Unit III: The Evolution of Cooperation. Can Selfishness Save the Environment? Repeated Games: the Folk Theorem Evolutionary Games A Tournament How to Promote Cooperation/Unit Review. 4/16. Today’s Agenda. Bounded Rationality Designing Repeated Game Strategies Finite Automata
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4/16
In the Repeated Prisoner’s Dilemma, it has been suggested that “uncooperative behavior is the result of ‘unbounded rationality’, i.e., the assumed availability of unlimited reasoning and computational resources to the players” (Papadimitrou, 1992: 122). If players are boundedly rational, on the other hand, the cooperative outcome may emerge as the result of a “muddling” process. They reason inductively and adapt (imitate or learn) locally superior strategies.
Thus, not only is bounded rationality a more “realistic” approach, it may also solve some deep analytical problems, e.g., resolution of finite horizon paradoxes.
Imagine a very simple decision making machine playing a repeated game. The machine has very little information at the start of the game: no knowledge of the payoffs or “priors” over the opponent’s behavior. It merely makes a choice, receives a payoff, then adapts its behavior, and so on.
The machine, though very simple, is able to implement a strategy against any possible opponent, i.e., it “knows what to do” in any possible situation of the game.
A repeated game strategy is a map from a history to an action.A history is all the actions in the game thus far ….
…T-3T-2T-1To
CCCCDCC
CCCDDCD
History at time To
?
A repeated game strategy is a map from a history to an action.A history is all the actions in the game thus far, subject to the constraint of a finite memory:
…T-3T-2T-1To
CCCCDCC
CCCDDCC
History of memory-4
?
TIT FOR TAT is a remarkably simple repeated game strategy. It merely requires recall of what happened in the last round (memory-1).
…T-3T-2T-1To
CCCCDDC
CCCDDCD
History of memory-1
?
A FINITE AUTOMATON (FA) is a mathematical representation of a simple decision-making process. FA are completely described by:
The output function determines an action, C or D, in each state.
The transition function determines how the FA changes states in
response to the inputs it receives (e.g., actions of other FA).
Rubinstein, “Finite Automata Play the Repeated PD” JET, 1986)
FA will implement a strategy against any possible opponent, i.e., they “know what to do” in any possible situation of the game.
FA meet in 2-player repeated games and make a move in each round (either C or D). Depending upon the outcome of that round, they “decide” what to play on the next round, and so on.
FA are very simple, have no knowledge of the payoffs or priors over the opponent’s behavior, and no deductive ability. They simply read and react to what happens. Nonetheless, they are capable of a crude form of “learning” — they receive payoffs that reinforce certain behaviors and “punish” others.
C D
D
C
D
C
“TIT FOR TAT”
C C D
D
D
C
C
D
C
“TIT FOR TWO TATS”
C,D
C,D
C
C D
Some examples:
D
D
C
D
C
D
D
START
C
“ALWAYS DEFECT” “TIT FOR TAT” “GRIM (TRIGGER)”
C
C CD
C
C
D
C
D
C
C
D
D
D
D
“PAVLOV” “M5”
Calculating Automata Payoffs
Time-average payoffs can be calculated because any pair of FA will achieve cycles, since each FA takes as input only the actions in the previous period (i.e., it is “Markovian”).
For example, consider the following pair of FA:
D
C C
C
D
C
C
C
C
C
D
D
D
D
D
“PAVLOV” “M5”
Calculating Automata Payoffs
PAVLOV:C
M5:D
D
C C
C
D
C
C
C
C
C
D
D
D
D
D
“PAVLOV” “M5”
Calculating Automata Payoffs
Payoff05105105AVG=2
PAVLOVCDDCDDCD
M5DCDDCDDC
Payoff5015015 AVG=2
cyclecycle cycle
D
C C
C
D
C
C
C
C
C
D
D
D
D
D
“PAVLOV” “M5”
The Indefinitely Repeated Prisoner’s Dilemma Tournament
Axelrod (1980a,b, Journal of Conflict Resolution).
A group of scholars were invited to design strategies to play indefinitely repeated prisoner’s dilemmas in a round robin tournament.
Contestants submitted computer programs that select an action, Cooperate or Defect, in each round of the game, and each entry was matched against every other, itself, and a control, RANDOM. . .
The Indefinitely Repeated Prisoner’s Dilemma Tournament
Axelrod (1980a,b, Journal of Conflict Resolution).
Contestants did not know the length of the games. (The first tournament lasted 200 rounds; the second varied probabilistically with an average of 151.)
The first tournament had 14 entrants, including game theorists, mathematicians, psychologists, political scientists, and others.
Results were published and new entrants solicited. The second tournament included 62 entrants . . .
The Indefinitely Repeated Prisoner’s Dilemma Tournament
TIT FOR TAT won both tournaments!
TFT cooperates in the first round, and then does whatever the opponent did in the previous round.
TFT “was the simplest of all submitted programs and it turned out to be the best!” (31).
TFT was submitted by Anatol Rapoport to both tournaments, even after contestants could learn from the results of the first.
The Indefinitely Repeated Prisoner’s Dilemma Tournament
This result has been so influential that “some authors use TIT FOR TAT as though it were a synonym for a self-enforcing, cooperative agreement” (Binmore, 1992, p. 433). And many have taken these results to have shown that TFT is the “best way to play” in IRPD.
TIT FOR TATis susceptible to 2 types of perturbations:
Mutations: random Cs can invade TFT (TFT is not ESS), which in turn allows exploiters to gain a foothold.
Noise: a “mistake” between a pair of TFTs induces CD, DC cycles (“mirroring” or “echo” effect).
TIT FOR TAT never beats its opponent; it wins because it elicits reciprocal cooperation. It never exploits “naively” nice strategies.
(See Poundstone: 242-248; Casti 76-84.)
Class Tournament
Imagine a population of strategies matched in pairs to play repeated PD, where outcomes determine the number of offspring each leaves to the next generation.
Then, strategies that do better than average will grow as a share of the population and those that do worse than average will eventually die-out. . .
Design 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 10,000 generations.
To 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.
C,D
C
C
C D
Some examples:
D
D
C
D
C
.9D
D
START
C
D
ALWAYS DEFECT TIT FOR TAT GENEROUS PAVLOV
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 9.
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.
Creating your automaton
The program prompts the user to:
The program also allows the user to specify probabilistic outputs
and transitions.
Submitting your automaton
After creating the fa, submit it by typing:
cp username.fa ~neugebor/ece08
chmod 744 ~neugebor/ece08/username.fa
where username is your user name. You may resubmit as often as you like before the submission deadline.
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 simulation
cp ~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 an iceserver.
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:
./fa
Name 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, AND RANDOM. Experiment with these and others as you like.
Axelrod offers two types of advice on how to promote cooperation (1984, pp.199-244):
How to Choose Effectively
(Axelrod, 1984: 109-123.)
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.
Sucker 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):
Repeated interactions provide the conditions necessary for cooperation by transforming the nature of the interaction in two ways:
d 1
d* = T-R
T-P
e0
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).
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).
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
The shaded area is the set of SPNE. The segment PP,RR is the set of “collectively stable” strategies,
for (d > d*).
(S,T)
(R,R)
(P,P)
(T,S)
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.
c1 = (Y – c2)/2
NE: c1 = c2 = Y/3
Social Optimum: c1 = c2 = Y/4
c2
Y/3
Y/4
c2 = (Y – c1)/2
Y/4Y/3 c1
Learning to Cooperate
Consider two fishermen deciding how many fish to remove from a commonly owned pond. There are Y fish in the pond.
c1 = (Y – c2)/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.
c2
Y/3
Y/4
c2 = (Y – c1)/2
Y/4Y/3 c1
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 D
C = 3 in the spring
D = 4 “ “
C9, 9 7.5,10
A Prisoner’s Dilemma
What would happen if the game were repeated?
D 10,7.5 8, 8
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)
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.
Can 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.
Repeated 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.
Pop. Share
0.140
0.100
0.060
0.020
0 200400600800Generations
1(TFT)
3
2
6
7,9
10
4
11
5
8
No. = Position after 1st Generation
18
14,12,15
13
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.
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.
·
Pop. Shares
PAV
TFT
GRIM (TRIGGER)
0.50
0.40
0.30
0.20
0.10
0.00
GTFT?
R
D
C
Generations
Population shares for 6 RPD strategies (including RANDOM), with noise at 0.01 level.
After 5000 generations
(as of 4/25/02)
Avg. Score (x10)
After 5000 generations
(10pm 4/27/02)
After 20000 generations
(7am 4/28/02)
After 600 generations
(4/22/05)
After 1000 generations
(4/27/05)
UNIT IV THINKING ABOUT THINKING
4/23Schelling, Choice and Consequence,
pp. 57-112; 328-46.