Mathematical Model of Cooperation Theory

InvasionAlliances are based upon mutual need, not love. – John Ringo

The Tit For Tat Cooperation Theory says, you should cooperate (vs. defect) with others, especially strangers, when you are likely to keep encountering them. This is because “the word” spreads about defectors, such that fewer people work with them and choose instead to work with cooperators.

Said another way, the individual reward for ripping someone off becomes less attractive when there are no victims left; whereas the cumulative reward for cooperation keeps getting bigger for cooperators. There is even a mathematical model for this principle.

Mathematical Model of Cooperation Theory

Payoff Matrix John  
Cooperates Defects
Jane Cooperates Jane gets R
John gets R
Jane gets S
John gets T
  Defects Jane gets T
John gets S
Jane gets P
John gets P

R: Reward for mutual cooperation
P: Punishment for mutual defection
S: Sucker’s payoff
T: Temptation to defect

d : Present Value Of Future Payoffs; 0<d<1
      0: There is no future – get all you can get, now!
      1: You will live forever!


T>R>P>S (temptation to defect is high but so is reward for cooperation)
R>(T+S)/2 (a detail applicable for two-only players)
d “is close to” 1 (high likelihood of further future contact / future is as important as present)

A “Tit For Tat” strategy, in which each player cooperates with another player unless the other player defects, cannot be bettered by anyone adopting a different strategy.

This is called a “collectively stable” strategy. A population of defectors can be successfully invaded by groups of cooperators if d is large enough and if the frequency with which cooperators interact with each other instead of defectors is sufficiently high.

Individual cooperators can not successfully invade a population of defectors  unless they cease to interact with defectors. A group of defectors can not successfully invade a population of cooperators using any collectively stable strategy.

Cooperation Theory (Sitearm)
Social Grooming: The Role of Gossip (Sitearm)
The_Evolution_of_Cooperation (Wikipedia)
Prisoner’s_Dilemma (Wikipedia)
The Artful Universe, 1994, John D. Barrow; pp. 88-89 (Amazon)


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