Washington University in St. Louis
L32 Political Science 506 Theories of Individual and Collective Choice II
John W. Patty Spring 2012
Syllabus
This course is an introduction to noncooperative game theory with an emphasis on its use in political science.
Midterm Exam # 2: Answers Graduate Game Theory Due April 12th, 2010 at 3pm
This is an open book and open notes exam. Do not discuss the exam with anyone other than Professor Patty prior to 3pm April 12th. The point value is listed for each question. There
Midterm Exam # 2 Graduate Game Theory Due April 12th, 2010 at 3pm
This is an open book and open notes exam. Do not discuss the exam with anyone other than Professor Patty prior to 3pm April 12th. The point value is listed for each question. There are 100
Midterm Exam # 1 Answers Graduate Game Theory Due March 15th, 2010 at 3pm This is an open book and open notes exam. Do not discuss the exam with anyone other than Professor Patty prior to 3pm March 15th. The point value is listed for each question. You ma
Midterm Exam # 1 Graduate Game Theory Due March 28th, 2011 This is an open book and open notes exam. Do not discuss the exam with anyone other than Professor Patty prior to the due date. The point value is listed for each question. 1. Consider the followi
Midterm Exam # 1 Graduate Game Theory Due March 15th, 2010 at 3pm This is an open book and open notes exam. Do not discuss the exam with anyone other than Professor Patty prior to 3pm March 15th. The point value is listed for each question. You may comple
Game Theory Lecture 7 Signaling Games
Prof. John Patty
1
Classical Signaling Games: Sender: knows Receiver's preferences state of nature , chooses message, m Receiver: knows Sender's preferences, distribution of Observes message, Infers state of nature, C
Repeated Games
Prof. John Patty
1
Stage game: N : finite set of players Ai: finite set of actions for i N A iN Ai: action profiles gi A R: stage payoff function for i N For t 1, H t = (A)t is the set of length-t histories
2
t Mixed (behavior) strategy for
Game Theory Lecture 7 Perfect and Sequential Equilibria
Prof. John Patty
1
Elimination of Weakly Dominated Strategies PBE (and SPNE) never involve strictly dominated strategies. But they may involve use of a weakly dominated strategy This is demonstrated
Nash Equilibrium
Prof. John Patty
1
Recall that ui maps outcomes into payoffs for player i. Let vi S R denote the (expected) payoff of player i. The difference between vi and ui is simply what they take as their inputs: outcomes (ui) or strategies (vi). F
Game Theory Lecture 5 Bayes Nash Equilibrium
Prof. John Patty
1
Bayes's Rule Let be a set of states of nature. An event is any subset of . Then, for any state and any event B , the conditional probability that has occurred, given that we know B has occurr
Backwards Induction
Prof. John Patty
1
A simple bargaining situation
Player 1: Proposer Propose Decline
Player 2: Responder ,1 Veto Player 1: Proposer 1,0 Override Accept Sign
-1,-1
0,2
2
A simple bargaining situation
Player 1: Proposer Propose Decline
Pl
Game Theory Lecture 8 Bargaining Models
Prof. John Patty
1
Rubinstein's Bargaining Game 2 players dividing a fixed pie Complete and perfect information bargaining Players alternate making offers After each offer, the other player accepts (a = 1) or reject
Game Theory Lecture 1 Decision Theory and Game Forms
Prof. John Patty
1
The Theory of Choice & Expected Utility Classical decision theory no uncertainty with finite choices: Weak orders with infinite choices: Weak orders + technical conditions Decision-ma
Homework Assignment #7 Graduate Game Theory Due Wednesday, April 18th, 2012 at 11:59pm. Prepare your answers in LaTeX and email to Gordon and Professor Patty. 1. Consider the following public good provision model with n players, N = cfw_1, n. At the begin
Homework Assignment #6 Graduate Game Theory Due March 29th. Note. In the bargaining games discussed below, a stage-game undominated voting strategy is one in which a player votes with probability one for any proposal that is strictly higher than his or he
Homework Assignment #6 Graduate Game Theory Due Wednesday, March 28th, 2012 at 11:59pm. Prepare your answers in LaTeX and email to Gordon and Professor Patty. Find at least one perfect Bayesian equilibrium for each of the following games.
=A 1 a 2 a 2 0 b
Homework Assignment #6: Signaling Games Graduate Game Theory Due Tuesday, April 5th at 11:59pm. 1. The sender, S, has two possible types: = cfw_H, L (with the probability that = H being denoted by p [0, 1]), and two messages, M = cfw_h, l, and the receive
Homework Assignment #6 Graduate Game Theory Due March 29th. Note. In the bargaining games discussed below, a stage-game undominated voting strategy is one in which a player votes with probability one for any proposal that is strictly higher than his or he
Homework Assignment #5 Graduate Game Theory
1. Consider the following 2 player game. What are the perfect Bayesian equilibria?
Nature Pr(=G)=0.4 1 L 2 x y L R x y x y x =G =B Pr(=B)=0.6 1 R 2 y
1 1
3 0
0 2
2 1
3 0
1 1
0 1
2 2
Answer. Let q denote the prob
Homework Assignment #5: Repeated Games Graduate Game Theory Due Wednesday, February 29th, 2012 at 11:59pm. Prepare your answers in LaTeX and email to Gordon and Professor Patty.
1. Consider the following two-player games. (Row player's payoff is listed fi
Homework Assignment #5 Graduate Game Theory
Consider the following infinitely repeated stage games: 1. L (7,10) (8,1) R (1,12) (2,5)
L R (a) Consider the following strategy: = L R
if t = 1 or neither player has played R previously. otherwise.
i. What is t
Homework Assignment #5 Graduate Game Theory
1. Consider the following 2 player game. What are the perfect Bayesian equilibria?
Nature Pr(=G)=0.4 1 L 2 x y L R x y x y x =G =B Pr(=B)=0.6 1 R 2 y
1 1
3 0
0 2
2 1
3 0
1 1
0 1
2 2
2. Consider the following 2 p
Homework Assignment #4 Graduate Game Theory
1. Consider the following 3 player game.
1 2 L 3 x 2 2 2 y 0 0 0 x 1 1 1 a R 3 y 0 0 0 x 3 3 3 3 y 1 0 -5 x 1 1 0 y 0 1 0 b 2 L R
(a) What are the subgame perfect Nash equilibria? Answer. In any subgame perfect
Homework Assignment #4 Graduate Game Theory Due Wednesday, February 22nd, 2012 at 11:59pm. Prepare your answers in LaTeX and email to Gordon and Professor Patty.
1. Consider the following game. Player 1 observes his or her type, t cfw_H, L as follows: wit
Homework Assignment #4 Graduate Game Theory
Consider the following two-player games. (Row player's payoff is listed first in each game, followed by the Column player's.) 1. L (0,3) (1,4) R (2,5) (1,3)
L R (a) What is the set of feasible payoffs?
(b) What
Homework Assignment #4 Graduate Game Theory
1. Consider the following 3 player game.
1 2 L 3 x 2 2 2 y 0 0 0 x 1 1 1 a R 3 y 0 0 0 x 3 3 3 3 y 1 0 -5 x 1 1 0 y 0 1 0 b 2 L R
(a) What are the subgame perfect Nash equilibria? (b) What are the set of Nash eq
Homework Assignment #3 Graduate Game Theory
1. Iterated Deletion of Dominated Strategies. Consider the following 3 player game. Each player i cfw_1, 2, 3 simultaneously names a real number ai [0, 100]. Then the set of 3 winners are the players whose choic
Homework Assignment #3 Graduate Game Theory Due Wednesday, February 17th, 2012 at 11:59pm. Prepare your answers in LaTeX and email to Gordon and Professor Patty. 1. Find all Nash equilibria of this game. L (3,5) (0,2) R (0,2) (2,3)
U D 2. Find all Nash eq
Homework Assignment #3 Graduate Game Theory
1. Find the subgame perfect Nash equilibria of the following games of complete and perfect information.
1 A 2 X 3 Y 3 3 X B 2 Y 3
L
R
L
R
L
R
L
R
(a)
10 10 5
0 0 4
0 3 2
1 6 8
3 1 5
3 0 7
2 4 6
4 2 4
1 A 2 X 3 Y