The linkage between
Social dilemma and
Indirect reciprocity
Rie Mashima (Kumamoto Gakuen University)
Nobuyuki Takahashi (Hokkaido University)
E-mail: [email protected]
Background
 The
exclusion of free-riders is necessary for the
emergence of cooperation in social exchange.
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).
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
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.
！
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!
Invasion by cooperative “no-link” players (1)
If the majority of a population are linkage players,
L
L
L
L
L
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
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
Invasion by cooperative “no-link” players (2)
If the frequency of cooperative no-link players
increases,
No-link
L
L
No-link
No-link
L
L
L
L
No-link
Invasion by cooperative “no-link” players (2)
If the frequency of cooperative no-link players
increases, the population will be invaded by SD freeriders who don’t cooperate in SD.
No-link
L
L
No-link
No-link
L
L
L
L
No-link
Invasion by cooperative “no-link” players (2)
If the frequency of cooperative no-link players
increases, the population will be invaded by SD freeriders who don’t cooperate in SD.
Mutual cooperation may collapse because of the
invasion by no-link players.
No-link
L
L
No-link
No-link
L
L
L
L
No-link
Invasion by cooperative “no-link” players (2)
If the frequency of cooperative no-link players
increases, the population will be invaded by SD freeriders 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
No-link
L
cooperative
no-link
players
try
to invade.
L
No-link
No-link
L
L
L
L
No-link
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
x
IR strategy
One generation
SD:1 round of social dilemma
IR: 1500 rounds of
indirect reciprocity game
IR round 1
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.
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
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.
Bad
Give
Good
Bad
Good
Good
Bad
Method (3)-2. IR strategy

Each player employs a strategy which assigns a
reputation to others based on the information about their
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)
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)
Behavior in SD
×
Cooperate（C）
or
Defect （D）
1st order info in IR
gave
or
Did not give
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)
Behavior in SD
×
Cooperate（C）
or
Defect （D）

1st order info in IR
gave
or
Did not give
×2
nd
order info in IR
Good
or
Bad
Others are classified into ①～⑧ (Table 1).
Table 1. Eight types of others
C in SD
D in SD
2nd order info in IR
1st order
info in IR
Gave
Didn’t give
Good
①
Bad
②
Good
⑤
Bad
⑥
③
④
⑦
⑧
Table 1. Eight types of others
C in SD
D in SD
2nd order info in IR
1st order
info in IR

Gave
Didn’t give
Good
①
Bad
②
Good
⑤
Bad
⑥
(G or B)
(G or B)
(G or B)
(G or B)
③
④
⑦
⑧
(G or B)
(G or B)
(G or B)
(G or B)
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.
Table 1. Eight types of others
C in SD
D in SD
2nd order info in IR
1st order
info in IR


Gave
Didn’t give
Good
①
Bad
②
Good
⑤
Bad
⑥
(G or B)
(G or B)
(G or B)
(G or B)
③
④
⑦
⑧
(G or B)
(G or B)
(G or B)
(G or B)
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
Full strategy (SD strategy x IR strategy)
（SD strategy,
IR strategy concerning Cooperator in SD,
IR strategy concerning Defector in SD）
(C or D, ①②③④, ⑤⑥⑦⑧)
2
x
24
24
x
C in SD
=512 strategies
D in SD
Good
Bad
Good
Bad
Gave
①
②
⑤
⑥
Didn’t give
③
④
⑦
⑧
Full strategy (SD strategy x IR strategy)
（SD strategy,
IR strategy concerning Cooperator in SD,
IR strategy concerning Defector in SD）
(C or D, ①②③④, ⑤⑥⑦⑧)
2
x
24
24
x
=512 strategies
e.g.,
ALLC=(C, GGGG, GGGG)
ALLC
(pure altruists)
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
G
G
Didn’t give
G
G
G
G
Full strategy (SD strategy x IR strategy)
（SD strategy,
IR strategy concerning Cooperator in SD,
IR strategy concerning Defector in SD）
(C or D, ①②③④, ⑤⑥⑦⑧)
2
x
24
24
x
=512 strategies
e.g.,
ALLC=(C, GGGG, GGGG), ALLD=(D, BBBB, BBBB)
ALLD
(hardcore freeriders)
C in SD
D in SD
Good
Bad
Good
Bad
Gave
B
B
B
B
Didn’t give
B
B
B
B
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)
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）
C in SD
Good
Bad
No-link STAND（C/D, GGBG, GGBG）
D in SD
Good
C in SD
Bad
Good
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）
C in SD
Good
Bad
C in SD
D in SD
Good
Good
Bad
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
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）
C in SD
Good
Bad
No-link STAND（C/D, GGBG, GGBG）
D in SD
Good
C in SD
Bad
Good
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）
C in SD
Good
Bad
C in SD
D in SD
Good
Good
Bad
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
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）
C in SD
Good
Bad
No-link STAND（C/D, GGBG, GGBG）
D in SD
Good
C in SD
Bad
Good
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）
C in SD
Good
Bad
C in SD
D in SD
Good
Good
Bad
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
Method
(4) Focused
strategies
reputation
to others
by using (2)
1st order and 2nd
assign
order information
【No-link
strategies】in IR only. (What kind of
discriminative
ruleorthey
adopt
varies.) on SD strategy).
In
SD, cooperate
defect
(depending
In IR, ignores other’s behaviors in SD.
No-link IS（C/D, GGBB, GGBB）
C in SD
No-link STAND（C/D, GGBG, GGBG）
D in SD
Good
Bad
Good
Bad
Gave
G
G
G
G
Didn’t
give
B
B
B
B
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
G
G
Didn’t
give
B
G
B
G
No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）
C in SD
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
G
B
Didn’t
give
B
B
B
B
D in SD
Good
Bad
Good
Bad
Gave
G
B
G
B
Didn’t
give
B
G
B
G
Method
(4) Focused
strategies
reputation
to others
by using (2)
1st order and 2nd
assign
order information
【No-link
strategies】in IR only. (What kind of
discriminative
ruleorthey
adopt
varies.) on SD strategy).
In
SD, cooperate
defect
(depending
In
don’t
the information
of other’s
SD behaviors.
IR, use
ignores
other’s behaviors
in SD.
No-link IS（C/D, GGBB, GGBB）
C in SD
No-link STAND（C/D, GGBG, GGBG）
D in SD
Good
Bad
Good
Bad
Gave
G
G
G
G
Didn’t
give
B
B
B
B
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
G
G
Didn’t
give
B
G
B
G
No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）
C in SD
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
G
B
Didn’t
give
B
B
B
B
D in SD
Good
Bad
Good
Bad
Gave
G
B
G
B
Didn’t
give
B
G
B
G
Method
(4) Focused
strategies
reputation
to others
by using (2)
1st order and 2nd
assign
order information
【No-link
strategies】in IR only. (What kind of
discriminative
ruleorthey
adopt
varies.) on SD strategy).
In
SD, cooperate
defect
(depending
In
don’t
the information
of other’s
SD behaviors.
IR, use
ignores
other’s behaviors
in SD.
No-link IS（C/D, GGBB, GGBB）
C in SD
No-link STAND（C/D, GGBG, GGBG）
D in SD
C in SD
Good
Bad
Good
Bad
Good
Gave
G
G
G
Didn’t
give
B
B
or
Didn’t
Bcooperated
B
Bad
D in SD
Good
IgnoreGave
whether
G
G others
G
G
give
defected
B
Gin SD.
B
Bad
G
G
No-link SDISC（C/D, GBBB, GBBB） No-link ES（C/D, GBBG, GBBG）
C in SD
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
G
B
Didn’t
give
B
B
B
B
D in SD
Good
Bad
Good
Bad
Gave
G
B
G
B
Didn’t
give
B
G
B
G
Method (4) Focused strategies (Linkage)
【Linkage strategies】 (cooperative linkage strategies)
Link IS（C, GGBB, BBBB）
C in SD
Good
Bad
Link STAND（C, GGBG, BBBB）
D in SD
Good
C in SD
Bad
Good
Gave
Gave
Didn’t
give
Didn’t
give
Link SDISC（C, GBBB, BBBB）
C in SD
Good
Bad
Good
Bad
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
D in SD
Good
Bad
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
Method (4) Focused strategies (Linkage)
【Linkage strategies】 (cooperative linkage strategies)
In SD, always cooperate.
Link IS（C, GGBB, BBBB）
C in SD
Good
Bad
Link STAND（C, GGBG, BBBB）
D in SD
Good
C in SD
Bad
Good
Gave
Gave
Didn’t
give
Didn’t
give
Link SDISC（C, GBBB, BBBB）
C in SD
Good
Bad
Good
Bad
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
D in SD
Good
Bad
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
Method (4) Focused strategies (Linkage)
【Linkage strategies】 (cooperative linkage strategies)
In SD, always cooperate. In IR,
Link IS（C, GGBB, BBBB）
C in SD
Good
Bad
Link STAND（C, GGBG, BBBB）
D in SD
Good
C in SD
Bad
Good
Gave
Gave
Didn’t
give
Didn’t
give
Link SDISC（C, GBBB, BBBB）
C in SD
Good
Bad
Good
Bad
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
D in SD
Good
Bad
Gave
Gave
Didn’t
give
Didn’t
give
Bad
D in SD
Good
Bad
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）
C in SD
Good
Bad
Gave
G
G
Didn’t
give
B
B
Link STAND（C, GGBG, BBBB）
D in SD
Good
Bad
Link SDISC（C, GBBB, BBBB）
C in SD
Good
Bad
Gave
G
B
Didn’t
give
B
B
C in SD
Good
Bad
Gave
G
G
Didn’t
give
B
G
Good
Bad
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
D in SD
Good
Bad
Gave
G
B
Didn’t
give
B
G
Bad
D in SD
Good
Bad
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, always regard defectors
always regard SD defectors as Bad.
in SD as Bad.
Link IS（C, GGBB, BBBB）
C in SD
Link STAND（C, GGBG, BBBB）
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t
give
B
B
B
B
Link SDISC（C, GBBB, BBBB）
C in SD
C in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t
give
B
G
B
B
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t
give
B
B
B
B
D in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t
give
B
G
B
B
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, always regard defectors
always regard SD defectors as Bad.
in SD as Bad.
Link IS（C, GGBB, BBBB）
C in SD
Link STAND（C, GGBG, BBBB）
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t
give
B
B
C in SD
Gave
D in SD
Good
Bad
Good
Bad
G
G
B
B
B
B
Didn’t
B
BSD free-riders
B from
G IR
exclude
give
Link SDISC（C, GBBB, BBBB）
C in SD
Link ES（C, GBBG, BBBB）
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t
give
B
B
B
B
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t
give
B
G
B
B
Method (5) Exploring the evolutionary stability
Can linkage players resist the invasion of ALLC, ALLD, and
no-link players?
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.
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
SDISC
SDISC
ES
ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
SD: C
IR: Link IS
Invader’s
SD
strategy
D
(C, GGBB, BBBB)
C
SD: C
IR: Link STAND
D
(C, GGBG, BBBB)
C
SD: C
IR: Link SDISC
D
(C, GBBB, BBBB)
C
SD: C
IR: Link ES
D
(C, GBBG, BBBB)
C
Invader’s
IR strategy
ALLD
No-link IS
ALLC
No-link IS
ALLD
Majority
No-link STAND
ALLC
No-link STAND
ALLD
No-link SDISC
ALLC
No-link SDISC
ALLD
No-link ES
ALLC
No-link ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
strategy
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
SDISC
SDISC
ES
ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
Invader
strategy
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
SDISC
SDISC
ES
ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
 ALLD
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
SDISC
SDISC
ES
ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
 ALLD
0
Robust
20
Invaded
 Defective
No-link
40
Invaded
strategy
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
SDISC
SDISC
ES
ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
 ALLD
0
Robust
20
Invaded
 Defective
No-link
40
Invaded
strategy
0
Robust
0
Robust
 ALLC
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
SDISC
SDISC
ES
ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
 ALLD
0
Robust
20
Invaded
 Defective
No-link
40
Invaded
strategy
0
Robust
0
Robust
 ALLC
80
Invaded
50
Invaded
 Cooperative
0
Robust
No-link
strategy
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
Invader’s
SD
strategy
Invader’s
IR strategy
ALLD
SD: C
No-link IS
IR: Link IS
ALLC
(C, GGBB, BBBB)
C
No-link IS
ALLD
D
Percentage
of replications
SD:
C
No-link STAND
IR:
Link STAND
where
the invader invaded
ALLC the
(C, GGBG, BBBB)
C
No-link STAND
majority.
ALLD
D
SD: C
No-link SDISC
IR: Link SDISC
ALLC
(C, GBBB, BBBB)
C
No-link SDISC
ALLD
D
SD: C
No-link ES
IR: Link ES
ALLC
(C, GBBG, BBBB)
C
No-link ES
D
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
SDISC
SDISC
ES
ES
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
Result
Majority
Invader’s
SD
strategy
Invader’s
IR strategy
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
ALLD

Link
SDISC
and
Link
ES
are
robust
against
D
SD: C
No-link IS
IR: Link
IS
the invasion by notALLC
only non-cooperative but
(C, GGBB, BBBB)
C
No-link IS
also cooperative players.
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
D
C
D
C
D
C
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
STAND
STAND
SDISC
SDISC
ES
ES
Result
Majority
Invader’s
SD
strategy
Invader’s
IR strategy
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
ALLD

Link
SDISC
and
Link
ES
are
robust
against
D
SD: C
No-link IS
IR: Link
IS
the invasion by notALLC
only non-cooperative but
(C, GGBB, BBBB)
C
No-link IS
also cooperative players.
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
D
C
D
C
D
C
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
STAND
STAND
SDISC
SDISC
ES
ES
Result
Majority
Invader’s
SD
strategy
Invader’s
IR strategy
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
ALLD

Link
SDISC
and
Link
ES
are
robust
against
D
SD: C
No-link IS
IR: Link
IS
the invasion by notALLC
only non-cooperative but
(C, GGBB, BBBB)
C
No-link IS
also cooperative players.
SD: C
IR: Link STAND
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
D
C
D
C
D
C
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
STAND
STAND
SDISC
SDISC
ES
ES
Result
Majority
Invader’s
SD
strategy
Invader’s
IR strategy
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
ALLD

Link
SDISC
and
Link
ES
are
robust
against
D
SD: C
No-link IS
IR: Link
IS
the invasion by notALLC
only non-cooperative but
(C, GGBB, BBBB)
C
No-link IS
also cooperative players.
SD: C
IR: Link STAND
D
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
STAND
Link SDISC and ES can be robust.
(C, GGBG, BBBB)
SD: C
IR: Link SDISC
(C, GBBB, BBBB)
SD: C
IR: Link ES
(C, GBBG, BBBB)
C
D
C
D
C
STAND
SDISC
SDISC
ES
ES
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
D strategies are robust against
other
SDISC
the invasion
by
SD defectors (ALLD, defective
(C, GBBB,
BBBB)
C
SDISC
no-link players),
 Although
SD:
C
IR: Link SDISC
SD: C
IR: Link ES
(C, GBBG, BBBB)
D
C
ES
ES
Result
Majority
SD: C
IR: Link IS
(C, GGBB, BBBB)
SD: C
IR: Link STAND
(C, GGBG, BBBB)
Invader’s
SD
strategy
D
C
D
C
Invader’s
IR strategy
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
ALLD
No-link
ALLC
No-link
IS
IS
STAND
STAND
of
% of invaded Robustness
replication majority
strategy
0
Robust
0
Robust
20
Invaded
40
Invaded
0
Robust
0
Robust
80
Invaded
50
Invaded
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
0
Robust
D strategies are robust against
other
SDISC
the invasion
by
SD defectors (ALLD, defective
(C, GBBB,
BBBB)
C
SDISC
no-link players),
 Although
SD:
C
IR: Link SDISC
SD: C
 They
IR:
Link ESare
D
ES
invaded by cooperative
players
(C, GBBG, BBBB)
C
(ALLC and cooperative
no-link players)
ES
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
Good Bad
D in SD
Good
Bad
STAND
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Gave
G
G
B
B
Didn’t give
B
B
B
B
Didn’t give
B
G
B
B
SDISC
C in SD
D in SD
ES
C in SD
Good Bad
Good
Bad
Good
Bad
Gave
G
B
B
B
Gave
G
Didn’t give
B
B
B
B
Didn’t give
B
D in SD
Good
Bad
B
B
B
G
B
B
Unsuccessful strategies
IS
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
B
B
B
STAND
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
G
B
B
Unsuccessful strategies can exclude SD defectors.
C in SD
IS
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
B
B
B
L
L
L
L
L
STAND
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
G
B
B
Unsuccessful strategies can exclude SD defectors.
However, they regard SD cooperators who helped
Bad (cooperative no-link players) as Good.
C in SD
IS
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
B
B
B
L
L
L
L
L
STAND
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
G
B
B
Unsuccessful strategies can exclude SD defectors.
However, they regard SD cooperators who helped
Bad (cooperative no-link players) as Good.
C in SD
IS
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
B
B
B
C in SD
STAND
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
G
B
B
Cooperative no-link players can increase.
No-link
L
L
No-link
L
No-link
L
L
D in SD
L
No-link
Unsuccessful strategies can exclude SD defectors.
However, they regard SD cooperators who helped
Bad (cooperative no-link players) as Good.
C in SD
IS
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
B
B
B
C in SD
STAND
D in SD
Good
Bad
Good
Bad
Gave
G
G
B
B
Didn’t give
B
G
B
B
Cooperative no-link players can increase.
The higher the frequency of no-link players becomes,
the easier it is for free-riders to invade.
No-link
L
L
← cannot
maintain
mutual
cooperation
in SD.
No-link
L
No-link
L
L
L
No-link
Successful strategies
SDISC
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
B
B
B
ES
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
G
B
B
Successful strategies can also exclude SD defectors.
SDISC
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
B
B
B
ES
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
G
B
B
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
Good
Bad
Gave
G
B
B
B
Didn’t give
B
B
B
B
ES
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
G
B
B
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
Good
Bad
Gave
G
B
B
B
Didn’t give
B
B
B
B
ES
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
G
B
B
Cooperative no-link players cannot invade.
L
No-link
L
L
L
L
ALLC
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
Good
Bad
Gave
G
B
B
B
Didn’t give
B
B
B
B
ES
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
G
B
B
Cooperative no-link players cannot invade.
SD free-riders cannot invade.
L
No-link
L
L
L
ALLC
Successful strategies can also exclude SD defectors.
Moreover, they regard SD cooperators who helped
Bad (cooperative no-link players) as Bad.
SDISC
Gave
C in SD
D in SD
Good
Bad
Good
Bad
G
B
B
B
ES
Gave
C in SD
D in SD
Good
Bad
Good
Bad
G
B
B
B
Didn’t give
G
B
B
B
B
B
key point for the maintenanceB of mutual
Didn’t give
B
The
cooperation
SD isstrategies
to exclude
notinvade.
only SD freeCooperative in
no-link
cannot
riders but also those who help SD free-riders.
L
No-link
L
L
L
ALLC
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.
Thank you
for your attention!
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
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
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）
C in SD
No-link SDISC（C/D, GBBB, GBBB）
D in SD
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t
give
B
B
B
B
C in SD
D in SD
Good
Bad
Good
Bad
Gave
G
B
G
B
Didn’t
give
B
B
B
B
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 = 0.025.
• 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.
Details of simulation settings (2)

After 10000 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.
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.
Characteristics of successful strategies
（C, GBBX, BBBB）
C in SD
Gave
Didn’t
give
D in SD
Good
Bad
Good
Bad
G
B
B
G or B
B
B
B
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
Invader’serrorInvader’s
β=perception
in SD
SD
IR
α=execution
error instrategy
SD
strategy
Majority
D
IS
C
D
STAND
C
D
SDISC
C
ES
D
C
ALLD
No-link IS
ALLC
No-link IS
ALLD
No-link
STAND
ALLC
No-link
STAND
ALLD
No-link
SDISC
ALLC
No-link
SDISC
ALLD
No-link ES
ALLC
No-link ES
β=0.0
α=0.0
β=0.0
α=0.025
β=0.025
α=0.0
β=0.025
α=0.025
Robust
Robust
Invaded
Invaded
Robust
Robust
Robust
Invaded
Invaded
Robust
Robust
Robust
Invaded
Invaded
Robust
Robust
Robust
Invaded
Invaded
Robust
Robust
Robust
Robust
Robust
Invaded
Invaded
Invaded
Invaded
Invaded
Invaded
Invaded
Invaded
Robust
Robust
Robust
Robust
Robust
Robust
Robust
Robust
Robust
Robust
Robust
Robust
Invaded
Robust
Robust
Robust
Robust
Robust
Robust
Invaded
Robust
Robust
Robust
Robust
Robust
Robust
Robust
Invaded
Robust
Robust
Robust
Robust
Results in random
matching situation
Majority
Invader’s
SD
strategy
D
Random matching
: The donor is randomly
matched with the
recipient.
IS
C
D
STAND
C
Errors (error of
execution / perception,
in SD / IR)=0.025
Number of
replication=30
D
SDISC
C
D
ES
C
Invader’s
IR
strategy
ALLD
No-link IS
ALLC
No-link IS
ALLD
No-link
STAND
ALLC
No-link
STAND
ALLD
No-link
SDISC
ALLC
No-link
SDISC
ALLD
No-link ES
ALLC
No-link ES
Invaded
rep (%)
0
0
96.7
100
0
0
0
13.3
0
0
0
10
0
0
0
0
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
Majority
Invader’s
SD
D
STAND
C
Invader’s
IR
ALLD
No-link STAND
ALLC
No-link STAND
Invaded
rep
(%)
0
0
0
47
Result that
P&B
(2004)
showed
Linkage STAND can resist the invasion by ALLC, but
cannot resist the invasion by cooperative no-link players.
“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.
C in SD
D in SD
SDISC
Good
Bad
Good
Bad
Gave
G
B
B
B
Didn’t give
B
B
B
B
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.
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 freeriders 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.
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.
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.
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
Good
I want to give
this Good person.
Bad
Bad
Bad
Good
Bad
Good
Unnatural settings: random matching situation
Random matching environment
Most of the theoretical studies in biology have
assumed this environment.
Give or
Not
give?
From the group, a donor and a recipient
is randomly chosen to form a pair.
Donor
Recipient
Bad
Good
Bad
Bad
Bad
Good
Bad
Bad
Good
A donor must act toward a designated opponent –
Give or Not give
Unnatural settings: random matching situation
Random matching environment
Most
of the theoretical
studies
in biology
 Question
from social
sciences:
Whyhave
is a donor
assumed this environment.
forced to play a game with a designated opponent?
Fromignore
the group,
a donoropponents?
and a recipient
Why not simply
undesirable
is randomly chosen to form a pair.
Give or
Not
 If a donor
knows
every other person’s reputation,
Donor
Recipient
give?
Badcan
isn’t it more natural to assume that
a donor
Good
Bad
choose freely his/her recipient he/sheBad
wants? (Aoki,
Good
Bad
2001)
Bad
Bad
Good
Selective-play
Hayashi,–
A donor mustenvironment
act toward a(Yamagishi
designated&recipient
1996)
be more natural.
Giveseems
or Not to
give
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.
Gave
Did not give
Good
Bad
GtoG
GtoB
NGtoG
NGtoB