Understanding and Solving Societal Problems with Modeling and Simulation

Transcription

1 Understanding and Solving Societal Problems with Modeling and Simulation Lecture 8: The Breakdown of Cooperation ETH Zurich April 15, 2013 Dr. Thomas Chadefaux

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3 Why Cooperation is Hard The Tragedy of the Commons

4 Tragedy of the Commons

5 Simulation activity (in R): tragedycommons.r Ten families herd goats that graze on 1 km 2 of common land. The milk a goat gives per day depends on how much grass it gets to eat milkpercow=e 1-1/10a milk.per.cow grass.per.cow

6 Optimal number of cows N, as determined by a social planner: max N Ne 1-1/(10(1/N)) so optimal N is 10, with a total production of 10 buckets total.production N

7 But suppose that the planner s decision cannot be enforced Then each family chooses its own number of goats g to maximize: ge (1-1/(10(1/(g+G))) Note that regardless of G, always better off choosing g=10 so each family keeps 10 goats, for a total production of 100e (1-1/(10*1/100)) = buckets! g family output G G

8 Solution? Problem: social planner generally does not know the production function; only knows: e 1-1/Aa Solution? Confiscate all the milk! (and redistribute it) Now each family solves: 1/10(g+G) e 1-1/A(1/g+G)), which is largest when g+g=a

9 Boiling it down The prisoner s dilemma

10 What Is Cooperation? Donor Pays a cost, c Recipient Gets a benefit, b

11 The Prisoner s Dilemma

12 Prisoner s dilemma I cooperate You cooperate b-c -c You defect b 0 b>c>0 Example: You cooperate I cooperate I defect I defect You defect

13 What is the dilemma? Even though everyone would be better off cooperating, defecting is always rational Everyone is worse off, but no one has an incentive to change

14 Natural selection favors defection C C C In any mixed population, defectors have a higher payoff than cooperators. C D D C C D D D D D D D

15 How to Achieve Cooperation? 5 mechanisms

16 5 mechanisms for cooperation Kin selection Direct reciprocity Indirect reciprocity Spatial selection Group selection

17 1. Kin selection Haldane: I would lay down my life for two brothers or eight cousins Kin selection is the evolutionary strategy that favors the reproductive success of one s relatives, even at the cost of one s own survival Mechanisms for kin selection: Kin recognition Viscous populations

18 1. Kin selection Hamilton s rule: r > c/b r coefficient of relatedness c cost of cooperation b benefit of cooperation

19 Empirical evidence: Red squirrels: surrogate mothers adopt related orphaned squirrel pups but not unrelated ones. c=risk to litter b= increased survival of pup. Females always adopted when rb>c Ground squirrels alerting others to danger is risky. Calls were made more frequently when relatives nearby Sea rocket: Competitive with strangers More considerate with siblings

20 2. Direct Reciprocity I help you, you help me

21 Repeated Prisoner s dilemma Instead of one-shot game, what happens if we play many times? How about 2 times? t1 t2 Me? D you? D t1 t2 Me D D you D D

22 How about 20 times? t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 Me??????????????????? D You??????????????????? D t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 Me D D D D D D D D D D D D D D D D D D D D You D D D D D D D D D D D D D D D D D D D D

23 In the finitely repeated PD, the only Nash equilibrium is always Defect. Any deviation from cannot be an equilibrium (why?) What if the game is infinite? Or if the end is unknown? Then cooperation is possible E.g.: grim trigger strategy: cooperate. If the other defects, always defect. Can we generalize?

24 The Folk Theorem (not covered in class) Folk theorem: that any feasible payoff profile that strictly dominates the minmax profile can be realized as a Nash equilibrium payoff profile, with sufficiently large discount factor. In the PD, both players cooperating is not a Nash equilibrium. The only Nash equilibrium is given by both players defecting, which is also a mutual minmax profile. The folk theorem says that, in the infinitely repeated version of the game, provided players are sufficiently patient, there is a Nash equilibrium such that both players cooperate on the equilibrium path.

25 Proof of the folk theorem (not covered in class) Intuitively, cooperation can always be enforced by employing a grim trigger strategy Let a be a pure strategy profile with payoff profile v which strictly dominates the minmax payoff profile. Then one can define a Nash eq. with v as resulting payoff profile as follows: 1. All players play a until a deviation occurs 2. If any one player, say I, deviates, then minimax her forever after 3. Ignore multilateral deviations Problem: not credible! Fix: If any player deviates, punish her for N periods (choose N large enough to remove incentives to deviate) If a player refused to punish, then target her

26 Good strategy to play repeated PD? Axelrod s tournament Tit-for-tat: Anatol Rapoport I start with cooperation If you C, I C If you D, I D

27 Repeated PD applet (Netlogo > Models library > Social sciences > Unverified >Prisoner s dilemma > PD N-person iterated)

28 Problem with TFT Errors destroy cooperation: TFT C C C C D C D C D C D D D D D D D TFT C C C D C D C D C D C D D D D D D Does not exploit ALLC TFT C C C C C C C C C C C C C C C C C ALL C C C C C C C C C C C C C C C C C C

29 Let natural selection design a strategy R,S,T,P -> response as a probability to cooperate (p1,p2,p3,p4) E.g.: TFT is (1,0,1,0) Start with population of (0.5,0.5,0.5,0.5) Introduce random mutations Strategies with higher payoffs have more offsprings

30 Win-stay, lose-switch Outcome: Win-stay, lose-switch is (1,0,0,1) Win-stay, lose-switch Corrects mistakes Exploites cooperators

31 Corrects mistakes WSLS C C C C D C C C C C C C C C C C C WSLS C C C D D C C C C C C C C C C C C Dominates ALLC: ALLC C C C C C C C C C C C C C C C C C WSLS C C C D D D D D D D D D D D D D D

32 Direct reciprocity can lead to the evolution of cooperation only if the probability, w, of another encounter between the same two individuals exceeds the cost-to-benefit ratio of the altruistic act: w > c/b

33 3. Indirect reciprocity People help those who have helped others. People refuse to help those who have refused to help others. Helpful people have a higher payoff in the end. give and you shall receive

34 Works via reputation A helps B A does not helps B A B A B The reputation of A increases The reputation of A decreases

35 Indirect reciprocity Gossip spreads reputation Experimental confirmation: People help those who help others

36 Indirect reciprocity is cognitively demanding: monitor the social network evolution of social intelligence Individuals must be able to talk to each other evolution of human language For direct reciprocity you need a face. For indirect reciprocity you need a name.

37 A rule for indirect reciprocity q > c / b q: probability to know someone s reputation c: cost of cooperation b: benefit of cooperation

38 4. Spatial selection Cooperators Defectors Nowak and May, Nature 1992 The graph describes a spatial structure or a social network

39 Cooperators pay a cost c for each neighbor to receive benefit b Graph selection favors cooperation if b/c>k, where k is the average number of neighbors

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41 Netlogo applet: evolutionary PD (Netlogo > Models library > Social sciences > Unverified >Prisoner s dilemma > PD Basic evolutionary)

42 Group selection There can be no doubt that a tribe including many members who [...] are always ready to give aid to each other and to sacrifice themselves for the common good, would be victorious over other tribes; and this would be natural selection. - Charles Darwin, The Descent of Man, 1871

43 Play the game with those in your group Offsprings are added to the group Groups divide when reaching a certain size Groups die

44 Although only individuals reproduce, there are two levels of selection. At the lower level, individuals compete with others in the same group. At the higher level, groups compete with each other This dynamic population structure favors the evolution of cooperation if the benefit-to-cost ratio of the altruistic act exceeds 1 plus the ratio of group size divided by the number of groups: b/c > 1 + (n/m).

45 The five mechanisms fall into 2 groups Kin selection Spatial selection Group selection Unconditional cooperation can win Direct reciprocity (uses own experience) Indirect reciprocity (uses experience of others) Conditional cooperation can win