Game Theory, Continued: From Zero-Sum to Non-Zero Sum

Game Theory, Continued: FromZero-Sum to Non-Zero SumProblem Set 3!due FRIDAY MARCH 23!Reading assignment (in PS3):http://www.wired.com/wired/archive/12.10/tail.html!

Game Theory: Basic Taxonomy • Zero- vs. non-zero sum • Two- vs. N-person games • Finite vs. infinite number of choices • Iterated vs. non-iterated games • Games of perfect information

Two-Person Zero Sum Games • The notion of a dominant choice • The solution of a zero-sum game – The value of a game – The minimax theorem • Pure vs. Mixed strategies

Blue 4 5 9 3 Worst I can do Red 8 4 3 7 7 6 8 9 7 2 4 6 Worst I can do

Scissors Paper Stone Scissors 0 1 -1 Paper -1 0 1 Stone 1 -1 0

A Non-Zero-Sum Game Blue Red 8 4 6 2 5 7 4 8

Blue Red 5 6 3 7 7 2 9 8

Clyde Clam up Bonnie Stool Pigeon Clam up 1 year 1 year 0 years 20 years Stool Pigeon 20 years 0 years 10 years 10 years

Blue Cooperate Red Defect Cooperate 3 3 5 0 0 5 1 1 Defect

Blue Red 6 6 7 4 4 7 -3 -3

Blue Red 3 3 3 1 1 3 0 0

Blue Red 4 4 3 1 1 3 0 0

The Ultimatum Game: addingintangibles to a utility function • Two players, Red and Blue. Red is given$1000 and is told that he can choose anamount to be taken out of this total to offerBlue. Blue then chooses whether to acceptthis amount or not. • If Blue rejects Reds offer, both get nothing.

Axelrods idea: hold a computer tournament in which each contestant is a program that will play a series of rounds of the Prisoners Dilemma game. Researchers can send in a program to enter in the tournament: each program will play all the others in a series of about 100 successive rounds of the PD game.

Some Simple Axelrod-Tournament-like strategies • All-Defect simply defects on every round • Poor-Trusting-Fool simply cooperates onevery round • Random is a test strategy that simplycooperates or defects randomly • Unforgiving cooperates initially until thefirst time that the other player defects; afterthat, Unforgiving defects forever

The Tit-for-Tat Strategy • Tit-for-Tat cooperates on the first round.Thereafter, on every subsequent round, itsimply imitates what the other player did onthe previous round. (That is, Tit-for-Tatdoes at round N what the other player did atround N-1.)

A British staff office on a tour of the trenchesremarked that he was astonished to observe German soldiers walking about withinrifle range behind their own line. Our men appeared to takeno notice. I privately made up my mind to do away with thatsort of thing when we took over; such things should not beallowed. These people evidently did not know there was awar on. Both sides apparently believed in the policy of liveand let live. Dugdale 1932, quoted in Axelrod, p. 74

The high commands of the two sides did notshare the view of the common soldier whosaid: The real reason for the quietness of some sections ofthe line was that neither side had any intention ofadvancing in that particular district... If the Britishshelled the Germans, the Germans replied, and thedamage was equal: if the Germans bombed anadvanced piece of trench and killed fiveEnglishmen, an answering fusillade killed fiveGermans.Belton Cobb 1916,quoted in Axelrod p. 76

The ethics that developed are illustrated in thisincident, related by a British officer recallinghis experience while facing a Saxon unit of theGerman Army. I was having tea with A Company when we heard alot of shouting and went out to investigate. We foundour men and the Germans standing on their respectiveparapets. Suddenly a salvo arrived but did no damage.Naturally both sides got down and our men startedswearing at the Germans, when all at once a braveGerman got on to his parapet and shouted out We arevery sorry about that; we hope no one was hurt. It isnot our fault, it is that damned Prussian artillery. Rutter 1934, quoted in Axelrod, p. 85