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. Let’s play a game. Experiment : The median game. 1. Guess an integer between 1 and 100, inclusive. 2. Write your number and name on your card.

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Social Networks 101

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Social networks 101

Social Networks 101

Prof. Jason Hartline and Prof. Nicole Immorlica


Social networks 101

Lecture Thirteen:

Normal form games

and equilibria notions.


Let s play a game

Let’s play a game

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.


The median game

The Median Game

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


The median game1

The Median Game

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


Social networks 101

How did you play?


Reasoning in games

Reasoning in games

Imagine what everyone else will do,

decide how to act based on that assumption.


Social networks 101

Bi-matrix games

Example:

prisoners’ dilemma

Mrs. Column

Confess

Deny

( -4 , -4 )

( 0 , -10 )

Confess

( -10 , 0 )

( -1 , -1 )

Deny

Mr. Row


Social networks 101

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


Social networks 101

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


Dominant strategies

Dominant strategies

Row’s best-response was Confess

no matter what Column did.

Confess is a dominant strategy for row.


Normal form games

Normal form games

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.


Best responses

Best responses

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.


Dominant strategies1

Dominant strategies

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.


Finding dominant strategies

Finding dominant strategies

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.


Social networks 101

Prisoners’ dilemma

Q. Is there a dominant strategy?

Mrs. Column

Confess

Deny

( -4 , -4 )

( 0 , -10 )

Confess

( -10 , 0 )

( -1 , -1 )

Deny

Mr. Row


Dominant strategy equilibria

Dominant strategy equilibria

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


Social networks 101

Another game

Q. Is there a dominant strategy?

Mrs. Column

High

Low

( 2 , 2 )

( 0 , 3 )

High

( 3 , 2 )

( 5 , 1 )

Low

Mr. Row


Nash equilibrium

Nash equilibrium

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

Chicken


Social networks 101

Chicken

Q. Is there a Nash equilibrium?

Mrs. Column

Swerve

Stay

( 1 , 1 )

( 0 , 2 )

Swerve

( 2 , 0 )

( -1 , -1 )

Stay

Mr. Row


Finding nash equilibria

Finding Nash equilibria

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.


Social networks 101

Chicken

Swerve

Stay

( 1 , 1 )

( 0 , 2 )

Swerve

(swerve, swerve)

( 2 , 0 )

( -1 , -1 )

Stay

(swerve, stay)

(stay, swerve)

(stay, stay)


Social networks 101

Chicken

Q. Is there a Nash equilibrium?

Mrs. Column

Swerve

Stay

( 1 , 1 )

( 0 , 2 )

Swerve

( 2 , 0 )

( -1 , -1 )

Stay

Mr. Row


Social networks 101

Matching pennies

Q. Is there a Nash equilibrium?

Mrs. Column

Heads

Tails

( -1 , 1 )

( 1 , -1 )

Heads

( 1 , -1 )

( -1 , 1 )

Tails

Mr. Row


Social networks 101

Matching pennies

Heads

Tails

( -1 , 1 )

( 1 , -1 )

Heads

(heads, heads)

( 1 , -1 )

( -1 , 1 )

Tails

(heads, tails)

(tails, heads)

(tails, tails)


Next time

Next time

Mixed Nash equilibria

and fixed points.


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