on game theory for wireless networking [1]

An introduction on game theory for wireless networking [1] An introduction on game theory for wireless networking [1] Ning Zhang 14 May, 2012 [1] Game Theory in Wireless Networks: A Tutorial 1 Roadmap 1 Introduction 2 Static games 3 Extensive-form games 4 Summary

More information

Sequential-move games with Nature s moves.

Sequential-move games with Nature s moves. Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 3. GAMES WITH SEQUENTIAL MOVES Game trees. Sequential-move games with finite number of decision notes. Sequential-move games with Nature s moves. 1 Strategies in

More information

Iterated Dominance and Nash Equilibrium

Iterated Dominance and Nash Equilibrium Chapter 11 Iterated Dominance and Nash Equilibrium In the previous chapter we examined simultaneous move games in which each player had a dominant strategy; the Prisoner s Dilemma game was one example.

More information

Sequential Rationality and Weak Perfect Bayesian Equilibrium

Sequential Rationality and Weak Perfect Bayesian Equilibrium Sequential Rationality and Weak Perfect Bayesian Equilibrium Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu June 16th, 2016 C. Hurtado (UIUC - Economics)

More information

PRISONER S DILEMMA. Example from P-R p. 455; also 476-7, Price-setting (Bertrand) duopoly Demand functions

PRISONER 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 information

Stochastic Games and Bayesian Games

Stochastic Games and Bayesian Games Stochastic Games and Bayesian Games CPSC 532l Lecture 10 Stochastic Games and Bayesian Games CPSC 532l Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games 4 Analyzing Bayesian

More information

Economics 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 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 information

Economic Management Strategy: Hwrk 1. 1 Simultaneous-Move Game Theory Questions.

Economic Management Strategy: Hwrk 1. 1 Simultaneous-Move Game Theory Questions. Economic Management Strategy: Hwrk 1 1 Simultaneous-Move Game Theory Questions. 1.1 Chicken Lee and Spike want to see who is the bravest. To do so, they play a game called chicken. (Readers, don t try

More information

Extensive-Form Games with Imperfect Information

Extensive-Form Games with Imperfect Information May 6, 2015 Example 2, 2 A 3, 3 C Player 1 Player 1 Up B Player 2 D 0, 0 1 0, 0 Down C Player 1 D 3, 3 Extensive-Form Games With Imperfect Information Finite No simultaneous moves: each node belongs to

More information

Exercises Solutions: Oligopoly

Exercises 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 information

M.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. 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 information

6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts

6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts 6.254 : Game Theory with Engineering Applications Lecture 3: Strategic Form Games - Solution Concepts Asu Ozdaglar MIT February 9, 2010 1 Introduction Outline Review Examples of Pure Strategy Nash Equilibria

More information

1 R. 2 l r 1 1 l2 r 2

1 R. 2 l r 1 1 l2 r 2 4. Game Theory Midterm I Instructions. This is an open book exam; you can use any written material. You have one hour and 0 minutes. Each question is 35 points. Good luck!. Consider the following game

More information

G5212: Game Theory. Mark Dean. Spring 2017

G5212: 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 information

(a) (5 points) Suppose p = 1. Calculate all the Nash Equilibria of the game. Do/es the equilibrium/a that you have found maximize social utility?

(a) (5 points) Suppose p = 1. Calculate all the Nash Equilibria of the game. Do/es the equilibrium/a that you have found maximize social utility? GAME THEORY EXAM (with SOLUTIONS) January 20 P P2 P3 P4 INSTRUCTIONS: Write your answers in the space provided immediately after each question. You may use the back of each page. The duration of this exam

More information

Player 2 L R M H a,a 7,1 5,0 T 0,5 5,3 6,6

Player 2 L R M H a,a 7,1 5,0 T 0,5 5,3 6,6 Question 1 : Backward Induction L R M H a,a 7,1 5,0 T 0,5 5,3 6,6 a R a) Give a definition of the notion of a Nash-Equilibrium! Give all Nash-Equilibria of the game (as a function of a)! (6 points) b)

More information

Simon Fraser University Fall Econ 302 D200 Final Exam Solution Instructor: Songzi Du Wednesday December 16, 2015, 8:30 11:30 AM

Simon Fraser University Fall Econ 302 D200 Final Exam Solution Instructor: Songzi Du Wednesday December 16, 2015, 8:30 11:30 AM Simon Fraser University Fall 2015 Econ 302 D200 Final Exam Solution Instructor: Songzi Du Wednesday December 16, 2015, 8:30 11:30 AM NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect

More information

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.

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. 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 information

ECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves

ECE 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 information

February 23, An Application in Industrial Organization

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 information

Game 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 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 information

Agenda. Game Theory Matrix Form of a Game Dominant Strategy and Dominated Strategy Nash Equilibrium Game Trees Subgame Perfection

Agenda. Game Theory Matrix Form of a Game Dominant Strategy and Dominated Strategy Nash Equilibrium Game Trees Subgame Perfection Game Theory 1 Agenda Game Theory Matrix Form of a Game Dominant Strategy and Dominated Strategy Nash Equilibrium Game Trees Subgame Perfection 2 Game Theory Game theory is the study of a set of tools that

More information

Chapter 8. Repeated Games. Strategies and payoffs for games played twice

Chapter 8. Repeated Games. Strategies and payoffs for games played twice Chapter 8 epeated Games 1 Strategies and payoffs for games played twice Finitely repeated games Discounted utility and normalized utility Complete plans of play for 2 2 games played twice Trigger strategies

More information

Introduction to Game Theory

Introduction to Game Theory Introduction to Game Theory What is a Game? A game is a formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence. By that, we mean that each

More information

SI 563 Homework 3 Oct 5, Determine the set of rationalizable strategies for each of the following games. a) X Y X Y Z

SI 563 Homework 3 Oct 5, Determine the set of rationalizable strategies for each of the following games. a) X Y X Y Z SI 563 Homework 3 Oct 5, 06 Chapter 7 Exercise : ( points) Determine the set of rationalizable strategies for each of the following games. a) U (0,4) (4,0) M (3,3) (3,3) D (4,0) (0,4) X Y U (0,4) (4,0)

More information

Continuing game theory: mixed strategy equilibrium (Ch ), optimality (6.9), start on extensive form games (6.10, Sec. C)!

Continuing game theory: mixed strategy equilibrium (Ch ), optimality (6.9), start on extensive form games (6.10, Sec. C)! CSC200: Lecture 10!Today Continuing game theory: mixed strategy equilibrium (Ch.6.7-6.8), optimality (6.9), start on extensive form games (6.10, Sec. C)!Next few lectures game theory: Ch.8, Ch.9!Announcements

More information

Economics 335 March 2, 1999 Notes 6: Game Theory

Economics 335 March 2, 1999 Notes 6: Game Theory Economics 335 March 2, 1999 Notes 6: Game Theory I. Introduction A. Idea of Game Theory Game theory analyzes interactions between rational, decision-making individuals who may not be able to predict fully

More information

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.

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. 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 information

1 Solutions to Homework 4

1 Solutions to Homework 4 1 Solutions to Homework 4 1.1 Q1 Let A be the event that the contestant chooses the door holding the car, and B be the event that the host opens a door holding a goat. A is the event that the contestant

More information

Economics 502 April 3, 2008

Economics 502 April 3, 2008 Second Midterm Answers Prof. Steven Williams Economics 502 April 3, 2008 A full answer is expected: show your work and your reasoning. You can assume that "equilibrium" refers to pure strategies unless

More information

CS 798: Homework Assignment 4 (Game Theory)

CS 798: Homework Assignment 4 (Game Theory) 0 5 CS 798: Homework Assignment 4 (Game Theory) 1.0 Preferences Assigned: October 28, 2009 Suppose that you equally like a banana and a lottery that gives you an apple 30% of the time and a carrot 70%

More information

Economics 51: Game Theory

Economics 51: Game Theory Economics 51: Game Theory Liran Einav April 21, 2003 So far we considered only decision problems where the decision maker took the environment in which the decision is being taken as exogenously given:

More information

Introduction to Game Theory

Introduction 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 information

13.1 Infinitely Repeated Cournot Oligopoly

13.1 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 information

Microeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 2017

Microeconomic 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 information

Mohammad Hossein Manshaei 1394

Mohammad Hossein Manshaei 1394 Mohammad Hossein Manshaei manshaei@gmail.com 1394 Let s play sequentially! 1. Sequential vs Simultaneous Moves. Extensive Forms (Trees) 3. Analyzing Dynamic Games: Backward Induction 4. Moral Hazard 5.

More information

MA200.2 Game Theory II, LSE

MA200.2 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

REPEATED GAMES. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Repeated Games. Almost essential Game Theory: Dynamic.

REPEATED GAMES. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Repeated Games. Almost essential Game Theory: Dynamic. Prerequisites Almost essential Game Theory: Dynamic REPEATED GAMES MICROECONOMICS Principles and Analysis Frank Cowell April 2018 1 Overview Repeated Games Basic structure Embedding the game in context

More information

Name. Answers Discussion Final Exam, Econ 171, March, 2012

Name. 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 information

Game Theory Week 7, Lecture 7

Game Theory Week 7, Lecture 7 S 485/680 Knowledge-Based Agents Game heory Week 7, Lecture 7 What is game theory? Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) who behave

More information

SF2972 GAME THEORY Infinite games

SF2972 GAME THEORY Infinite games SF2972 GAME THEORY Infinite games Jörgen Weibull February 2017 1 Introduction Sofar,thecoursehasbeenfocusedonfinite games: Normal-form games with a finite number of players, where each player has a finite

More information

In the Name of God. Sharif University of Technology. Microeconomics 2. Graduate School of Management and Economics. Dr. S.

In the Name of God. Sharif University of Technology. Microeconomics 2. Graduate School of Management and Economics. Dr. S. In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 2 44706 (1394-95 2 nd term) - Group 2 Dr. S. Farshad Fatemi Chapter 8: Simultaneous-Move Games

More information

Copyright 2008, Yan Chen

Copyright 2008, Yan Chen Unless otherwise noted, the content of this course material is licensed under a Creative Commons Attribution Non-Commercial 3.0 License. http://creativecommons.org/licenses/by-nc/3.0/ Copyright 2008, Yan

More information

Appendix: Common Currencies vs. Monetary Independence

Appendix: 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 information

LECTURE NOTES ON GAME THEORY. Player 2 Cooperate Defect Cooperate (10,10) (-1,11) Defect (11,-1) (0,0)

LECTURE NOTES ON GAME THEORY. Player 2 Cooperate Defect Cooperate (10,10) (-1,11) Defect (11,-1) (0,0) LECTURE NOTES ON GAME THEORY September 11, 01 Introduction: So far we have considered models of perfect competition and monopoly which are the two polar extreme cases of market outcome. In models of monopoly,

More information

Game Theory with Applications to Finance and Marketing, I

Game 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 information

Warm Up Finitely Repeated Games Infinitely Repeated Games Bayesian Games. Repeated Games

Warm Up Finitely Repeated Games Infinitely Repeated Games Bayesian Games. Repeated Games Repeated Games Warm up: bargaining Suppose you and your Qatz.com partner have a falling-out. You agree set up two meetings to negotiate a way to split the value of your assets, which amount to $1 million

More information

Econ 101A Final Exam We May 9, 2012.

Econ 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 information

University of Hong Kong ECON6036 Stephen Chiu. Extensive Games with Perfect Information II. Outline

University of Hong Kong ECON6036 Stephen Chiu. Extensive Games with Perfect Information II. Outline University of Hong Kong ECON6036 Stephen Chiu Extensive Games with Perfect Information II 1 Outline Interpretation of strategy Backward induction One stage deviation principle Rubinstein alternative bargaining

More information

Early PD experiments

Early PD experiments REPEATED GAMES 1 Early PD experiments In 1950, Merrill Flood and Melvin Dresher (at RAND) devised an experiment to test Nash s theory about defection in a two-person prisoners dilemma. Experimental Design

More information

Answers to Odd-Numbered Problems, 4th Edition of Games and Information, Rasmusen

Answers to Odd-Numbered Problems, 4th Edition of Games and Information, Rasmusen ODD Answers to Odd-Numbered Problems, 4th Edition of Games and Information, Rasmusen Eric Rasmusen, Indiana University School of Business, Rm. 456, 1309 E 10th Street, Bloomington, Indiana, 47405-1701.

More information

m 11 m 12 Non-Zero Sum Games Matrix Form of Zero-Sum Games R&N Section 17.6

m 11 m 12 Non-Zero Sum Games Matrix Form of Zero-Sum Games R&N Section 17.6 Non-Zero Sum Games R&N Section 17.6 Matrix Form of Zero-Sum Games m 11 m 12 m 21 m 22 m ij = Player A s payoff if Player A follows pure strategy i and Player B follows pure strategy j 1 Results so far

More information

Subgame Perfect Cooperation in an Extensive Game

Subgame Perfect Cooperation in an Extensive Game Subgame Perfect Cooperation in an Extensive Game Parkash Chander * and Myrna Wooders May 1, 2011 Abstract We propose a new concept of core for games in extensive form and label it the γ-core of an extensive

More information

ECON402: Practice Final Exam Solutions

ECON402: Practice Final Exam Solutions CO42: Practice Final xam Solutions Summer 22 Instructions There is a total of four problems. You must answer any three of them. You get % for writing your name and 3% for each of the three best problems

More information

preferences of the individual players over these possible outcomes, typically measured by a utility or payoff function.

preferences of the individual players over these possible outcomes, typically measured by a utility or payoff function. Leigh Tesfatsion 26 January 2009 Game Theory: Basic Concepts and Terminology A GAME consists of: a collection of decision-makers, called players; the possible information states of each player at each

More information

Solution to Tutorial 1

Solution to Tutorial 1 Solution to Tutorial 1 011/01 Semester I MA464 Game Theory Tutor: Xiang Sun August 4, 011 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are

More information

Topics in Contract Theory Lecture 1

Topics 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 information

Solution to Tutorial /2013 Semester I MA4264 Game Theory

Solution to Tutorial /2013 Semester I MA4264 Game Theory Solution to Tutorial 1 01/013 Semester I MA464 Game Theory Tutor: Xiang Sun August 30, 01 1 Review Static means one-shot, or simultaneous-move; Complete information means that the payoff functions are

More information

UC Berkeley Haas School of Business Economic Analysis for Business Decisions (EWMBA 201A) Fall 2012

UC Berkeley Haas School of Business Economic Analysis for Business Decisions (EWMBA 201A) Fall 2012 UC Berkeley Haas School of Business Economic Analysis for Business Decisions (EWMBA 01A) Fall 01 Oligopolistic markets (PR 1.-1.5) Lectures 11-1 Sep., 01 Oligopoly (preface to game theory) Another form

More information

Preliminary Notions in Game Theory

Preliminary Notions in Game Theory Chapter 7 Preliminary Notions in Game Theory I assume that you recall the basic solution concepts, namely Nash Equilibrium, Bayesian Nash Equilibrium, Subgame-Perfect Equilibrium, and Perfect Bayesian

More information

Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2015

Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2015 Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2015 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.

More information

Beliefs and Sequential Rationality

Beliefs and Sequential Rationality Beliefs and Sequential Rationality A system of beliefs µ in extensive form game Γ E is a specification of a probability µ(x) [0,1] for each decision node x in Γ E such that x H µ(x) = 1 for all information

More information

Elements of Economic Analysis II Lecture X: Introduction to Game Theory

Elements of Economic Analysis II Lecture X: Introduction to Game Theory Elements of Economic Analysis II Lecture X: Introduction to Game Theory Kai Hao Yang 11/14/2017 1 Introduction and Basic Definition of Game So far we have been studying environments where the economic

More information