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Social Networks 101

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Social Networks 101

Prof. Jason Hartline and Prof. Nicole Immorlica

Lecture Thirteen:

Normal form games

and equilibria notions.

Experiment: The median game.

1. Guess an integer between 1 and 100, inclusive.

2. Write your number and name on your card.

P R I Z E : The people whose numbers are closest to 2/3 of the median win 5 points.

25

45

0

50

69

Alok

Brent

Casey

Dirk

Ela

Calculating the winner:

1. Sort the numbers: 0, 25, 45, 50, 69

2. Pick the middle one (the median): 45

3. Compute 2/3 of the median: 30

Median is 45, and Alok wins because his guess is closest to 2/3 of the median, or 30.

25

45

0

50

69

Alok

Brent

Casey

Dirk

Ela

How did you play?

Imagine what everyone else will do,

decide how to act based on that assumption.

Bi-matrix games

Example:

prisoners’ dilemma

Mrs. Column

Confess

Deny

( -4 , -4 )

( 0 , -10 )

Confess

( -10 , 0 )

( -1 , -1 )

Deny

Mr. Row

Prisoners’ dilemma

Q. If Row confesses, what should Column do?

Mrs. Column

Confess

Deny

( -4 , -4 )

( 0 , -10 )

Confess

( -10 , 0 )

( -1 , -1 )

Deny

Mr. Row

Prisoners’ dilemma

Q. If Row denies, what should Column do?

Mrs. Column

Confess

Deny

( -4 , -4 )

( 0 , -10 )

Confess

( -10 , 0 )

( -1 , -1 )

Deny

Mr. Row

Row’s best-response was Confess

no matter what Column did.

Confess is a dominant strategy for row.

Definition. A normal form game for a set N of n players is described by

1. A set of strategies Si for each player i.

2. A payoff function ¼i for each player i and each profile of strategies (s1, …, sn) indicating player i’s reward for every strategy profile.

Definition. A strategy si* is a best-response to strategies sj of players i ≠ j if

¼(s1, …, si*, …, sn) ¸¼(s1, …, si, …, sn)

for all strategies si in Si.

Definition. A strategy si is a dominant strategy for player i if it is a best-response to all strategy profiles of the other players.

To find a dominant strategy for a row player, compare vectors of payoffs in each row.

If (and only if) some row vector dominates coordinate-wise, it is a dominant strategy for the row player.

Prisoners’ dilemma

Q. Is there a dominant strategy?

Mrs. Column

Confess

Deny

( -4 , -4 )

( 0 , -10 )

Confess

( -10 , 0 )

( -1 , -1 )

Deny

Mr. Row

Definition. A strategy profile (s1, …, sn) is a dominant strategy equilibrium if, for each player i, si is a dominant strategy.

Another game

Q. Is there a dominant strategy?

Mrs. Column

High

Low

( 2 , 2 )

( 0 , 3 )

High

( 3 , 2 )

( 5 , 1 )

Low

Mr. Row

Definition: A strategy profile (s1, …, sn) is a Nash equilibrium (NE)if for each player i, si is a best-response to strategies sj of players j ≠ i.

Chicken

Q. Is there a Nash equilibrium?

Mrs. Column

Swerve

Stay

( 1 , 1 )

( 0 , 2 )

Swerve

( 2 , 0 )

( -1 , -1 )

Stay

Mr. Row

Method: Best-response (directed) graph

1. For each strategy profile s create a node su.

2. Connect node su to node sv if for some player i, his strategy sviin v is a best response to the other players’ strategies in u and for all other players j, suj= svj.

3. Search for a node with no out-going links.

Chicken

Swerve

Stay

( 1 , 1 )

( 0 , 2 )

Swerve

(swerve, swerve)

( 2 , 0 )

( -1 , -1 )

Stay

(swerve, stay)

(stay, swerve)

(stay, stay)

Chicken

Q. Is there a Nash equilibrium?

Mrs. Column

Swerve

Stay

( 1 , 1 )

( 0 , 2 )

Swerve

( 2 , 0 )

( -1 , -1 )

Stay

Mr. Row

Matching pennies

Q. Is there a Nash equilibrium?

Mrs. Column

Heads

Tails

( -1 , 1 )

( 1 , -1 )

Heads

( 1 , -1 )

( -1 , 1 )

Tails

Mr. Row

Matching pennies

Heads

Tails

( -1 , 1 )

( 1 , -1 )

Heads

(heads, heads)

( 1 , -1 )

( -1 , 1 )

Tails

(heads, tails)

(tails, heads)

(tails, tails)

Mixed Nash equilibria

and fixed points.