Game Theory: From Zero-Sum to Non-Zero-Sum CSCI 3202, Fall 2010
Assignments Reading (should be done by now): Axelrod (at website) Problem Set 3 due Thursday next week
Two-Person Zero Sum Games The notion of a dominant choice Tricks for reducing game matrices The solution of a zero-sum game The value of a game The minimax theorem Pure vs. Mixed strategies Symmetric games
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
The Minimax Theorem in Game Theory Applies to zero-sum, two-person finite games. The minimax theorem says that in such a game, there is a value V for the game (the same value V for both players). Given an optimal strategy (possibly a mixed strategy), each player can be assured (on average) of obtaining at least V for the game regardless of what the other player does. What this means, essentially, is that both players can examine such a matrix and determine beforehand (and regardless of the other player s plan) what they need to do to ensure receiving an average of V for the game.
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 : adding intangibles to a utility function Two players, Red and Blue. Red is given $1000 and is told that he can choose an amount to be taken out of this total to offer Blue. Blue then chooses whether to accept this amount or not. If Blue rejects Red s offer, both get nothing.
Some Simple Axelrod- Tournament-like strategies All-Defect simply defects on every round Poor-Trusting-Fool simply cooperates on every round Random is a test strategy that simply cooperates or defects randomly Unforgiving cooperates initially until the first time that the other player defects; after that, Unforgiving defects forever
The Tit-for-Tat Strategy Tit-for-Tat cooperates on the first round. Thereafter, on every subsequent round, it simply imitates what the other player did on the previous round. (That is, Tit-for-Tat does at round N what the other player did at round N-1.)
Blue Cooperate Red Defect Cooperate 3 3 5 0 0 5 1 1 Defect
Simulating Evolution in a PD World Consider a world of animals, each one playing in a PD tournament against representative rules from the original tournament. Each animal is completely determined by its actions in the PD game. Each animal has a memory of the past three rounds only. Since each round has four possibilities (DC, CC, CD, CC), there are sixty-four responses that each animal must remember. Thus, each animal is essentially determined by a 64-entry table (one entry for each of the possible preceding three-round sets).
PD World, Continued It seems, then, that we need sixty-four bits to specify an animal. To be a bit more precise, since each animal must begin on the first round by making some choice, we include an additional six bits to specify which of the sixtyfour table choices we will begin with. Thus, each animal requires seventy bits total. In each generation, a population of our animals plays against the representative rules. The best-scoring of these animals are preferentially allowed to survive to the next generation. We also include new animals created by mating between the successful animals. New animals are derived via crossover (between two animals) and occasional mutations.
Rules that Evolve in the PD system Don t rock the boat: continue to cooperate after three mutual cooperations (cooperate after CC, CC, CC) Be provocable: defect when the other player defects out of the blue (defect after CC, CC, CD) Accept an apology: continue to cooperate after cooperation has been restored (cooperate after DC, CD, CC) Forget: cooperate when mutual cooperation has been restored after an exploitation (cooperate after CD, CC, CC) Accept a rut: defect after three mutual defections (defect after DD, DD, DD)
Some Additional Topics in Non- Zero-Sum Games Noise Altruism and evolutionary theory Variants of the prisoner s dilemma
Blue Red 5 8.2 4 7.9-100 8 4.5 7.8
Altruism and Animal Behavior Prisoner s dilemma-type cooperation Ground squirrel alarm calls Bee-eater altruism in child-raising
Mother Father 1/2 1/2 1/2 1/2 A child shares 1/2 its genetic endowment with each parent. Siblings share 1/2 their genetic endowment with each other.
Hymenoptera (bees, wasps, ants) Mother Father (haploid) 1 1/2 1/2 1/2 1 son daughter1 daughter2 Sisters share 75% of their genetic endowment, but only 25% with their brothers.
Red Blue
Red Blue