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The linkage between Social dilemma and Indirect reciprocity

The linkage between Social dilemma and Indirect reciprocity. Rie Mashima (Kumamoto Gakuen University) Nobuyuki Takahashi (Hokkaido University). E-mail: mashima@kumagaku.ac.jp. Background. The exclusion of free-riders is necessary for the emergence of cooperation in social exchange.

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The linkage between Social dilemma and Indirect reciprocity

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  1. The linkage between Social dilemmaandIndirect reciprocity Rie Mashima (Kumamoto Gakuen University) Nobuyuki Takahashi (Hokkaido University) E-mail: mashima@kumagaku.ac.jp

  2. Background • The exclusion of free-riders is necessary for the emergence of cooperation in social exchange.

  3. Background • The exclusion of free-riders is necessary for the emergence of cooperation in social exchange. • Individual exchange: Discriminative altruism is an effective solution (e.g., reciprocal altruism, indirect reciprocity).

  4. Background • The exclusion of free-riders is necessary for the emergence of cooperation in social exchange. • Individual exchange: Discriminative altruism is an effective solution (e.g., reciprocal altruism, indirect reciprocity). On the contrary… • Social Dilemma (SD) : The emergence of cooperation has been a theoretical puzzle because to exclude a specific person is impossible. Public goods

  5. New solution: “Linkage strategy” • Recent theoretical studies focused on various ”linkage” strategies, which link SD and individual exchange, as the solution for SD. （e.g., Aoki 2001; Panchanathan & Boyd, 2004; Takagi, 1999; Watabe et al, 2008） “Linkage” strategies : exclude SD free-riders from individual exchange. • make SD free-riders maladaptive because they drive SD free-riders into isolation in individual exchange. ！

  6. Limitation of previous studies • Previous studies showed that “linkage” players can resist the invasion by SD free-riders. • However, there can also be no-link strategies. don’t care about other’s behaviors in SD • Can linkage strategies resist the invasion by other cooperative no-link strategies? • If not, the invasion by cooperative no-link players may cause the collapse of mutual cooperation!

  7. Invasion by cooperative “no-link” players (1) If the majority of a population are linkage players, L L L L L

  8. Invasion by cooperative “no-link” players (1) If the majority of a population are linkage players, • SD free-riders cannot invade. L L L L L

  9. may not be excluded by linkage players Invasion by cooperative “no-link” players (1) If the majority of a population are linkage players, • SD free-riders cannot invade. L L L L L • However, cooperative “no-link” players may invade. • In SD, always cooperate. • In individual exchange, ignore other’s behaviors in SD. (don’t exclude SD free-riders) L No-Link L L L L

  10. No-link No-link No-link No-link L L L L L L Invasion by cooperative “no-link” players (2) • If the frequency of cooperative no-link players increases,

  11. No-link No-link No-link No-link L L L L L L Invasion by cooperative “no-link” players (2) • If the frequency of cooperative no-link players increases, the population will be invaded by SD free-riders who don’t cooperate in SD.

  12. No-link No-link No-link No-link L L L L L L Invasion by cooperative “no-link” players (2) • If the frequency of cooperative no-link players increases, the population will be invaded by SD free-riders who don’t cooperate in SD. • Mutual cooperation may collapse because of the invasion by no-link players.

  13. No-link No-link No-link No-link L L L L L L Invasion by cooperative “no-link” players (2) • If the frequency of cooperative no-link players increases, the population will be invaded by SD free-riders who don’t cooperate in SD. • Mutual cooperation may collapse because of the invasion by no-link players. Purpose:to explore whether and what kind of linkage strategies can maintain mutual cooperation when cooperative no-link players try to invade.

  14. IR: 1500 rounds of indirect reciprocity game IR round 1 One generation IR round 2 ・・・ IR round 1500 Selection：Natural selection determines the distribution of strategies in the next generation, based on the fitness in the current generation. Method (1) Simulation • Simulation: On each generation, 300 players play a one-shot social dilemma game (SD) and many rounds of indirect reciprocity game (IR). (10000 generations per 1 replication) Strategy: SD strategy xIR strategy SD:1 round of social dilemma

  15. Public goods Method (2) SD stage • Players play a one-shot social dilemma game. • SD strategy: • “Cooperate（Ｃ）” : pay a cost to produce the benefit for others. • “Defect (D)”: don’t pay a cost. Public goods C or D C or D C or D C or D C or D

  16. Method (3)-1. IR stage (one-sided PD) • On each round, a randomly selected “donor” decides whether to give his resource to a recipient with a cost of c (recipient receives a benefit b: b>c) . • The donor chooses his recipient based on reputations of other members (chooses one whom the donor believes that he has a Good reputation). When there is no Good player among the population, the donor doesn’t give. • IR strategy determines how to assign a reputation (Good or Bad) to others by using three types of information. Good Give Bad Good Bad Good Bad

  17. Method (3)-2. IR strategy • Each player employs a strategy which assigns a reputation to others based on the information about their

  18. Behavior in SD Cooperate（C） or Defect （D） Method (3)-2. IR strategy • Each player employs a strategy which assigns a reputation to others based on the information about their • Behaviors in SD (Ｃ or D)

  19. Behavior in SD Cooperate（C） or Defect （D） Method (3)-2. IR strategy • Each player employs a strategy which assigns a reputation to others based on the information about their • Behaviors in SD (Ｃ or D) • 1st order information in IR (whether gave or did not give in the previous IR round) × 1st order info in IR gave or Did not give

  20. Behavior in SD 2nd order info in IR Cooperate（C） Good or or Defect （D） Bad Method (3)-2. IR strategy • Each player employs a strategy which assigns a reputation to others based on the information about their • Behaviors in SD (Ｃ or D) • 1st order information in IR (whether gave or did not give in the previous IR round) • 2nd order information in IR （the reputation of their previous recipients) × × 1st order info in IR gave or Did not give • Others are classified into ①～⑧ (Table 1).

  21. Table 1. Eight types of others 1st order info in IR

  22. Table 1. Eight types of others • 8 genes determine how to assign a reputation （Good or Bad） to each of type of others (①-⑧) . 1st order info in IR e.g., Gene 1 determines whether • SD cooperator who • gave • to Good person in previous IR round is Good or Bad.

  23. Table 1. Eight types of others • 8 genes determine how to assign a reputation （Good or Bad） to each of type of others (①-⑧) . • IR strategies are represented by the sets of 8 genes. ←IR strategies are represented as (①②③④, ⑤⑥⑦⑧） 1st order info in IR *①-⑧:Good or Bad

  24. （SD strategy, IR strategy concerning Cooperator in SD, IR strategy concerning Defector in SD） Full strategy (SD strategy x IR strategy) (C or D, ①②③④, ⑤⑥⑦⑧) 2 x x 24 24 =512 strategies

  25. （SD strategy, IR strategy concerning Cooperator in SD, IR strategy concerning Defector in SD） Full strategy (SD strategy x IR strategy) (C or D, ①②③④, ⑤⑥⑦⑧) 2 x x 24 24 =512 strategies e.g.,ALLC=(C, GGGG, GGGG)

  26. （SD strategy, IR strategy concerning Cooperator in SD, IR strategy concerning Defector in SD） Full strategy (SD strategy x IR strategy) (C or D, ①②③④, ⑤⑥⑦⑧) 2 x x 24 24 =512 strategies e.g.,ALLC=(C, GGGG, GGGG), ALLD=(D, BBBB, BBBB)

  27. Focused strategies • We picked up 4 linkage and no-link strategies based on IR strategies, which were proposed as successful ones in indirect reciprocity research. • IS (Image Scoring, Nowak & Sigmund, 1998) • STAND (Standing, Leimar & Hammerstein, 2001; Panchanathan & Boyd, 2003) • SDISC (Strict Discriminator, Takahashi & Mashima, 2006) • ES (Extra Standing, Takahashi & Mashima, 2006)

  28. Method (4) Focused strategies (No-link) 【No-link strategies】 In SD, cooperate or defect (depending on SD strategy). In IR, ignores other’s behaviors in SD. No-link IS（C/D, GGBB, GGBB） No-link STAND（C/D, GGBG, GGBG） No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）

  29. Method (4) Focused strategies (No-link) 【No-link strategies】 In SD, cooperate or defect (depending on SD strategy). In IR, ignores other’s behaviors in SD. No-link IS（C/D, GGBB, GGBB） No-link STAND（C/D, GGBG, GGBG） No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）

  30. Method (4) Focused strategies (No-link) 【No-link strategies】 In SD, cooperate or defect (depending on SD strategy).In IR, ignores other’s behaviors in SD. No-link IS（C/D, GGBB, GGBB） No-link STAND（C/D, GGBG, GGBG） No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）

  31. Method (4) Focused strategies (2) • assign reputation to others by using 1st order and 2nd order information in IR only. (What kind of discriminative rule they adopt varies.) 【No-link strategies】 In SD, cooperate or defect (depending on SD strategy). In IR, ignores other’s behaviors in SD. No-link IS（C/D, GGBB, GGBB） No-link STAND（C/D, GGBG, GGBG） No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）

  32. Method (4) Focused strategies (2) • assign reputation to others by using 1st order and 2nd order information in IR only. (What kind of discriminative rule they adopt varies.) • don’t use the information of other’s SD behaviors. 【No-link strategies】 In SD, cooperate or defect (depending on SD strategy). In IR, ignores other’s behaviors in SD. No-link IS（C/D, GGBB, GGBB） No-link STAND（C/D, GGBG, GGBG） No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）

  33. Method (4) Focused strategies (2) • assign reputation to others by using 1st order and 2nd order information in IR only. (What kind of discriminative rule they adopt varies.) • don’t use the information of other’s SD behaviors. 【No-link strategies】 In SD, cooperate or defect (depending on SD strategy). In IR, ignores other’s behaviors in SD. No-link IS（C/D, GGBB, GGBB） No-link STAND（C/D, GGBG, GGBG） Ignore whether others cooperated or defected in SD. No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）

  34. Method (4) Focused strategies (Linkage) 【Linkage strategies】 (cooperative linkage strategies) Link IS（C, GGBB, BBBB） Link STAND（C, GGBG, BBBB） Link SDISC（C, GBBB, BBBB） Link ES（C, GBBG, BBBB）

  35. Method (4) Focused strategies (Linkage) 【Linkage strategies】 (cooperative linkage strategies) In SD, always cooperate. Link IS（C, GGBB, BBBB） Link STAND（C, GGBG, BBBB） Link SDISC（C, GBBB, BBBB） Link ES（C, GBBG, BBBB）

  36. Method (4) Focused strategies (Linkage) 【Linkage strategies】 (cooperative linkage strategies) In SD, always cooperate.In IR, Link IS（C, GGBB, BBBB） Link STAND（C, GGBG, BBBB） Link SDISC（C, GBBB, BBBB） Link ES（C, GBBG, BBBB）

  37. Method (4) Focused strategies (Linkage) • how to regard 4 types of SD cooperators varies. 【Linkage strategies】 (cooperative linkage strategies) In SD, always cooperate.In IR, Link IS（C, GGBB, BBBB） Link STAND（C, GGBG, BBBB） Link SDISC（C, GBBB, BBBB） Link ES（C, GBBG, BBBB）

  38. Method (4) Focused strategies (Linkage) • how to regard 4 types of SD cooperators varies. • always regard SD defectors as Bad. 【Linkage strategies】 (cooperative linkage strategies) In SD, always cooperate.In IR, always regard defectors in SD as Bad. Link IS（C, GGBB, BBBB） Link STAND（C, GGBG, BBBB） Link SDISC（C, GBBB, BBBB） Link ES（C, GBBG, BBBB）

  39. Method (4) Focused strategies (Linkage) • how to regard 4 types of SD cooperators varies. • always regard SD defectors as Bad. 【Linkage strategies】 (cooperative linkage strategies) In SD, always cooperate.In IR, always regard defectors in SD as Bad. Link IS（C, GGBB, BBBB） Link STAND（C, GGBG, BBBB） exclude SD free-riders from IR Link SDISC（C, GBBB, BBBB） Link ES（C, GBBG, BBBB）

  40. Method (5) Exploring the evolutionary stability Can linkage players resist the invasion of ALLC, ALLD, and no-link players?

  41. Method (5) Exploring the evolutionary stability Can linkage players resist the invasion of ALLC, ALLD, and no-link players? • The population initially consists of 99% of one majority strategy and 1% of one invader strategy. • Majority strategies: four types of linkage strategies (IS, STAND, SDISC, ES) whose SD strategy=C. • Invader strategies: ALLC, ALLD, cooperative no-link strategy (SD=C), defective no-link strategy (SD=D). • 10000 generations per 1 replication * Mutation rate=0.0001, C=5, B=10, c=2, b=4, n of IR rounds=1500, error of execution in SD =0.025, error of perception in SD=0.025, error of execution in IR =0.025, error of perception in IR=0.025, n of replications=10.

  42. Result Majority

  43. Result Majority Majority strategy

  44. Result Majority Invader strategy

  45. Result Majority • ALLD

  46. Result Majority • ALLD • ALLD • Defective No-link strategy

  47. Result Majority • ALLD • ALLD • Defective No-link strategy • ALLC

  48. Result Majority • ALLD • ALLD • Defective No-link strategy • ALLC • Cooperative No-link strategy

  49. Result Majority Percentage of replications where the invader invaded the majority.

  50. Result Majority

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