CMU 15-896 Noncooperative games 1: Basic concepts Teacher: Ariel - - PowerPoint PPT Presentation

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CMU 15-896 Noncooperative games 1: Basic concepts Teacher: Ariel - - PowerPoint PPT Presentation

Jan 23, 2023 380 likes •631 views

CMU 15-896 Noncooperative games 1: Basic concepts Teacher: Ariel Procaccia Normal-Form Game A game in normal form consists of: Set of players o Strategy set o For each , utility function : if o each plays the strategy

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SLIDE 1

CMU 15-896

Noncooperative games 1: Basic concepts

Teacher: Ariel Procaccia

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SLIDE 2

15896 Spring 2016: Lecture 17

Normal-Form Game

  • A game in normal form consists of:
  • Set of players
  • Strategy set
  • For each

, utility function

  • : if

each plays the strategy , the utility of player is

  • Next example created by taking

screenshots of http://youtu.be/jILgxeNBK_8

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SLIDE 3

15896 Spring 2016: Lecture 17

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Selling ice cream at the beach. One day your cousin Ted shows up. His ice cream is identical! You split the beach in half; you set up at 1/4. 50% of the customers buy from you. 50% buy from Teddy. One day Teddy sets up at the 1/2 point! Now you serve only 37.5%!

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SLIDE 4

15896 Spring 2016: Lecture 17

  • The Ice Cream Wars
  • To be continued…

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SLIDE 5

15896 Spring 2016: Lecture 17

The prisoner’s dilemma

  • Two men are charged with a crime
  • They are told that:
  • If one rats out and the other does not, the

rat will be freed, other jailed for nine years

  • If both rat out, both will be jailed for six

years

  • They also know that if neither rats out,

both will be jailed for one year

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SLIDE 6

15896 Spring 2016: Lecture 17

The prisoner’s dilemma

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  • 1,-1
  • 9,0

0,-9

  • 6,-6

Cooperate Defect Cooperate Defect

What would you do?

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SLIDE 7

15896 Spring 2016: Lecture 17

Prisoner’s dilemma on TV

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http://youtu.be/S0qjK3TWZE8

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SLIDE 8

15896 Spring 2016: Lecture 17

The professor’s dilemma

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106,106

  • 10,0

0,-10 0,0

Make effort Slack off Listen Sleep

Dominant strategies?

Professor Class

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SLIDE 9

15896 Spring 2016: Lecture 17

Nash equilibrium

  • Each player’s strategy

is a best response to strategies of others

  • Formally, a Nash

equilibrium is a vector of strategies

  • such that
  • 9
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SLIDE 10

15896 Spring 2016: Lecture 17

Nash equilibrium

http://youtu.be/CemLiSI5ox8

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SLIDE 11

15896 Spring 2016: Lecture 17

Russel Crowe was wrong

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SLIDE 12

15896 Spring 2016: Lecture 17

End of the Ice Cream Wars

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Day 3 of the ice cream wars… Teddy sets up south of you! You go south of Teddy. Eventually…

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SLIDE 13

15896 Spring 2016: Lecture 17

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R P S R

0,0

  • 1,1

1,-1

P

1,-1 0,0

  • 1,1

S

  • 1,1

1,-1 0,0

Rock-paper-scissors

Nash equilibrium?

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SLIDE 14

15896 Spring 2016: Lecture 17

Mixed strategies

  • A mixed strategy is a probability

distribution over (pure) strategies

  • The mixed strategy of player

is , where

  • The utility of player

is

, … ,

  • , … , ⋅
  • ,…,∈

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SLIDE 15

15896 Spring 2016: Lecture 17

Nash’s Theorem

  • Theorem [Nash, 1950]: if everything is

finite then there exists at least one (possibly mixed) Nash equilibrium

  • We’ll talk about computation some other

time

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SLIDE 16

15896 Spring 2016: Lecture 17

Does NE make sense?

  • Two players, strategies are
  • If both choose the same number, that is

what they get

  • If one chooses , the other , and

, the former player gets , and the latter gets

  • Poll 1: what would you choose?

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100 99 98 97 96 95

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SLIDE 17

15896 Spring 2016: Lecture 17

Correlated equilibrium

  • Let

for simplicity

  • A mediator chooses a pair of strategies
  • according to a distribution
  • ver
  • Reveals to player

and to player

  • When player

gets , he knows that the distribution over strategies of is

  • ∈

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SLIDE 18

15896 Spring 2016: Lecture 17

Correlated equilibrium

  • Player

is best responding if for all

  • ∈

∈

  • Equivalently,
  • ∈

∈

  • is a correlated equilibrium (CE) if both

players are best responding

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SLIDE 19

15896 Spring 2016: Lecture 17

Game of chicken

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http://youtu.be/u7hZ9jKrwvo

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SLIDE 20

15896 Spring 2016: Lecture 17

Game of chicken

  • Social welfare is the sum of

utilities

  • Pure NE: (C,D) and (D,C),

social welfare = 5

  • Mixed NE: both

social welfare = 4

  • Optimal social welfare = 6

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Dare Chicken Dare

0,0 4,1

Chicken

1,4 3,3

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SLIDE 21

15896 Spring 2016: Lecture 17

Game of chicken

  • Correlated equilibrium:
  • (D,D):
  • (D,C):
  • (C,D):
  • (C,C):
  • Social welfare of CE =
  • 21

Dare Chicken Dare

0,0 4,1

Chicken

1,4 3,3

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SLIDE 22

15896 Spring 2016: Lecture 17

Implementation of CE

  • Instead of a mediator, use a hat!
  • Balls in hat are labeled with

“chicken” or “dare”, each blindfolded player takes a ball

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C D D C C C D C

Which balls implement the distribution of the previous slide?

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SLIDE 23

15896 Spring 2016: Lecture 17

CE vs. NE

  • Poll 2: What is the relation between CE

and NE?

1.

CE NE

2.

NE CE

3.

NE CE

4.

NE CE

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SLIDE 24

15896 Spring 2016: Lecture 17

CE As LP

  • Can compute CE via linear programming

in polynomial time!

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find

  • s.t.
  • , 1

,∈

  • , , ,

, ∈ ∈

, , , ,

  • ∈

∈