Unit III: The Evolution of Cooperation

•Can Selfishness Save the Environment?•Repeated Games: the Folk Theorem•Evolutionary Games•A Tournament•How to Promote Cooperation4/13

How to Promote Cooperation

• Advice to Participants• Advice to Reformers• The Role of Institutions• Learning to Cooperate • Tournament Instructions• Unit Review

How to Promote Cooperation

Axelrod offers two types of advice on how to promote cooperation (1984, pp.199-244):

• Advice to Participants - the players in the game

• Advice to Reformers - the rules makers

How to Choose Effectively(Axelrod, 1984: 109-123.)

• Don’t be envious• Don’t be the first to defect• Reciprocate both cooperation and defection• Don’t be too clever

These are intended as the ingredients of a strategy that will, in the long range and against a wide range of opponents, advance the player’s interests.

Advice to Participants

Nice: Never be the first to defect. A nice strategy signals a willingness to cooperate and may induce reciprocal

cooperation. Nice strategies did best in Axelrod’s tournaments.

Provocable: Punish defection. Don’t get fleeced by an exploiter.

Forgiving: Reciprocate cooperation. Triggers may be susceptible to misunderstandings, mistakes, etc, that can lead

otherwise cooperative players into spirals of alternating or mutual defection.

Clear: Be easily understood and predictable. Don’t be too clever. A simple rule works best.

Advice to Participants

Advice to ParticipantsSucker the Simple?

Recall that while TIT FOR TAT never beats is opponent, PAVLOV always defects against a naïve cooperator. Hence, the success of PAVLOV in newer tournaments may suggest it is wise to exploit the weak, both

(i) for “egoistic” benefit; and (ii) to increase the overall fitness of the population.

Either the simple will learn (not to let themselves be exploited), or they will be winnowed.

Axelrod offers five concrete suggestions on how “the strategic setting itself can be transformed in order to promote cooperation among the players” (124-141):

• Enlarge the “shadow of the future”• Change the payoffs• Teach people to care about each other• Teach reciprocity• Improve recognition abilities

Advice to Reformers

Repeated interactions provide the conditions necessary for cooperation by transforming the nature of the interaction in two ways:

· “Enlarge the shadow of the future”

· Increase the amount of information in the system. This may reduces strategic uncertainty (e) and allow players to coordinate their expectations and behavior on mutually beneficial outcomes.

d 1

e 0

Advice to Reformers

d* = T-RT-P

The Role of Institutions

Axelrod and Keohane (1986) apply the lessons from The Evolution of Cooperation to international relations, arguing that “not only can actors in world politics pursue different strategies within an established context of interaction, they may also seek to alter the context through building institutions embodying particular principles, norms, rules, or procedures for the conduct of international relations” (p. 228).

The Role of Institutions

Building an institution implies changing the context within which states make their decisions, and this may make it possible to achieve cooperation where it had been inaccessible.

Hence, institutions “contribute to cooperation (...) by changing the context within which states make decisions based on self-interests” (Keohane, 1984, p. 13).

The Folk Theorem

(R,R)

(T,S)

(S,T)

(P,P)

The shaded area is the set of SPNE. The segment PP,RR is the set of “collectively stable” strategies,for (d > d*).

Learning to Cooperate

We have seen that whereas cooperation is irrational in a one-shot Prisoner’s Dilemma, it may be rational (i.e., achieved in a SPNE), if the game is repeated and “the shadow of the future” is sufficiently large:

d > (T-R)/(T-P) (i)

Repeated interaction is a necessary but not a sufficient condition for cooperation. In addition, players must have reason to believe the other will reciprocate.

This involves judging intentions, considerations of fairness, (mis)communication, trust, deception, etc.

Learning to Cooperate

Consider two fishermen deciding how many fish to remove from a commonly owned pond. There are Y fish in the pond.

• Period 1 each fishery chooses to consume (c1, c2).• Period 2 remaining fish are equally divided (Y –

(c1+c2))/2). 

 

c1 = (Y – c2)/2  

 

Y/4Y/3 c1

c2

Y/3Y/4 c2 = (Y – c1)/2

NE: c1 = c2 = Y/3

Social Optimum: c1 = c2 = Y/4

Learning to Cooperate

Consider two fishermen deciding how many fish to remove from a commonly owned pond. There are Y fish in the pond.

• Period 1 each fishery chooses to consume (c1, c2).• Period 2 remaining fish are equally divided (Y –

(c1+c2))/2). 

 

c1 = (Y – c2)/2  

 

Y/4Y/3 c1

c2

Y/3Y/4 c2 = (Y – c1)/2

If there are 12 fish in the pond, each will consume (Y/3) 4 in the spring and 2 in the fall in a NE. Both would be better off consuming (Y/4) 3 in the fall, leaving 3 for each in the spring.

Learning to Cooperate

If there are 12 fish in the pond, each will consume (Y/3) 4 in the spring and 2 in the fall in a NE. Both would be better off consuming (Y/4) 3 in the fall, leaving 3 for each in the spring.

C 9, 9 7.5,10

D 10,7.5 8, 8

C D

A Prisoner’s Dilemma

What would happen if the game were

repeated?

Learning to Cooperate

C = 3 in the springD = 4 “ “

Imagine the fisherman make the following deal: Each will Cooperate (consume only 3) in the spring as long as the other does likewise; as soon as one Defects, the other will Defect for ever, i.e., they adopt trigger strategies.

This deal will be stable if the threat of future punishment makes both unwilling to Defect, i.e., if the one period gain from Defect is not greater than the discounted future loss due to the punishment:

(T – R) < (dR/(1-d) – dP/(1-d)) (ii)

Learning to Cooperate

Imagine there are many fishermen, each of whom can adopt either D(efect), C(ooperate), or T(rigger). In every generation, each fisherman plays against every other. After each generation, those that did poorly can switch to imitate those that did better. Eventually, C will die out, and the population will be dominated by either D or T, depending on the discount parameter.

Noise (miscommunication) can also affect the outcome.

Learning to Cooperate

Tournament AssignmentDesign a strategy to play an Evolutionary Prisoner’s Dilemma Tournament.

Entries will meet in a round robin tournament, with 1% noise (i.e., for each intended choice there is a 1% chance that the opposite choice will be implemented). Games will last at least 1000 repetitions (each generation), and after each generation, population shares will be adjusted according to the replicator dynamic, so that strategies that do better than average will grow as a share of the population whereas others will be driven to extinction. The winner or winners will be those strategies that survive after at least 100,000 generations. 

Tournament AssignmentTo design your strategy, access the programs through your fas Unix account. The Finite Automaton Creation Tool (fa) will prompt you to create a finite automata to implement your strategy. Select the number of internal states, designate the initial state, define output and transition functions, which together determine how an automaton “behaves.” The program also allows you to specify probabilistic output and transition functions. Simple probabilistic strategies such as GENEROUS TIT FOR TAT have been shown to perform particularly well in noisy environments, because they avoid costly sequences of alternating defections that undermine sustained cooperation.

Some examples:

C CD D .9D

C,D

C D

D

C

CC

DSTART

ALWAYS DEFECT TIT FOR TAT GENEROUS PAVLOV

Tournament Assignment

A number of test runs will be held and results will be distributed to the class. You can revise your strategy as often as you like before the final submission date. You can also create your own tournament environment and test various designs before submitting.

Entries must be submitted by 5pm, Friday, May 6.

D

Creating your automaton

To create a finite automaton (fa) you need to run the fa creation program. Log into your unix account and at the % prompt, type:

~neugebor/simulation/fa

From there, simply follow the instructions provided. Use your user name as the name for the fa. If anything goes wrong, simply press “ctrl-c” and start over.

Computer Instructions

You must log in to a “nice” server.

Creating your automaton

The program prompts the user to:

• specify the number of states in the automaton, with an upper limit of 50. For each state, the program asks:

• “choose an action (cooperate or defect);” and • “in response to cooperate (defect), transition to what state?”

Finally, the program asks:• specify the initial state.

The program also allows the user to specify probabilistic outputsand transitions.

Computer Instructions

Submitting your automaton

After creating the fa, submit it by typing:cp name.fa ~neugebor/ece1040.11chmod 744 ~neugebor/ece1040.11/name.fa

where name is the name you’ve given to your automaton. You may resubmit as often as you like before the submission deadline.

Computer Instructions

You must log in to a “nice” server.

Testing your automaton

You may wish to test your fa before submitting it. You can do this by running sample tournaments with different fa’s you’ve created. To run the tournament program, you must copy it into your own account. You can do this by typing:mkdir simulationcp ~neugebor/simulation/* simulation

To change into the directory with the tournament program type:cd simulation

Then, to run the tournament type:./tournament

NOTE: To run the tournament, you must be logged on to a nice server.

Computer Instructions

Testing your automaton

Follow the instructions provided. Note that running a tournament with many fa’s can be computationally intensive and may take a long time to complete. Use your favorite text editor to view the results of the tournament (“less” is an easy option if you are unfamiliar with unix -- type “less textfilename” to open a text file).To create extra automaton to test in your tournament type: ./faName each fa whatever you want by entering the any name you wish to use instead of your user name. Initially six different kinds of fa’s are in the directory: D, C, TFT, GRIM, PAVLOV, GTFT, AND RANDOM. Experiment with these and others as you like.

Computer Instructions

Simulating EvolutionPAV

TFT

GRIM (TRIGGER)D

R

C

Population shares for 6 RPD strategies (including RANDOM), with noise at 0.01 level.

Pop. Shares 0.50

0.40

0.30

0.20

0.10

0.00Generations

GTFT?

Preliminary Tournament Results

Test007.b

0

0.2

0.4

0.6

0.8

1

Generations

Popu

latio

n Sh

ares

defectcooperategrimtit4tatpavlovrandomataubbrillcgerrydanielsdaniels1daniels2daranowdemashk

After 5000 generations(as of 4/25/02)

Avg. Score (x10)

Preliminary Tournament ResultsTest.009

0

0.2

0.4

0.6

0.8

1

1.2

Generations (x50)

Popu

latio

n Sh

ares

defectcooperategrimtit4tatpavlovrandomataubbjweissbmartinbrillcgerrydanielsdaniels1daniels2

After 5000 generations(10pm 4/27/02)

Preliminary Tournament ResultsAfter 30000 generations

(4/20/09)fs40q.10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 95

1000

1950

2900

3850

4800

5750

6700

7650

8600

9550

1050

0

1145

0

1240

0

1335

0

1430

0

1525

0

1620

0

1715

0

1810

0

1905

0

2000

0

2095

0

2190

0

2285

0

2380

0

2475

0

2570

0

2665

0

2760

0

2855

0

2950

0

Generations

Popu

latio

n Sh

ares

defectcooperategrimtit4tatpavlovgpavlovrandomtf2tgtft32keymetaoascicarus1manghamtrialcrimson11tft85ngoekenmafiapagDicarus2muffy1meta-key

Unit ReviewCan Selfishness Save the Environment?

Common property resources (e.g., clean air, clean water), will often be overconsumed and public goods (e.g., legal system, public radio) undersupplied because of the incentive to defect, or free-ride. These are examples of n-person Prisoner’s Dilemmas.

Viewed as a one-shot interaction, the Prisoner’s Dilemma has pessimistic implications for rational behavior.

Yet examples from biology and elsewhere suggest that strictly selfish behavior may take on a socially cooperative form in the long run e.g., the “selfish gene” gives rise to kinship relations and altruism.

Unit ReviewRepeated Games: The Folk Theorem

Analysis of repeated games suggest it may be possible for cooperation to emerge over the course of a long-term interaction.

Theorem: Any payoff that pareto-dominates the one-shot NE can be supported in a SPNE of the repeated game, if the discount parameter is sufficiently high.

Axelrod (1984) argued that when the PD is repeated and the “shadow of the future” is large (d > d*), players will have an incentive to cooperate.

The Indefinitely Repeated Prisoner’s Dilemma Tournament

The success of TIT FOR TAT in tournaments of the repeated prisoner’s dilemma has led many to the optimistic conclusion that whereas rational players will choose to defect in a one-shot prisoner’s dilemma, repeated interaction may allow the evolution of cooperation over time. This is because TIT FOR TAT does well against other cooperative strategies and hence can grow as a proportion of the population over repeated plays of the game.

Yet in evolutionary games, there is reason to believe that TIT FOR TAT will perform less well in noisy environments and over substantially longer (and more complex) time horizons.

Unit Review

Pop. Share0.140

0.100

0.060

0.020

0 200 400 600 800Generations

Simulating Evolution

1(TFT)326

7,9

10411

5

81814,12,1513

No. = Position after 1st Generation

Source:Axelrod 1984, p. 51.

The Indefinitely Repeated Prisoner’s Dilemma Tournament

The success of TIT FOR TAT in tournaments of the repeated prisoner’s dilemma has led many to the optimistic conclusion that whereas rational players will choose to defect in a one-shot prisoner’s dilemma, repeated interaction may allow the evolution of cooperation over time. This is because TIT FOR TAT does well against other cooperative strategies and hence can grow as a proportion of the population over repeated plays of the game.

Yet in evolutionary games, there is reason to believe that TIT FOR TAT will perform less well in noisy environments and over substantially longer (and more complex) time horizons.

Unit Review

Thus, while repetition is necessary, it is not a sufficient condition for the evolution of cooperation. The Folk theorem tell us that when the game is repeated, almost anything can happen.

Evolutionary stability is a much finer selection criterion, but it remains an open question whether there is an ESS in RPD. If it exists, is it unique? Efficient?

An alternative view holds that the winner of any evolutionary tournament will depend upon the set of strategies submitted. 

When cooperation doesn’t emerge “spontaneously,” it may be possible to change the rules of the game. However, changing the rules of the game requires an agreement (consensus), and this is a different sort of solution than an equilibrium.

  ·      

 

Unit Review

Simulating EvolutionPAV

TFT

GRIM (TRIGGER)D

R

C

Population shares for 6 RPD strategies (including RANDOM), with noise at 0.01 level.

Pop. Shares 0.50

0.40

0.30

0.20

0.10

0.00Generations

GTFT?

For Next Time

UNIT IV THINKING ABOUT THINKING

4/20 Schelling, Choice and Consequence, pp. 57-112; 328-46.