1. The linkage between Social dilemma and Indirect reciprocity
Rie Mashima (Kumamoto Gakuen University)
Nobuyuki Takahashi (Hokkaido University)

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.
Linkage strategies are the strategies which exclude free riders in social dilemma from individual exchange.
Previous studies argued that linkage strategies can solve the social dilemma problem because they drive social dilemma free-riders into isolation in important 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. 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.
may not be excluded by linkage players
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. Invasion by cooperative “no-link” players (2)
If the frequency of cooperative no-link players increases, L
No-link
No-link
No-link L
No-link

11. 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. L
No-link
No-link
No-link L
No-link

12. 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. L
No-link
No-link
No-link L
No-link

13. 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. L
No-link
No-link
No-link L
No-link

14. Method (1) Simulation SD:1 round of social dilemma
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 x IR strategy
One generation
SD:1 round of social dilemma
IR round 1
IR round 2
・・・
IR round 1500
IR: 1500 rounds of indirect reciprocity game
Selection：Natural selection determines the distribution of strategies in the next generation, based on the fitness in the current generation.

15. Method (2) SD stage Players play a one-shot social dilemma game.
Public goods
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. Each player employs a strategy which assigns a reputation to others based on the information about their
Method (3)-2. IR strategy

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

19. × or or Method (3)-2. IR strategy Behaviors in SD (Ｃ or D)
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)
Method (3)-2. IR strategy ×
Behavior in SD
Defect （D） or
Cooperate（C）
1st order info in IR
gave or
Did not give

20. × × Good or or or 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)
Method (3)-2. IR strategy × ×
Behavior in SD
Defect （D） or
Cooperate（C）
1st order info in IR
2nd order info in IR
Good or
Bad
gave or
Did not give
Others are classified into ①～⑧ (Table 1).

21. C in SD D in SD Table 1. Eight types of others ① ② ⑤ ⑥ ③ ④ ⑦ ⑧
2nd order info in IR
Good
Bad
Gave ① ② ⑤ ⑥
Didn’t give ③ ④ ⑦ ⑧
1st order info in IR

22. e.g., Gene 1 determines whether SD cooperator who gave
Table 1. Eight types of others
C in SD
D in SD
2nd order info in IR
Good
Bad
Gave
① (G or B)
② (G or B)
⑤ (G or B)
⑥ (G or B)
Didn’t give
③ (G or B)
④ (G or B)
⑦ (G or B)
⑧ (G or B)
1st order info in IR
8 genes determine how to assign a reputation （Good or Bad） to each of type of others (①-⑧) .
e.g., Gene 1 determines whether
SD cooperator who
gave
to Good person in previous IR round
is Good or Bad.

23. (①②③④, ⑤⑥⑦⑧） *①-⑧: Good or Bad
Table 1. Eight types of others
C in SD
D in SD
2nd order info in IR
Good
Bad
Gave
① (G or B)
② (G or B)
⑤ (G or B)
⑥ (G or B)
Didn’t give
③ (G or B)
④ (G or B)
⑦ (G or B)
⑧ (G or B)
1st order info in IR
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）
*①-⑧: Good or Bad

24. (C or D, ①②③④, ⑤⑥⑦⑧) Full strategy (SD strategy x IR strategy)
IR strategy concerning Cooperator in SD,
IR strategy concerning Defector in SD）
(C or D, ①②③④, ⑤⑥⑦⑧) 2 x 24 x 24
=512 strategies
C in SD
D in SD
Good
Bad
Gave ① ② ⑤ ⑥
Didn’t give ③ ④ ⑦ ⑧

25. (C or D, ①②③④, ⑤⑥⑦⑧) Full strategy (SD strategy x IR strategy)
IR strategy concerning Cooperator in SD,
IR strategy concerning Defector in SD）
(C or D, ①②③④, ⑤⑥⑦⑧) 2 x 24 x 24
=512 strategies
e.g., ALLC=(C, GGGG, GGGG)
ALLC (pure altruists)
C in SD
D in SD
Good
Bad
Gave G
Didn’t give

26. (C or D, ①②③④, ⑤⑥⑦⑧) Full strategy (SD strategy x IR strategy)
IR strategy concerning Cooperator in SD,
IR strategy concerning Defector in SD）
(C or D, ①②③④, ⑤⑥⑦⑧) 2 x 24 x 24
=512 strategies
e.g., ALLC=(C, GGGG, GGGG), ALLD=(D, BBBB, BBBB)
ALLD (hardcore free-riders)
C in SD
D in SD
Good
Bad
Gave B
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give
No-link SDISC（C/D, GBBB, GBBB）
No-link ES（C/D, GBBG, GBBG）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give
No-link SDISC（C/D, GBBB, GBBB）
No-link ES（C/D, GBBG, GBBG）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give
No-link SDISC（C/D, GBBB, GBBB）
No-link ES（C/D, GBBG, GBBG）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
No-link SDISC（C/D, GBBB, GBBB）
No-link ES（C/D, GBBG, GBBG）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
No-link SDISC（C/D, GBBB, GBBB）
No-link ES（C/D, GBBG, GBBG）
As you can see, the left side and right side are identical.
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
Ignore whether others cooperated or defected in SD.
No-link SDISC（C/D, GBBB, GBBB）
No-link ES（C/D, GBBG, GBBG）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

34. Method (4) Focused strategies (Linkage)
【Linkage strategies】 (cooperative linkage strategies)
Link IS（C, GGBB, BBBB）
Link STAND（C, GGBG, BBBB）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give
Link SDISC（C, GBBB, BBBB）
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give
Link SDISC（C, GBBB, BBBB）
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give

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）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give
Link SDISC（C, GBBB, BBBB）
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Gave
Didn’t give
C in SD
D in SD
Good
Bad
Gave
Didn’t give

37. Method (4) Focused strategies (Linkage)
【Linkage strategies】 (cooperative linkage strategies)
how to regard 4 types of SD cooperators varies.
In SD, always cooperate. In IR,
Link IS（C, GGBB, BBBB）
Link STAND（C, GGBG, BBBB）
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
C in SD
D in SD
Good
Bad
Gave G
Didn’t give B
Link SDISC（C, GBBB, BBBB）
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

38. Method (4) Focused strategies (Linkage)
【Linkage strategies】 (cooperative linkage strategies)
how to regard 4 types of SD cooperators varies.
always regard SD defectors as Bad.
In SD, always cooperate. In IR, always regard defectors in SD as Bad.
Link IS（C, GGBB, BBBB）
Link STAND（C, GGBG, BBBB）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
Link SDISC（C, GBBB, BBBB）
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

39. Method (4) Focused strategies (Linkage)
【Linkage strategies】 (cooperative linkage strategies)
how to regard 4 types of SD cooperators varies.
always regard SD defectors as Bad.
In SD, always cooperate. In IR, always regard defectors in SD as Bad.
Link IS（C, GGBB, BBBB）
Link STAND（C, GGBG, BBBB）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
exclude SD free-riders from IR
Link SDISC（C, GBBB, BBBB）
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

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. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority

43. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Majority strategy

44. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Invader strategy

45. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
ALLD

46. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
ALLD
Defective No-link strategy
ALLD

47. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
ALLD
Defective No-link strategy
ALLC
ALLD

48. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
ALLD
Defective No-link strategy
ALLC
Cooperative No-link strategy
ALLD

49. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Percentage of replications where the invader invaded the majority.

50. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority

51. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Link SDISC and Link ES are robust against the invasion by not only non-cooperative but also cooperative players.

52. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Link SDISC and Link ES are robust against the invasion by not only non-cooperative but also cooperative players.

53. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Link SDISC and Link ES are robust against the invasion by not only non-cooperative but also cooperative players.

54. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Link SDISC and Link ES are robust against the invasion by not only non-cooperative but also cooperative players.
Link SDISC and ES can be robust.

55. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Although other strategies are robust against the invasion by SD defectors (ALLD, defective no-link players),

56. % of invaded replication
Invader’s SD strategy
Invader’s IR strategy
% of invaded replication
Robustness of majority strategy
SD: C
IR: Link IS
(C, GGBB, BBBB) D
ALLD
Robust
No-link IS C
ALLC 20
Invaded 40
IR: Link STAND
(C, GGBG, BBBB)　
No-link STAND 80 50
IR: Link SDISC
(C, GBBB, BBBB)　
No-link SDISC
IR: Link ES
(C, GBBG, BBBB)　
No-link ES
Result
Majority
Although other strategies are robust against the invasion by SD defectors (ALLD, defective no-link players),
They are invaded by cooperative players (ALLC and cooperative no-link players)

57. What made the difference?
Among 4 types of linkage players, only SDISC and ES are robust whereas other ones are invaded by cooperative no-link players.
What made the difference?
The key is to exclude altruists who help SD free-riders. IS
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
STAND
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give ES
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

58. Unsuccessful strategiesIS
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
STAND
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

59. Unsuccessful strategies can exclude SD defectors.IS
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
STAND
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give L L L L L

60. Unsuccessful strategies can exclude SD defectors.
However, they regard SD cooperators who helped Bad (cooperative no-link players) as Good. IS
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
STAND
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give L L L L L

61. Unsuccessful strategies can exclude SD defectors.
However, they regard SD cooperators who helped Bad (cooperative no-link players) as Good. IS
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
STAND
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
Cooperative no-link players can increase. L
No-link L
No-link L
No-link L L L
No-link

62. ← cannot maintain mutual cooperation in SD.
Unsuccessful strategies can exclude SD defectors.
However, they regard SD cooperators who helped Bad (cooperative no-link players) as Good. IS
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
STAND
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
Cooperative no-link players can increase.
The higher the frequency of no-link players becomes, the easier it is for free-riders to invade.
← cannot maintain mutual cooperation in SD. L
No-link L
No-link L
No-link L L L
No-link

63. Successful strategies
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give ES
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

64. Successful strategies can also exclude SD defectors.
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give ES
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

65. Successful strategies can also exclude SD defectors.
Moreover, they regard SD cooperators who helped Bad (cooperative no-link players) as Bad.
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give ES
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

66. Successful strategies can also exclude SD defectors.
Moreover, they regard SD cooperators who helped Bad (cooperative no-link players) as Bad.
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give ES
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
Cooperative no-link players cannot invade. L
No-link L L L L
ALLC

67. Successful strategies can also exclude SD defectors.
Moreover, they regard SD cooperators who helped Bad (cooperative no-link players) as Bad.
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give ES
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
Cooperative no-link players cannot invade.
SD free-riders cannot invade. L
No-link L L L
ALLC

68. Successful strategies can also exclude SD defectors.
Moreover, they regard SD cooperators who helped Bad (cooperative no-link players) as Bad.
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give ES
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
The key point for the maintenance of mutual cooperation in SD is to exclude not only SD free-riders but also those who help SD free-riders.
Cooperative no-link strategies cannot invade. L
No-link L L L
ALLC

69. Conclusion
As previous studies showed, linkage players can resist the invasion by SD free-riders and can maintain mutual cooperation.
However, sooner or later, such a mutual cooperation should collapse unless linkage players have the characteristic which excludes cooperative no-link players.
To regard the benevolence toward free-riders as Bad is necessary for the maintenance of mutual cooperation in both SD and IR.

70. Thank you
for your attention!

71. Appendices Details of SD stage
Characteristics of linkage and no-link strategies
Details of simulation settings
Evolutionary stability
Characteristics of successful strategies
Results under various error rate (in SD)
Results in random matching situation
Results under the settings in Panchanathan & Boyd (2004)
“SD cooperators who helped Bad” as no-link players
Details of the process of invasion of defectors
How linkage strategies avoid the 2nd order problem ?
Random matching & Selective play situation in IR
Unnatural settings: random matching situation
Four types of target in selective play

72. Details of SD stage
Each player decides whether or not to cooperate (with a cost of C) to produce a benefit B. (B is divided equally among all other members)
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

73. Characteristics of linkage and no-link strategies
left half (strategy concerning with SD cooperators) : same between in linkage and in no-link strategies.
right half (strategy concerning with SD defectors): while linkage players regard all of SD defectors as Bad, no-link players do not distinguish SD cooperators from SD defectors.
Link SDISC（C, GBBB, BBBB）
No-link SDISC（C/D, GBBB, GBBB）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

74. Details of simulation settings (1)
Parameters
Mutation rate=0.0001
SD stage: C=5, B=10, error of execution in SD=0.025, error of perception in SD =
IR stage: c=2, b=4, n of IR rounds=1500, error of execution in IR =0.025, error of perception in IR=0.025
←The weight of SD (the amount of players’ profit which derive from mutual cooperation in SD) is 2 times as large as the weight of IR.
Selective play giving game in IR
Number of replications=10.

75. Details of simulation settings (2)
After generations, if the frequency of invader is…
>10% : The invader succeeded.
(The majority is Invaded)
≤ 10% : The invader failed.
(The majority is Robust )
If the number of invaded replication=0, we consider the situation that the majority strategy is robust against the invasion by the invader strategy.

76. Evolutionary stability
Our simulations purposed to examine whether and what kind of linkage strategies are evolutionarily stable strategies (ESS).
Suppose a population is consisted of one strategy (majority). If a few people who adopt another strategy (invader) cannot increase in such a situation, the majority is ESS against the invader.
If the strategy is not ESS, there is no guarantee that the equilibrium by the strategy can be maintained.

77. Characteristics of successful strategies
（C, GBBX, BBBB）
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give
G or B
Regard SD cooperator who gave to Good as Good
Regard SD defectors and SD cooperator who didn’t give to Good as Bad
Regard SD cooperator who gave to Bad (cooperative no-link players) as Bad

78. β=perception error in SD α=execution error in SD
Invader’s SD strategy
Invader’s IR strategy
β=0.0
α=0.0
α=0.025
β=0.025 IS 　D
ALLD
Robust
No-link IS 　C
ALLC
Invaded
STAND
No-link STAND
SDISC
No-link SDISC ES
No-link ES
β=perception error in SD
α=execution error in SD
Majority

79. Results in random matching situation
Invader’s SD strategy
Invader’s IR strategy
Invaded rep (%) IS 　D
ALLD
No-link IS 　C
ALLC
96.7
100
STAND
No-link STAND
13.3
SDISC
No-link SDISC 10 ES
No-link ES
Results in random matching situation
Majority
Random matching
: The donor is randomly matched with the recipient.
Errors (error of execution / perception, in SD / IR)=0.025
Number of replication=30

80. Results under the settings in Panchanathan & Boyd (2004)
Random matching
Error of execution in IR 0.05, other errors (error of execution/ perception in SD, error of perception in IR)=0.
Number of replication=30
Invader’s SD
Invader’s IR
Invaded rep (%)
STAND 　D
ALLD
No-link STAND 　C
ALLC 47
Result that P & B (2004) showed
Majority
Linkage STAND can resist the invasion by ALLC, but cannot resist the invasion by cooperative no-link players.

81. “SD cooperators who helped Bad” as no-link players
Linkage players regard SD defectors as Bad.
Therefore, “players who gave toward Bad” include those who gave toward SD defectors.
Players who gave toward SD defectors are players without linkage trait (no-link players, ALLC) because they gave to Bad (including SD defectors).
Then, “players who gave toward Bad” correspond the feature of cooperative no-link players.
SDISC
C in SD
D in SD
Good
Bad
Gave G B
Didn’t give

82. Details of the process of invasion of defectors
Cooperative no-link players (who cooperate in SD, help SD defectors in IR) can invade the population of cooperative linkage players.
Non-cooperative no-link players (who defect in SD, help SD defector in IR) can invade.
The strategy which would evolve is non-cooperative no-link players.
The situation where people defect in SD and engage in IR independently of SD would emerge.
While mutual cooperation in IR would occurs (by discriminative altruism), wouldn’t occur in SD.

83. How linkage strategies avoid the 2nd order problem ?
Punishing free-riders or rewarding cooperators to induce cooperation require additional cost…So who pay the cost?
Linkage strategies can avoid paying additional cost because they can exclude SD free-riders from IR by merely not giving resources to free-riders in IR.
Moreover, their “appropriate” behaviors (not giving to free-riders but giving to cooperators) increases one’s reputation and increases the chance to receive resources from the other linkage players. ！

84. How linkage strategies avoid the 2nd order problem ?
Although linkage players need to observe others’ behaviors, anyhow, it is necessary for one’s survival in IR.

85. Random matching & Selective play situation(1)
Random matching: A donor has to interact with the designated recipient.
A potential recipient is randomly chosen.
Give or
Not give?
donor
recipient
Bad
Good
Bad
Bad
Bad
Good
Bad
Bad
Good
Results of simulations of random matching showed that only SDISC makes indirect reciprocity possible.

86. Random matching & Selective play situation(2)
Selective play: A donor choose his recipient based on reputations of other members.
Observe reputations of other members
A donor choose a recipient to give by himself.
donor
recipient
Bad
Good
Bad
Good
Good
Bad
I want to give this Good person.
Bad
Bad
Good

87. Unnatural settings: random matching situation
Random matching environment
Most of the theoretical studies in biology have assumed this environment.
From the group, a donor and a recipient is randomly chosen to form a pair.
Give or Not give?
Donor
Recipient
Bad
Good
Bad
Bad
Bad
Good
Bad
Bad
Good
A donor must act toward a designated opponent – Give or Not give

88. Unnatural settings: random matching situation
Random matching environment
Most of the theoretical studies in biology have assumed this environment.
Question from social sciences: Why is a donor forced to play a game with a designated opponent? Why not simply ignore undesirable opponents?
From the group, a donor and a recipient is randomly chosen to form a pair.
Give or Not give?
If a donor knows every other person’s reputation, isn’t it more natural to assume that a donor can choose freely his/her recipient he/she wants? (Aoki, 2001)
Donor
Recipient
Bad
Good
Bad
Bad
Bad
Good
Bad
Bad
Good
Selective-play environment (Yamagishi & Hayashi, 1996) seems to be more natural.
A donor must act toward a designated recipient – Give or Not give

89. Four types of targets in selective play
As “not giving” means “not to choose anyone as the recipient” in selective play, “3) Non-giver to Giver” and “4) Non-giver to Non-giver” in selective play are defined as following:
Unjustifiable not-giving under situations where there was a chance to give to Good
Justifiable not-giving under situations where there was no chance to give to Good
“Non-giver to Good” is a person who didn’t give his resource although he had a chance to give to Good if he wanted to do so.
“Non-giver to Bad” is a person who didn’t give resources in situations where there was no Good player in the group.
Good
Bad
Gave
GtoG
GtoB
Did not give
NGtoG
NGtoB