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Game Theory: Nash Equilibrium and Cooperation in Prisoner's Dilemma and Stag Hunt, Slides of Artificial Intelligence

Bidhan Chandra Krishi ViswavidyalayaArtificial Intelligence

The concepts of nash equilibrium and cooperation in the context of the prisoner's dilemma and stag hunt games. The interpretation of the prisoner's dilemma, the concept of utility, and the existence of nash equilibrium in various interaction scenarios. It also covers efforts to recover cooperation through altruism, reciprocity, and the shadow of the future. Axelrod's tournament is introduced as an example of iterated prisoner's dilemma, and the success of tit-for-tat strategy is explained.

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CSCI 100
Think Like Computers
Lecture 25
Fall 2008
Last Time …
•Multi player interactions
•A bit of game theory
•Payoff matrix
•Nash Equilibrium
•Prisoner’s Dilemma:
The interpretation
The Prisoner’ Dilemma
What should A (or B) do?
3
3
5
0
A cooperates
0
5
2
2
A defects
B cooperatesB defects
The Utility (not the same as years in prison)
Nash Equilibrium
•Example: which side of the road you drive?
•Two strategies are in Nash Equilibrium if
If A plays s1, then B can do no better than pla y s2,
and
If B plays s2, then A can do no better than pla y s1
•“Locked in” – Neither agent has any incentive to
deviate from a Nash equilibrium (bad for
unilateralism)
Nash Equilibrium
•Seems very good!
•But the sad news:
Not every interaction scenario (game) has a
Nash equilibrium; and
Some interaction scenarios (games) have
more than 1 Nash equilibrium.
Efforts to “recover” cooperation
•1. We are not all Machiavelli – some
altruism
Given up your seat on the bus to …
Altruism? Or being afraid of punishments?
Consider a public transport system relying on
every one honestly paying (not verified)
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Download Game Theory: Nash Equilibrium and Cooperation in Prisoner's Dilemma and Stag Hunt and more Slides Artificial Intelligence in PDF only on Docsity!

CSCI 100

Think Like Computers

Lecture 25

Fall 2008

Last Time …

  • Multi player interactions
  • A bit of game theory
  • Payoff matrix
  • Nash Equilibrium
  • Prisoner’s Dilemma: Š The interpretation

The Prisoner’ Dilemma

What should A (or B) do?

3 3

5 0

A cooperates

0 5

2 2

A defects

B defects B cooperates

The Utility (not the same as years in prison)

Nash Equilibrium

  • Example: which side of the road you drive?
  • Two strategies are in Nash Equilibrium if Š If A plays s1, then B can do no better than play s2, and Š If B plays s2, then A can do no better than play s
  • “Locked in” – Neither agent has any incentive to deviate from a Nash equilibrium (bad for unilateralism)

Nash Equilibrium

  • Seems very good!
  • But the sad news: Š Not every interaction scenario (game) has a Nash equilibrium; and Š Some interaction scenarios (games) have more than 1 Nash equilibrium.

Efforts to “recover” cooperation

    1. We are not all Machiavelli – some

altruism

Š Given up your seat on the bus to … Š Altruism? Or being afraid of punishments?

Š Consider a public transport system relying on every one honestly paying (not verified)

Recover Cooperation

  • The other prisoner is my twin! Š “Think alike” ƒ Cooperation is the best outcome. Š What if everyone were to behave like that … ƒ Joseph Heller’s Catch 22 ƒ You’d be a fool to behave any other way

Recover Cooperation

  • People are not rational … Š We’d risk cooperation when the sucker’s payoff really does not matter very much Š Paying a bus fare of a few pennies does not hurt much, even if everybody else is defecting

Recover Cooperation

  • The shadow of the future Š What if we play the game more than once? Š Iterated game: many rounds Š In fact, infinite rounds
  • If you defect, your opponent can punish

you by also defecting (not possible in one-

round game)

  • If you test the water by cooperating? Š Cooperation becomes rational

Axelrod’s Tournament

  • For iterated Prisoner’s dilemma: Š 1980 Š Many strategies were submitted
  • Examples: Š All-D Š Random Š Tit-for-Tat (the overall winner, only 5 lines) Š …

Tit-for-Tat

  • Tit-for-tat is not the optimal strategy
  • It was able to succeed because it had the

opportunity to play against other programs

that were also inclined to cooperate.

Tit-for-Tat

  • Reasons for its success: Š Do not be envious. Š Do not be the first to defect. Š Reciprocate cooperation and defection. ƒ Balance between punishing and forgiving Š Do not be too clever!

The Game of Chicken

  • To establish who is bravest out of two

young thugs.

  • Both drive their cars at high speed towards

a cliff.

  • The least brave of the two (the ‘chicken’)

will be the first to drop out of the game by

steering away from the cliff.

Chicken

2 2

3 1

C (steer away)

1 3

0 0

D

D C (steer away)

Nash equilibrium?