The Prisoner’s Dilemma or Life With My Brother and Sister John CT

play with my cool brother, or … … spend time with my loving sister

Prisoner’s 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 Prisoner’s Dilemma –based on two choices: confess or don’t confess –shows up in many different places

An example of the prisoner’s 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 don’t confess

How to Play (Billy confesses) + (Bob confesses) = both owe 5 years in jail (Billy confesses) + (Bob doesn’t confess) = Bob owes 10 years in jail & Billy goes free (Billy doesn’t confess) + (Bob doesn’t confess) = both owe 1 year in jail (Billy doesn’t 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?

Billy confessesBilly doesn’t confess Bob confesses (5,5)(0,10) Bob doesn’t confess (10,0)(1,1) Bob’s penalty, years in jail Billy’s penalty, years in jail Cost Table: “Normal Form”

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 doesn’t confess, Bob’s outcome is better than he was expecting

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

Pareto Optimality Pareto optimal solution concept: any change in strategy that causes improvement for one player must make it worse for the other player Billy confesses Billy doesn’t confess Bob confesses (5,5) NASH (0,10) Bob doesn’t confess (10,0)(1,1) PARETO Vilfredo Pareto

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 doesn’t matter what Amy does.

The Iterated Prisoner’s Dilemma the iterated* prisoner’s 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 don’t benefit as much as more altruistic strategies * a.k.a. “irritated”

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 –Joe’s “most” sack is compared to John’s “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)

{1,1,4}{1,2,3}{2,2,2} {1,1,4}DRAW {1,2,3}DRAW {2,2,2}DRAW Nash equilibrium

TTTTTBBBBBBB 129 TTTTTTBBBBBB 138 TTTTTTTBBBBB 147 TTTTTBTBBDTT 156 TTTTTBBTTTDD 228 DTTDDTTTTBBB 237 DDTTDTTTTTBB 246 DDDDTTTTTTTT 255 DDDDTTTTTBTD 336 DDDBTDTTDTTB 345 DDDTBDDTTTTT 444 DDDTBDDTBDTT Colonel Blotto with 12 armies – he has only 12 choices

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 don’t understand this P.S. Dude goes down, and Blotto goes across

Conclusions A Prisoner’s 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 Prisoner’s Dilemma is an easy way to beat family Wikipedia is very useful

Bibliography { type in Prisoners Dilemma} { type in Nash Equilibrium} { type in Pareto} / Jonathan Partington’s Colonel Blotto page