Cooperation

Evolution of Cooperation The importance of being suspicious

Do we see cooperation in Nature?

Do we see cooperation in Nature?

United Nations

Big Picture

Do we see cooperation in Nature?

Do we see cooperation in Nature?

Do we see cooperation in Nature?

Do we see cooperation in Nature?

Do we see cooperation in Nature? If I give you some DNA will you give me some? Ya, Sure.

Just promise it won’t get complicated between us

Small Picture: Bacteria Sex

Martin A. Nowak (2006): 

Genes cooperate in genomes.

Martin A. Nowak (2006):  

Genes cooperate in genomes. Chromosomes cooperate in eukaryotic cells.

Martin A. Nowak (2006):   

Genes cooperate in genomes. Chromosomes cooperate in eukaryotic cells. Cells cooperate in multicellular organisms.

Martin A. Nowak (2006):



Genes cooperate in genomes. Chromosomes cooperate in eukaryotic cells. Cells cooperate in multicellular organisms.



There are many examples of cooperation among

 

animals.

Martin A. Nowak (2006):     

Genes cooperate in genomes. Chromosomes cooperate in eukaryotic cells. Cells cooperate in multicellular organisms. There are many examples of cooperation among animals. Humans are the champions of cooperation: From hunter-gatherer societies to nation-states, cooperation is the decisive organizing principle of human society.

Martin A. Nowak (2006):     



Genes cooperate in genomes. Chromosomes cooperate in eukaryotic cells. Cells cooperate in multicellular organisms. There are many examples of cooperation among animals. Humans are the champions of cooperation: From hunter-gatherer societies to nation-states, cooperation is the decisive organizing principle of human society. The question of how natural selection can lead to cooperative behavior has fascinated evolutionary biologists for several decades.

Cooperation as a “paradox”: The Tragedy of the Commons 

Take a fishing lake where there is an upper limit on how much harvest can be taken in a sustainable manner.



Above this limit, the fish pop. eventually crashes and everyone is worse off.

Cooperation as a “paradox”: The Tragedy of the Commons 

And they have to wait for someone to come and give them fish...

Cooperation as a “paradox”: The Tragedy of the Commons 

And they have to wait for someone to come and give them fish...

Tragedy of the Commons What should you do? Best: Everyone fishes below the limit, but you cheat and fish more.

Tragedy of the Commons What should you do? Best: Everyone fishes below the limit, but you cheat and fish more. Next best: Everyone fishes below the limit, and you do too.

Tragedy of the Commons What should you do? Best: Everyone fishes below the limit, but you cheat and fish more. Next best: Everyone fishes below the limit, and you do too. Pretty bad: Everyone fishes above the limit, and you do too.

Tragedy of the Commons What should you do? Best: Everyone fishes below the limit, but you cheat and fish more. Next best: Everyone fishes below the limit, and you do too. Pretty bad: Everyone fishes above the limit, and you do too. Worst: Everyone fishes above the limit, but you don’t for some reason.

Tragedy of the Commons What should you do? Results. Lesson: No matter what everyone else is doing, you always do better by cheating.

Tragedy of the Commons What should you do? Results. Lesson: No matter what everyone else is doing, you always do better by cheating. Conclusion: Everyone cheats. Everyone does pretty bad.

Tragedy of the Commons Assigning score (5) Best (T): temptation to cheat (3) Next best (R): reward for cooperating (1) Pretty bad (P): punishment for everyone cheating (0) Worst (S): suckers payoff for cooperating against cheaters **Scores are arbitrary, while obeying T > R > P > S, and an additional condition: (T+P)/2 > R. These scores are the convention.

Tragedy of the Commons Simplified to two people Payoffs: (p1,p2) p1↓ p2→

Cooperator

Defector

Cooperator (Fish below limit)

(3,3)

(0,5)

Defector (Fish above limit)

(5,0)

(1,1)

Tragedy of the Commons Simplified to two people Payoffs: (p1,p2) p1↓ p2→

Cooperator

Defector

Cooperator (Fish below limit)

(3,3)

(0,5)

Defector (Fish above limit)

(5,0)

(1,1)

***This is the Prisoner’s Dilemma

The Prisoner’s Dilemma (PD) Your payoff you↓ Cooperator Defector

Cooperator

Defector

3 5

0 1

If you are playing a cooperator, you can do best by defecting

The Prisoner’s Dilemma (PD) Your payoff you↓ Cooperator

Cooperator

Defector

3

Defector

5

0 1

If you are playing a cooperator, you can do best by defecting If you are playing a defector, you can do best by defecting

The Prisoner’s Dilemma (PD) 

No matter what type of strategists are in a population, the best response is always to defect.

The Prisoner’s Dilemma (PD) 



No matter what type of strategists are in a population, the best response is always to defect. If we consider score to be a measure of fitness, then we should expect defectors to leave more offspring.

The Prisoner’s Dilemma (PD) 





No matter what type of strategists are in a population, the best response is always to defect. If we consider score to be a measure of fitness, then we should expect defectors to leave more offspring. Defectors take over, and can’t be invaded by a cooperator.

Nice guys finish last... 

So defection dominates, even though everyone does worse than if everyone cooperated.

Nice guys finish last... 



So defection dominates, even though everyone does worse than if everyone cooperated. “Everyone cooperating” is an optimal strategy for the population, but it is unstable. Defectors invade and take over.

Nice guys finish last... 





So defection dominates, even though everyone does worse than if everyone cooperated. “Everyone cooperating” is an optimal strategy for the population, but it is unstable. Defectors invade and take over. How can we explain the emergence of cooperation?

Achieving Cooperation: Direct Reciprocity 

When there is a potential for future rewards, cooperation could evolve via reciprocity (Trivers).

Achieving Cooperation: Direct Reciprocity 



When there is a potential for future rewards, cooperation could evolve via reciprocity (Trivers) We could have two agents repeat the game. Call this the Iterated PD (IPD).

Achieving Cooperation: Direct Reciprocity 





When there is a potential for future rewards, cooperation could evolve via reciprocity (Trivers) We could have two agents repeat the game. Call this the Iterated PD (IPD). Axelrod (1980a, b) hosted two roundrobin tournaments of the IPD. A wide range of complex strategies were submitted...

Achieving Cooperation: Direct Reciprocity 

Amazingly, the winner of both tournaments was the simplest strategy entered: tit-for-tat (TFT).

Achieving Cooperation: Direct Reciprocity 



Amazingly, the winner of both tournaments was the simplest strategy entered: tit-for-tat (TFT). TFT cooperates on the first turn then copies its opponent’s previous move.

Achieving Cooperation: Direct Reciprocity 





Amazingly, the winner of both tournaments was the simplest strategy entered: tit-for-tat (TFT). TFT cooperates on the first turn then copies its opponent’s previous move. TFT can be considered as a special case of a “reactive strategy.”

Reactive Strategies for the IPD 

Reactive strategies are given by an ordered triple (y,p,q) that define their behaviour in the IPD.

Reactive Strategies for the IPD Reactive strategies are given by an ordered triple (y,p,q) that define their behaviour in the IPD. y – probability of C on the 1st turn p – probability of C following a C q – probability of C following a D 

Reactive Strategies for the IPD Thus TFT is (1,1,0). Other interesting strategies at the vertices are: Always defect AllD = (0,0,0) Always cooperate AllC = (1,1,1) 

Reactive Strategies for the IPD Thus TFT is (1,1,0). Other interesting strategies at the vertices are: Always defect AllD = (0,0,0) Always cooperate AllC = (1,1,1)  (0,1,0) is “Suspicious TFT” since it defects on the first turn (nervous of strangers) then has TFT behaviour. 

Evolution of TFT in the IPD. 

Many models consider the infinitely iterated version, or a sufficiently long version of the IPD (Nowak & Sigmund, 1992; 1994; Imhof et al., 2005)

Evolution of TFT in the IPD. 



Many models consider the infinitely iterated version, or a sufficiently long version of the IPD (Nowak & Sigmund, 1992; 1994; Imhof et al., 2005) This completely discounts the effects of the first turn, which allows for the reduction of strategy space from (y,p,q) to a strategy square: (p,q).

Is this biologically reasonable? 

At some levels of organization, the assumption of long games may be founded.

Is this biologically reasonable? 



At some levels of organization, the assumption of long games may be founded. For multi-cellular organisms, this assumption seems hard to justify.

Is this biologically reasonable? 

 

At some levels of organization, the assumption of long games may be founded. For multi-cellular organisms, this assumption seems hard to justify. Also, if encounters are infrequent the agents may not recognize each other when they play again (and remember their opponents “last move”). Or end interactions early with defectors.

Let’s make a model 



 

Let ‘N’ individuals play the PD iterated ‘m’ times (m = 10 for results). Let each individual be given by (y,p). ‘y’ matters in short games. Start the population always defecting. Have many generations of: selection, reproduction, mutation, death.

Selection

Selection: Pairing

Selection: Playing

Selection: First Play Probability y2 Probability 1 - y2

Probability 1 – y4 Probability y4 

Probability of cooperating on the first turn is defined by each player’s ‘y’ value

Selection: Subsequent Plays Probability p2 Probability 1 - p2



Probability of p2 cooperating on round i given that p4 cooperated on round i-1 is p2.



p4 defects in round i if p2 defected in round i - 1

Reproduction, Mutation, Death 

 



Based on their cumulative scores, an individual is selected stochastically for reproduction. Another individual is selected randomly to be replaced. The reproducing individual produces an offspring with the same ‘y’ and ‘p’ value with a small chance of a random mutation. All results are for population size N = 30, number of iterations m = 10, number of populations D = 50, and number of generations = 10000

Results: no noise, weak selection No noise, weak selection (1) 1 0.9

p (reciprosity)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

y (generosity)

0.7

0.8

0.9

1

Results: no noise, intermediate selection No Noise, Intermediate Selection (5) 1 0.9

p (reciprosity)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

y (generosity)

0.7

0.8

0.9

1

Results: no noise, strong selection No Noise, Strong Selection (10) 1 0.9

p (reciprosity)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

y (generosity)

0.7

0.8

0.9

1

Results: noise = 0.00001, strong selection (10) Noise = 0.0001, Strong Selection (10) 1 0.9

p (reciprosity)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

y (generosity)

0.7

0.8

0.9

1

Results: noise = 0.0001, strong selection (10) Noise = 0.0001, Strong Selection (10) 1 0.9

p (reciprosity)

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

y (generosity)

0.7

0.8

0.9

1

Results: noise = 0.0001, very strong selection (30) Noise = 0.0001, Very Strong Selection (30) 1 0.9 0.8

p (reciprosity)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

y (generosity)

0.7

0.8

0.9

1

I have time for discussion? 



Without noise, a population can evolve toward TFT for sufficiently strong selection – even though the game is iterated a short amount With even a modest amount of noise, selection must be increased in strength to see natural selection (as opposed to drift)

I have time for discussion? 



For high noise (0.1%) A population must be under very strong selection to reach TFT from always defect A population accomplishes this using a trajectory close to STFT.

Thanks, 



Students, organizers, and mentors for your discussions! Special thanks to Alex and Lou for your help and patience.