Clicker Question-A Chicken Game 0, 0 0, 1 1, 0 - 10, -10 Swerve Hang Tough Swerve Hang Tough Player 2 Pllayer 1 oes either player have a dominant stra A) Yes B) No
Dec 14, 2015
Clicker Question-A Chicken Game
0, 0 0, 1
1, 0 -10, -10
Swerve Hang Tough
Swerve
Hang Tough
Player 2
Pllayer 1
Does either player have a dominant strategy?
A) YesB) No
Nash Equilibrium in Chicken Game?
0, 0 0, 1
1, 0 -10, -10
Swerve Hang Tough
Swerve
Hang Tough
Player 2
Pllayer 1
How many Nash equilibria does this game have?
A) NoneB) Exactly oneC) At least two
Monte Hall Problem
Let’s Draw a Game Tree…
Problem 6, chapter 2
Information sets in Monte Hall game
• In last move, contestant knows which door he chose and which Monte opened. The door that Monte opened is neither the one that contestant opened nor the one with the good prize.
• Six things contestant might see.1/ 2, 1/ 3, 2/ 1, 2/ 3, 3/ 1, 3/ 2.
Finding information sets
• If contestant sees 1/ 2, then the node might be 1/ 1 /2 or it might be 3/ 1/ 2. Why couldn’t it be 2/ 1/ 2?
• So one information set is {1/1/2,3/1/2}.• What are the others?
HW Problem 2.11
Perfect Recall?
Problem 8
Figure PR2.8Harrington: Games, Strategies, and Decision Making, First EditionCopyright © 2009 by Worth Publishers
Nash Equilibrium
Clicker Question 1:
6,0 3,2 4,12,3 2,3 2,81,1 2,0 5,2Player 1
Strategy A1
Strategy B1
Strategy C1
Strategy A2 Stategy B2 Strategy C2
Is the outcome where Player 1 plays B1 and Player 2 plays C2 a Nash equilibrium?A) YesB) No
Clicker Question 2:
6,0 3,2 4,12,3 2,3 2,81,1 2,0 5,2Player 1
Strategy A1
Strategy B1
Strategy C1
Strategy A2 Stategy B2 Strategy C2
Is the outcome where Player 1 plays A1 and Player 2 plays B2 a Nash equilibrium?A) YesB) No
Definition
A strategy profile is a Nash Equilibrium if each player’s strategy maximizes his payoff given the strategies used by the other players.
Best response mapping
Best response for a player is a mapping from actions by the others to the action (or actions) that maximizes the player’s payoffs given the actions of the others.
In Nash equilibrium, every player is doing the best response to what the other players are doing.
Prisoners’ Dilemma Game
10, 10 0, 11
11, 0 1, 1
Cooperate Defect
Cooperate
Defect
PLAyER 1
Player 2
Battle of Sexes
2,1 0,0
0,0 1,2
Bob
Alice
Movie A
Movie B
Movie A
Movie B
BRA(A)=ABRA(B)=B
BRB(A)=ABRB(B)=B
Best Responses and Nash Equilibria for this game?
BR2(a)=z BR1(w)=bBR2(b)={w,x,z} BR1(x)=bBR2(c)=y BR1(y)=bBR2(d)={y,z} BR1(z)={a,d}
Find Nash equilibria for these games
• Chicken• Pure coordination (Driving Game)
How many Nash equilibria(in pure strategies)?
There might be just one.There might be more than one.There might not be any.
Rock, Paper Scissors, Where is Nash equilibrium?
When is Nash equilibrium “the right answer”?
1. Players are “rational”. Each player’s strategy maximizes his payoff, given his beliefs about the strategies used by the other players.
2. Each player’s beliefs about the other players’ strategies are correct.
When is 2) a reasonable assumption?
3-Hunter Stag Hunt
2,2,2 0,1,0
1,0,0 1,1,0
Hunter 3 does Stag Hunter 3 does Hare
Hunter 2
Stag Hare
Stag
Hare
Hunter 1
0,0,1 0,1,1
1,0,1 1,1,1
Hunter 2
Stag
Hare
Stag Hare
Find the Nash equilibria
Hunter 1
Weakly dominated strategies?
• Nobody will use a strictly dominated strategy in Nash equilibrium.
• If there is a strictly dominant strategy for all players, it is a Nash equilibrium. (example Prisoners’ Dilemma.)
• Nash equilibrium does not exclude possibility of using a weakly dominated strategy. (A voting example with unanimous preferences.)
Nash and domination
Every Nash equilibrium survives the iterated elimination of strictly dominated strategies.
Not every outcome that satisfies the iterated elimination of strictly dominated strategies is a Nash equilibrium.
Clicker Question 3:
6,0 3,2 4,12,3 2,3 2,81,1 2,0 5,2Player 1
Strategy A1
Strategy B1
Strategy C1
Strategy A2 Stategy B2 Strategy C2
Does this game have more than one Nash equilibrium?A) YesB) No
Clicker Question:What are the Nash Equilibria for this game?
A) Player 1 plays a and Player 2 plays z.B) Player 1 plays d and Player 2 plays z.C) Player 1 plays b and Player 2 plays y.D) Both outcomes A) and B) are Nash equilibria.E) There are no Nash equilibria
4,1