Mixed Strategy Equilibrium
Levent Kokesen
1
Introduction
Up to now we have assumed that the only choice available to players was to pick an action
from the set of available actions. In some situations a player may want to randomize between
several actions
Nash Equilibrium: Applications
Prof. Levent Kokesen
Columbia University
and
Prof. Efe A. Ok
New York University
Introduction
In this section, we shall consider several economic scenarios which are modeled well by
means of strategic games. We shall also ex
Chapter 1
Introduction
1.1
W HAT IS G AME T HEORY ?
We, humans, cannot survive without interacting with other humans, and ironically, it sometimes seems that we have survived despite those interactions. Production and exchange require
cooperation between
Auctions
Many economic transactions are conducted through auctions. Governments sell treasury
bills, foreign exchange, publicly owned companies, mineral rights, and more recently airwave
spectrum rights via auctions. Art work, antiques, cars, and houses a
Game Theory
Solutions to Problem Set 8
Levent Kokesen
1. Let A and N denote attack and do not attack, and F and R denote ght and
retreat. The following payo functions models the situation described:
(u1 (N ) ; u2 (N ) = (1; 2)
(u1 (A; F ) ; u2 (A; F ) = (
Game Theory
Solutions to Problem Set 7
Levent Kokesen
1. The backward induction equilibrium of the game is (SSS; SSS ), i.e., each player plays
S whenever it is her turn to move.
2. Given the childs action the parent maximizes the following payo function
Game Theory
Solutions to Problem Set 6
Levent Kokesen
1. The Bayesian game is:
N = f1; 2g
Ai = fF; Y g ; i = 1; 2
2 = fS; W g
p1 (S ) =
payos are given by:
F
Y
F
1; 1
0; 1
Y
1; 0
0; 0
F
Y
Player 2 is strong
F
1; 1
0; 1
Y
1; 0
0; 0
Player 2 is weak
L
Game Theory
Problem Set 5 Solutions
Levent Kokesen
1. Find all the pure and mixed strategy equilibria of the following games by constructing
the best response correspondences of the players:
(a) Matching Pennies:
H
T
1; 1 1; 1
1; 1 1; 1
H
T
Let 1 (H ) = p
Game Theory
Problem Set 4 Solutions
1. Assuming that in the case of a tie, the object goes to person 1, the best response
correspondences for a two person first price auction are:
cfw_b2 , b2 < v1
undefined , b1 < v 2
B1 (b2 ) = [0, b2 ], b2 = v1
B2 (b1
Solutions to Problem Set #3
1. a) We obtain the best response functions by maximizing each firms profit
function taking the other firms output as given, subject to the constraint that
output cannot be negative. The best response functions are:
a c1
a c1
Game Theory
Solutions to Problem Set 2
1. Question 1: The set of Nash equilibria = f(0; 0); (0:25; 0:25); (0:5; 0:5); (0:75; 0:75); (1; 1); (0; 1)g. Note
that these are not payos but action proles.
Question 2: The set of Nash equilibria = f(U; M ); (D; L)
Game Theory
Solutions to Problem Set 1
1.
N = f1; 2g
A1 = A2 = f0; :25; :50; :75; 1g
u1 (a1 ; a2 ) =
u2 (a1 ; a2 ) =
100 a1 ; if a1 a2
0;
otherwise
a1 ; if a1 a2
:
0; otherwise
We can conveniently represent the strategic form with the following bimatrix
Game Theory
Problem Set 9
Levent Kokesen
1. Consider the following prisoners dilemma game.
C
D
C
2; 2
3; 0
D
0; 3
1; 1
For what values of ; if any, the following strategies constitute subgame perfect equilibria?
(a) Tit-For-Tat: Choose C in period 1 and t