The Prisoners Dilemma or Life With My Brother and Sister

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The Prisoners DilemmaorLife With My Brother
and Sister

• John Willett
• Age 9
• G.H. Robertson
• Coventry, CT

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spend time with my loving sister
play with my cool brother, or
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Prisoners Dilemma

• last year I did a presentation on blackjack and
  craps
• these are simple games
• there is no opponent with a strategy to play
  against
• poker is a more interesting game
• adversarial
• repeated
• because poker is so complicated, I will look at a
  simple example of an adversarial game
• the Prisoners Dilemma
• based on two choices confess or dont confess
• shows up in many different places

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An example of the prisoners dilemma

• The police have arrested two people, Billy and
  Bob, for a crime that they have committed.
• Both can talk to the police, or stay quiet
• confess, or dont confess

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How to Play

• (Billy confesses) (Bob confesses)
• both owe 5 years in jail
• (Billy confesses) (Bob doesnt confess)
• Bob owes 10 years in jail Billy goes free
• (Billy doesnt confess) (Bob doesnt confess)
  
• both owe 1 year in jail
• (Billy doesnt confess) (Bob confesses)
• Billy owes 10 years in jail Bob goes free
• Billy and Bob don't know what the other person
  did (or will do), and they are not allowed to
  talk to each other.
• What should they do?

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Cost Table Normal Form
Bobs penalty, years in jail
Billys penalty, years in jail
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The Answer

• the best answer is for neither Bob nor Billy to
  confess
• both would get only 1 year in jail
• but this would require cooperation
• suppose Bob decides not to confess
• if Billy confesses, Bob is in trouble
• on the other hand, suppose Bob confesses,
  reasoning that Billy will confess, too
• then if Billy doesnt confess, Bobs outcome is
  better than he was expecting

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Nash Equilibrium

• the both confess strategy pair (or solution
  concept) has the property that neither Billy nor
  Bob can do better by changing his strategy
• that is, neither can do better without
  coordinating a change in strategy with the other
  player
• this solution concept is a Nash Equilibrium
• named after John Forbes Nash, who discovered it
  in his 1950 dissertation
• the movie A Beautiful Mind was about Nash

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Pareto Optimality

• Pareto optimal solution concept any change in
  strategy that causes improvement for one player
  must make it worse for the other player

Vilfredo Pareto 1848-1923
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Another example Friend or Foe

• Friend or Foe was a game show aired in the 1990s
• The contestants answer a series of questions, and
  close to the end of the show, there are two left
• Say the two last contestants are named Amy and
  Emma
• (Emma friend) (Amy friend) split
  winnings
• (Emma foe) (Amy foe) empty handed
• (Emma foe) (Amy friend) Emma gets all
• (Emma friend) (Amy foe) Amy gets all
• The strategy foe/foe is a weak equilibrium,
  since if Emma votes foe it doesnt matter what
  Amy does.

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The Iterated Prisoners Dilemma

• the iterated prisoners dilemma can occur when
  both players remember the game or games before
• under certain conditions they will most likely
  interact to a cooperative strategy
• even if the agreement is to defect
• Robert Axelrod wrote a book called The Evolution
  of Cooperation
• states that greedy strategies dont benefit as
  much as more altruistic strategies

a.k.a. irritated
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Colonel Blotto Games

• Joe and John each have 6 rocks and 3 sacks
• both put at least one rock in each sack, ordered
  least to most
• Joes most sack is compared to Johns most,
  and so on
• the object is to win the most match-ups
• John (2,2,2) vs. Joe (1,1,4)
• John wins 2, Joe wins 1
• only 3 possibilities (1,2,3) (2,2,2) (1,1,4)

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Nash equilibrium
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Colonel Blotto with 12 armies he has only 12
choices
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The T

• T is for tie
• B means Colonel Blotto wins
• D means Colonel Dude wins
• T is the Nash Equilibrium of this game
• It is the Nash Equilibrium because
• You are Colonel Blotto, and you pick 246
• If Colonel Dude does the same you will tie (same
  with some other choices) if he does anything
  else, you win
• With 13 armies there is no deterministic Nash
• optimal solution is randomized
• I dont understand this
• P.S. Dude goes down, and Blotto goes across

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Conclusions

• A Prisoners Dilemma can be found in many areas
• it uses game theory
• the stable solution is a Nash Equilibrium
• It can only be used when all the players know the
  payoffs
• Prisoners Dilemma is an easy way to beat family
• Wikipedia is very useful

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Bibliography

• www.wikipedia.com type in Prisoners Dilemma
• www.wikipedia.com type in Nash Equilibrium
• www.wikipedia.com type in Pareto
• http//pespmc1.vub.ac.be/PRISDIL.html
• http//serendip.brynmawr.edu/playground/pd.html
• http//prisonersdilemma.groenefee.nl/
• Jonathan Partingtons Colonel Blotto page