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6 Modeling Strong
Reciprocity

Armin Falk and Urs
Fischbacher




6.1 Introduction

In this chapter, we discuss how preferences for reciprocity can be mod-
eled in a game-theoretic framework. The fact that people have a taste
for fairness and reciprocity implies that their utility does not only de-
pend on their own monetary payoff but also on the payoffs of the other
players. This means that we have to distinguish between the payoff
subjects receive (for instance, in an experimental game) and the utility,
which not only consists of the own payoff but also on the payoff rela-
tive to the payoffs of the other players. We denote by pi the material
payoff player i gets while Ui denotes utility.
The basic structure of reciprocal behavior consists of the reward of
kind actions and the punishment of unkind ones. This structure can be
expressed in the following formula:

Ui ¼ pi þ ri js °1Þ
According to de¬nition (1) player i™s utility is the sum of the follow-
ing two terms: The ¬rst summand is simply player i™s material payoff
pi . This material payoff corresponds to the material payoffs that are
induced by the experimenter. The second summand”which we call
reciprocity utility”is composed of the following terms:

The positive constant ri is called the reciprocity parameter. This con-
0


stant is an individual parameter that captures the strength of player i™s
reciprocal preferences. The higher ri , the more important is the reci-
procity utility as compared to the utility arising from the material pay-
off. Note that if ri equals zero, player i™s utility is equal to his material
payoff. Put differently, if ri ¼ 0, the player has Homo economicus prefer-
ences just as assumed in standard game theory.
194 Falk and Fischbacher



The kindness term j measures the kindness player i experiences from
0


another player j™s actions. The kindness term can be positive or nega-
tive. If j is positive, the action of player j is considered as kind. If j is
negative, the action of player j is considered as unkind.
The reciprocation term s measures player i™s reciprocal response. As a
0


¬rst approximation, s is simply player j™s payoff.
The product of the kindness term (j) and the reciprocation term (s)
0


measures the reciprocity utility. If the kindness term is greater than
zero, player i can ceteris paribus increase his utility if he chooses an
action that increases player j™s payoff. The opposite holds if the kind-
ness term is negative. In this scenario, player i has an incentive to re-
duce player j™s payoff. As an example of this phenomenon, consider
¨
the ultimatum game (Guth, Schmittberger, and Schwarze 1982). If the
proposer™s offer was very low, the kindness term is negative and a
reciprocal player i can increase his utility by rejecting the offer, which
reduces player j™s payoff.

To make the strong reciprocity model more concrete, we have to
clarify two questions: How do people evaluate whether a treatment is kind
or unkind and how do people react towards that kindness or unkindness?
Both questions are much more subtle than it may seem on the surface.
For example, in order to evaluate kindness, do people care only for the
consequences of an action or do they also look at the motives imputed
to that action? Or, what exactly is the primary aim of a reciprocating
subject, to retaliate or to decrease payoff differences between him and
the other person? And, who is the relevant other person? These and
many other questions need to be answered before a concise modeling
of reciprocity is possible.
To better understand the determination of kindness and the aim of
the reciprocal act, we present new questionnaire data and report on a
series of experiments that were designed to answer the most relevant
questions:

1. What determines the sign of j? This question is intimately related to
the question of what reference standard is applied for the comparison of
payoffs. An intuitive anchor for comparisons is an equitable share of
payoffs. In fact, we show that many people consider an equitable share
as an important reference standard.
2. How important are the fairness intentions attributed to other
players™ actions? This question has attracted a lot of attention. It
Modeling Strong Reciprocity 195



touches on the fundamental issue of consequentialistic versus non-
consequentialistic fairness judgments: Is it only the material payoff
consequence that determines the perception of fairness, or do people
also take into account the motivation that underlies a particular action?
The answer to this question is not only important for the correct mod-
eling of fairness preferences, but also for the consequentialistic practice
in standard economic theory that de¬nes the utility of an action solely
in terms of the consequences of this action.
3. What is the aim of a reciprocating agent? Do people punish in order
to reduce distributional inequity between themselves and their oppo-
nent(s) or in order to reduce the other players™ payoffs”that is, to retal-
iate? This question allows a clear distinction between so-called inequity
aversion approaches and reciprocity approaches. According to the
former, reciprocal actions are triggered by the desire to reduce an un-
fair payoff distribution. This implies, for example, that we should ob-
serve no reciprocal actions if the rewarding or punishing act cannot
reduce inequity. Reciprocity, on the other hand, understands recipro-
cal actions as the desire to reward or to retaliate”meaning one should
observe punishments and rewards even in situations where inequity
cannot be reduced.
4. Who is the relevant reference agent? This question arises immediately
if persons interact in groups and not only in bilateral relationships. Do
people evaluate fairness towards individuals separately or towards the
group as a whole? This question is of obvious importance, for example,
in the context of social dilemma situations. If kindness is evaluated
towards each of the other group members, reciprocal reactions are also
targeted individually. Only if this dynamic holds it is guaranteed that
the ˜˜right™™ persons (the defectors) receive the punishments. As a con-
sequence, reciprocally driven sanctions can function as a disciplinary
device to promote cooperative outcomes.
The chapter is organized as follows. In the next section, we discuss
the determinants of kindness and reciprocation with the help of a ques-
tionnaire study and various experiments. Section 6.3 summarizes the
arguments and presents a formal model of reciprocity that takes the
facts presented in section 6.2 into account. The ¬nal section discusses
related literature and gives a short conclusion.
196 Falk and Fischbacher



6.2 Determinants of Kindness and Reciprocation

6.2.1 Equity as a Reference Standard
To investigate how people perceive the kindness or unkindness of an
action, it is a natural starting point to simply ask people. We therefore
conducted a questionnaire study with 111 students from the University
of Zurich and the Polytechnical University of Zurich. The study was
performed under anonymous conditions in our experimental computer
lab during the months of May and June in 1998.
In the study, each subject i was in a hypothetical bilateral exchange
situation with another subject j. Subjects i were asked to indicate how
kind or unkind they perceive different divisions of an endowment of 10
Swiss Francs, where it was always j who divides the pie between her-
self and i. To measure kindness, subjects could use numbers ranging
from À100 (very unkind) to þ100 (very kind). In total, subjects were
given nine different decision situations with different strategy sets
for j.
In this section we concentrate on the ¬rst decision situations, where j
could choose among 11 different allocations (see ¬gure 6.1). If j offers 0
to i (and keeps everything to herself), i perceives this as very unkind
(À95.4). If j offers 1 (and keeps 9 to herself), this is regarded as slightly
less unkind (À84.5). This progression continues to the situation where


100.0
80.0
Estimation of second mover




60.0
40.0
20.0
0.0
-20.0
-40.0
-60.0
-80.0
-100.0
0 1 2 3 4 5 6 7 8 9 10
Offer

Figure 6.1
Kindness dependent on player j™s offer.
Modeling Strong Reciprocity 197



j keeps nothing to herself and offers i 10, which is viewed as very kind
(þ72.3). Figure 6.1 reveals various important insights. First, it demon-
strates kindness is monotonically increasing in the offer. The more j is
willing to share with i, the more kind this is perceived by i. Second,
an equitable share serves as an anchor for the determination of a fair
or unfair offer. This can be concluded from the fact that as the offer
approaches the equitable ¬gure of 5, the perception changes from ˜˜un-
kind™™ to ˜˜kind.™™
A similar questionnaire study was conducted by Loewenstein,
Thompson, and Bazerman (1989). They also found strong evidence for
the importance of an equity based reference standard.

6.2.2 How Important Are Intentions?
There is an ongoing debate whether intentions are behaviorally rele-
vant. The consequentialistic perspective claims that intentions are irrel-
evant for the evaluation of kindness. According to this conception, the
distributive consequences of an action are suf¬cient to trigger behavior
and no consideration of intentions is needed. Alternatively, it has been
argued that the kindness or unkindness of an action crucially depends
on the motives and intentions that can be attributed to that action.
According to this perspective, actions that cause the same consequen-
ces might be perceived very differently, depending on the underlying
intentions. As a result, they also might be reciprocated very differently.
To examine whether intentions matter, Falk, Fehr, and Fischbacher
(2003) conducted four mini-ultimatum games (see ¬gure 6.2). As ¬gure
6.2 indicates, the proposer can choose between two allocations, x and
y. In all four games, the allocation x is the same while the allocation
y (the ˜˜alternative™™ to x) differs from game to game. If the proposer
chooses x and the responder accepts this offer, the proposer gets 8
points while the responder receives 2 points. In game (a), the alterna-
tive offer y is (5/5). This game is therefore called the (5/5)-game and
so forth.
Let us concentrate on the (8/2)-offer. The standard model with
sel¬sh preferences predicts that in all games, the offer (8/2) is never
rejected. Fairness theories that rely on the consequentialistic perspec-
tive predict that this offer may be rejected, but that the rejection rate
of the (8/2)-offer is the same across all games. Since outcomes follow-
ing the (8/2)-offer are identical across games, different rejection rates
of the (8/2)-offer are impossible to reconcile with a consequentialistic
perspective.
198 Falk and Fischbacher


P P

y y
x x
R R
R R
a a
r r r r
a a


8 0 5 0 8 0 2 0
2 0 5 0 2 0 8 0

(a) (5/5)-game (b) (2/8)-game



P P

y y
x x
R R
R R
a a
r r r r
a a


8 0 8 0 8 0 10 0
2 0 2 0 2 0 0 0

(c) (8/2)-game (d) (10/0)-game
Figure 6.2
The mini-ultimatum games that test the importance of intentions.


Intuitively, one would expect that in the (5/5)-game, a proposal of
(8/2) is clearly perceived as unfair because the proposer could have
proposed the egalitarian offer (5/5). In the (2/8)-game offering, (8/2)
may still be perceived as unfair”but probably less so than in the
(5/5)-game, because the only alternative available to (8/2) gives the
proposer much less than the responder. In a certain sense, therefore,
the proposer has an excuse for not choosing (2/8) because one cannot
unambiguously infer from his unwillingness to propose an unfair offer
to himself that he wanted to be unfair to the responder.
Thus, one could expect that the rejection rate of the (8/2)-offer in the
(5/5)-game is higher than in the (2/8)-game. In the (8/2)-game, the
proposer has no choice at all so that the proposer™s behavior cannot be
judged in terms of fairness. Responders can only judge the fairness of
the outcome (8/2), and if they exhibit suf¬cient aversion against ineq-
uitable distributions, they will reject this distribution of money. Since
any attribution of unfairness to the proposer™s behavior is ruled out in
Modeling Strong Reciprocity 199



this situation, one might expect an even lower rejection rate compared
to the (2/8)-game. Finally, offering (8/2) in the (10/0)-game may even
be perceived as fair (or less unfair) since, after all, the proposer could
have chosen an even more unfair payoff. Therefore, the rejection rate
of (8/2) is likely to be the lowest in this game.
In fact, the rejection rate in the (5/5)-game is highest: 44.4 percent
rejected the (8/2)-offer in this game. Meanwhile, 26.7 percent rejected
the (8/2)-offer in the (2/8)-game, 18 percent in the (8/2)-game and 8.9
percent in the (10/0)-game. These results clearly reject the consequen-
tialistic perspective and suggest that intentions play a major role in
the determination of kindness. Since rejection rates are not zero in the
(8/2)-game (where intentions cannot play a role because the proposer
has no choice), it seems, however, that kindness comprises both inten-
tions and consequences of an action. This ¬nding is corroborated by
experiments by Brandts and Sola (2001), McCabe, Rigdon, and Smith
(2003), Blount (1995), and Charness (forthcoming). The latter two
studies investigate experiments in which the ¬rst mover™s choice is
made by a random device. This excludes any intention from this
choice. They show that the reciprocal response of the second mover is
weaker but not absent. For a dissenting view on the role of intentions
see Cox (2000), and Bolton, Brandts, and Ockenfels (1998).
The discussion of the experiment shown in ¬gure 6.2 has pointed
at the question how people infer intentions from their opponents. We
have argued that the alternatives the opponent can choose from are es-
sential in this determination. To better understand how the set of alter-
native actions of an opponent j alters i™s perception of j™s kindness, we
conducted the questionnaire study mentioned earlier in this section. In
particular, we changed the set of alternatives of j could choose from
and asked players i how kind they perceived different actions of j
to be. Table 6.1 contains all variants. As a benchmark for our discus-
sion, we use column (i) where player j has a rather unlimited action
space”where j™s action set allows the choice between fair and unfair
actions, and therefore each action clearly signals (fair or unfair) inten-
tions. In columns (ii) to (ix), we systematically vary the strategy set of
j. In column (ii), for example, j can offer only 2, 5, or 8 to player i,
while in column (iii), j can offer only 2, and so on. Table 6.1 reveals
¬ve interesting observations. In our discussion we focus primarily on
the two payoff combinations (2/8) and (8/2).
First, if j™s strategy set contains only one element”that is, if j has no
alternative to choose, the kindness of an advantageous offer and the
200 Falk and Fischbacher


Table 6.1
Player i™s estimation of j™s kindness (average values, n ¼ 111).
(p j , pi ) i ii iii iv v vi vii viii ix

(0, 10) 72.3 79.9 73.4 80.3
(1, 9) 68 73.3 62 72.5
(2, 8) 62 75.3 41.1 61.2 61.9 40.8 62.2
(3, 7) 51.4
(4, 6) 40
(5, 5) 29.4 33.4 27.9
À23.2
(6, 4)
À52.9
(7, 3)
À71.9 À70.6 À31.5 À47.7 À50.5 À9.1 À60.9
(8, 2)
À84.5 À80.3 À56.4 À82.6
(9, 1)
À95.4 À97.3 À88.8 À97.3
(10, 0)



unkindness of a disadvantageous offer is much weaker, compared to a
situation where j can choose between fair and unfair offers. This can be
seen by comparing the perceived kindness of the (2/8)-offer in col-
umns (i) and (iv) (þ62 versus þ41.1), and the unkindness of the (8/2)-
offer in columns (i) and (iii) (À71.9 versus À31.5). The fact that the
same payoff consequences are perceived differently, depending on the
strategy set of player j, clearly contradicts the consequentialistic view
of fairness.
Second, even if j has no alternatives and therefore cannot signal any
intentions, perceived kindness or unkindness is not zero (see col-
umns (iii) and (iv)). People dislike the disadvantageous (8/2)-outcome
(À31.5) and like the advantageous (2/8)-outcome (þ41.1), even if this
outcome was unintentionally caused. This ¬nding is in contradiction
to a purely intention-based notion of fairness.
Third, even if j™s strategy space is limited, the kind (2/8)-offer is
viewed as similarly kind as in column (i), as long as j could have
made less kind offers to i (compare columns (i) with columns (ii), (v),
(vi), and (ix)). This means that a fair offer signals fair intentions if j
could have been less fair. By the same token, the kindness of the (2/8)-
offer is lower than in column (i) and similarly low as in column (iv) if
player j does not have the chance to make a less fair offer (compare
column (vii)). The intuition for the latter result is straightforward. If j
has no chance to behave more ˜˜opportunistically,™™ how should i infer
from a fair action that j really wanted to be fair? After all, he took the
least fair action.
Modeling Strong Reciprocity 201



Fourth, a comparison between columns (viii) and (i) shows that the
unkindness of an offer depends on j™s possibility to choose a less un-
kind offer. In column (viii), the (8/2)-offer is the least unfriendly offer
and players i indicate a much lower perceived unkindness compared
to column (i). The intuition for the different kindness scores parallels
the one given in case of a kind offer: You cannot blame a person for
being mean if”after all”he did the best he could.
Fifth, there is an asymmetry between the perception of kind and un-
kind offers. We saw that as long as j could have made a worse offer to
i, the kind (2/8)-offer is viewed as similarly kind as in column (i) (see
our third observation). Things look different for unkind offers, how-
ever. Take a look at column (ii). In this situation, j could have made
more friendly offers than to offer (8/2)”namely, (5/5) and (2/8). The
perceived unkindness of the (8/2) offer is very similar to the one in
column (i) (À70.6 and À71.9, respectively). If we move on to columns
(v) and (vi), however, the perceived unkindness drops to À47.7 and
À50.5, respectively”even though player j could have made better
offers to i.
The difference between the perceived unkindness in columns (ii)
and (v) and (vi) points at the question of ˜˜how reasonable™™ it is to
expect that j chooses an offer that puts herself in a disadvantageous
position. In column (ii), j has the option to offer (5/5), an offer that
is reasonable to expect. In columns (v) and (vi) on the other hand, a
more friendly offer for i than (8/2) implies that player j puts herself in
a very disadvantageous position (8 to i and 2 to j). In this situation,
players i seem to understand that it is an unreasonable sacri¬ce for j to
make a more friendly offer. Therefore, they consider the (8/2) as less
unkind compared to a situation where j does have a reasonable alter-
native (as in column (ii)). In our formal model later in this chapter, we
use these ¬ve observations to formalize intentions.
The results of this questionnaire data match very nicely the rejection
behavior in the mini ultimatum games (UGs) in ¬gure 6.2. Consider for
instance the (8/2)-offer in the (8/2)-game. The rejection rate of this of-
fer is lower in the (8/2)-game than in the (5/5)-game. This corresponds
to the lower indicated unkindness of the corresponding offer in table
6.1 (À31.5 in column (iii) compared to À70.6 in column (ii)). However,
an (8/2)-offer is also perceived as unkind if there is no alternative
(À31.5 in column (iii) is negative). This explains why there are rejec-
tions of the (8/2)-offer in the (8/2)-game.
202 Falk and Fischbacher



6.2.3 Reciprocation Target
So far, we have analyzed important motives for the evaluation of kind-
ness. In this section, we address the question how people react to per-
ceived kindness, that is, to what end do they reward or punish? There
are two principal motives that may account for performing punish-
ments: (i) retaliation and (ii) inequity aversion. According to the latter,
a person will punish another person only if this reduces the inequity
between the person and his opponent(s). Retaliation, on the other
hand, dictates to punish in order to reciprocate an unkind act. In re-
taliation, the aim of the reciprocating subject is not to reduce dis-
tributional inequity but to lower the (unkind) opponent™s payoff.
Retaliation-driven punishments are therefore not restricted to situa-
tions where inequity can be reduced. Instead it occurs whenever a per-
son is treated unkindly and is given a chance to ˜˜pay back.™™
With the help of the following three experiments, Falk, Fehr, and
Fischbacher (2001) directly tested the importance of inequity aversion
and retaliation motives for the performance of punishments. In the ¬rst
experiment, the prediction depends on the way inequity is measured.
In discussing this experiment, we also bring attention to the question
whether inequity aversion should be measured as a difference between
payoffs or in terms of relative payoff shares. The ¬rst two games
described here are simple ultimatum games, and the third is a public
goods game with a subsequent punishment stage. The two ultimatum
games are presented in ¬gure 6.3.
In both games presented in ¬gure 6.3, the proposer can choose be-
tween the offer x and y, where x is the unfair offer (8/2) and y the
fair offer (5/5). The consequences arising from the rejection of an offer

P P

y y
x x

R R
R R
r r r r
a a a a


6
8 .8 5 .5 8 5 3
2 .2 5 .5 2 5 3
0
UG with constant relative share UG with constant difference

Figure 6.3
Ultimatum games (UGs) where a rejection does not reduce inequity.
Modeling Strong Reciprocity 203



are different, however. In the game UG with constant relative share, a
rejection of an offer leaves the ratio between the players™ payoffs un-
changed. In the game UG with constant difference, it is the difference be-
tween the payoffs that remains unchanged following a rejection. The
standard economic prediction for the two ultimatum games dictates
the proposers to choose the offer x which is accepted by the res-
ponders. Assuming that people have fairness preferences, the predic-
tions depend on the nature of these preferences.
Let us start our discussion with the UG with constant relative share.
In this game, the retaliation motive predicts that the unfair offer x
is rejected. After all, the offer (8/2) is very unkind and the proposer
˜˜deserves™™ the punishment. The predictions put forward by a fairness
concept based on inequity aversion depend on the way inequity is
measured. If inequity is measured as the difference between the own
payoff and the payoff of the other player °pi À pj Þ, inequity aversion
predicts rejection because the difference between 0.8 and 0.2 is smaller
than the difference between 8 and 2. If inequity aversion is measured
on the basis of the relative share of own payoff to (the sum of all)
others™ payoffs °pi =Spj Þ, no rejections are expected”since the share of
0.2/1 is exactly the same as that of 2/10. In the UG with constant differ-
ences, inequity aversion predicts no punishments independent on how
inequity aversion is measured. The retaliation motive, on the other
hand, does not preclude punishments in this game because the offer
(8/2) is quite unfair and the responder™s payoff can be reduced by a
rejection.
The results of the UG with constant relative share cast serious doubts
on the validity of the inequity aversion motive if modeled as relative
share. Even though the relative share argument predicts a rejection
rate of zero, 38 percent of all players reject the unfair offer (8/2). Ineq-
uity aversion as measured by the difference between payoffs seems to
¬t the data better. However, in the UG with constant differences, the re-
jection rate of the unfair offer (8/2) is still 19 percent. Thus, even
if modeled in terms of differences of payoffs, inequity aversion does
not account for about 20 percent of the observed punishment in our
reduced ultimatum game.
To further test the importance of inequity aversion as a motivational
factor that drives punishments, we studied a public goods game with
a subsequent punishment opportunity (Falk, Fehr, and Fischbacher
2001). In the ¬rst stage of this one-shot game, three players ¬rst decide
simultaneously on a voluntary provision to a linear public good. The
204 Falk and Fischbacher



decision amounts to an investment of either 0 or 20 points to the public
good. Payoffs in the ¬rst stage are 20 À own provision þ 0.6 Ã sum of all
provisions. During the second stage, each player has to decide whether
or not he wants to sanction the other players in his group. Deducting
points is costly”deducting 1 point from another player is associated
with a cost of 1 point.
Assuming sel¬sh preferences, the predicted outcome is straightfor-
ward. Since deducting points is costly, the second stage is basically
irrelevant and hence nobody invests into the public good. As a conse-
quence, total group income amounts to 60 points. Social surplus, on
the other hand, is maximized if each player invests his 20 points. In
this case, total group income equals 60 Ã 1:8 ¼ 108. What do we expect
if people have a preference for fairness? The most obvious prediction
can be derived for the situation where a cooperator faces two defectors.
In this situation, a cooperator cannot reduce inequality by punishing the
defectors (independent of how inequality is measured). In fact, to re-
duce the payoff of each of the two defectors by one point, a cooperator
has to invest two points. Hence, inequity aversion predicts zero pun-
ishments in this situation. Quite to the contrary, retaliation is compati-
ble with punishments in this situation since a defector has acted in an
unkind fashion and ˜˜deserves™™ a punishment.
As it was the case with the two UGs discussed in the previous para-
graphs, inequity aversion cannot account for the observed reciprocal
actions in the public goods game with costly punishments: here 46.8
percent of the cooperators punish even if they are facing two defectors,
where as the inequity aversion motive predicts zero punishment. We
therefore conclude that much of the observed punishment behavior is
incompatible with inequity aversion and should rather be understood
as a desire to retaliate. Even if cooperators cannot reduce the inequity
between cooperators and defectors, they want to lower the payoffs of
the unkind defectors.

6.2.4 Is Kindness Evaluated towards Individuals or towards the
Group?
In the questionnaire study outlined above, as well as in many bilateral
bargaining situations, the question ˜˜who is the relevant reference
agent™™ is trivial. This does not hold, however, if players interact in a
group. Here it is far from obvious whether people determine kindness
towards each of the other group members or, for example, towards
the group on average. To investigate this question, Falk, Fehr, and
Fischbacher (2001) conducted a variant of the public goods game with
Modeling Strong Reciprocity 205



punishment opportunity discussed in the previous section. The only
difference concerns the cost of punishment: The cost of deducting
points now depends on the ¬rst stage behavior. Deducting one point
from another player who has cooperated in the ¬rst stage is slightly
more expensive (0.4 points per point deducted) than deducting a point
from a defector (0.3 points per point deducted). Thus, punishing a
cooperator is more expensive than punishing a defector.
Defecting in this game is clearly an unfair act. If people are suf¬-
ciently reciprocally motivated, they will therefore punish this unfair be-
havior. The precise punishment pattern, however, depends on whether
people evaluate kindness towards each other as individuals or towards
the group on average. Let us concentrate on the situation where player
i cooperated while one of the other two players cooperated and the
other defected. Theories that assume individual comparisons predict
that if cooperators punish in this situation, they will only punish defec-
tors: The other cooperator acted in a fair way and therefore there is no
reason to punish him. The defector, however, acted in an unfair man-
ner and therefore deserves the punishment.
Theories that rest on group comparison, however, predict that if
cooperators punish, they will punish the other cooperator. The reason
is simple: Before punishing, the cooperator has a lower payoff than the
average group payoff. This is so because defectors have a higher pay-
off than cooperators. If the cooperator wants to reduce the disadvanta-
geous inequity between his payoff and the average group payoff, it is
cheapest to punish the cooperator because punishing the other cooper-
ator is cheaper (0.3 per deduction point) than punishing the defector
(0.4 per deduction point). Thus, if the cooperator punishes, he strictly
prefers to punish the cooperator.
The experimental results clearly indicate that people perform individ-
ual comparisons. In the situation where a cooperator faces a defector
and another cooperator, cooperators punish defectors quite frequently
and almost never punish other cooperators. On average, cooperators
allocate 6.6 punishment points to defectors and 0.3 punishment points
to cooperators. This is in clear contradiction to the idea that coopera-
tors want to improve their situation towards the group average, which
would have dictated them to punish cooperators.

6.3 Modeling Reciprocity

The preceding sections suggest that a theory of reciprocity should in-
corporate the following four motives: (i) Equitable shares serve as a
206 Falk and Fischbacher



reference standard. (ii) The evaluation of kindness rests on intentions
and consequences of an action. (iii) The desire to retaliate is much
more important for a reciprocating person than the desire to reduce in-
equity. (iv) People evaluate kindness not towards a group average, but
individually towards each person with whom they interact.
In the following section, we sketch a model of reciprocity that tries to
take account of motives (i) through (iv). The purpose of the presenta-
tion in this section is only expositional. We restrict our presentation to
the key aspects of our model and omit all technical details. The reader
interested in technical details should refer to Falk and Fischbacher
(1999) where the model is explained and discussed in detail.

6.3.1 Notations
Recall the sketch of the reciprocity model in section 6.1. We pointed
out that reciprocity consists of a kind (or unkind) treatment by another
person (represented by the kindness term j) and a behavioral reaction to
that treatment (represented by the reciprocation term s). We will now
de¬ne these terms and start the outline of the model with the two-
player case.
Consider a two-player extensive form game with a ¬nite number of
stages and with complete and perfect information. Let i and j be the
two players in the game. Ni denotes the set of nodes where player i
has the move. Let n be a node of this player. Let An be the set of actions
in node n. Let F be the set of end nodes of the game. The payoff func-
tion for player i is given by pi : F ! R.
Let Si be the set of mixed behavior strategies of player i. For si A Si
and sj A Sj and for a player k A fi; jg, we de¬ne pk °si ; sj Þ as player k™s
expected payoff, given strategies si and sj . Furthermore, we de¬ne
pk °n; si ; sj Þ as the expected payoff conditional on node n: It is the
expected payoff of player k in the subgame starting from node n, given
that the strategies si and sj are played.

6.3.2 The Kindness Term
Let n A Ni . The kindness term jj °n; si ; sj Þ is the central element of our
model. It measures how kind a player i perceives the action by a
player j and depends on both outcomes and intentions of this ac-
tion. The outcome is measured with the outcome term Dj °n; si ; sj Þ, where
Dj °n; si ; sj Þ > 0 expresses an advantageous outcome and Dj °n; si ; sj Þ < 0
expresses a disadvantageous outcome. In order to determine the
overall kindness, Dj °n; si ; sj Þ is multiplied with the intention factor
Qj °n; si ; sj Þ b 0. This factor is a number between a small and positive e
Modeling Strong Reciprocity 207



and 1, where Qj °n; si ; sj Þ ¼ 1 captures a situation where Dj °n; si ; sj Þ is
induced fully intentionally, and Qj °n; si ; sj Þ < 1 expresses a situation
where intentions are absent or where it was not fully intentionally
fair or unfair. The kindness term jj °n; si ; sj Þ is simply the product of
Dj °n; si ; sj Þ and Qj °n; si ; sj Þ.
First, we de¬ne the outcome term:

Dj °n; si ; sj Þ :¼ pi °n; si ; sj Þ À pj °n; si ; sj Þ °2Þ
For a given Qj °n; si ; sj Þ, the outcome term Dj °n; si ; sj Þ measures the kind-
ness of player j towards player i. It captures the knowledge of player i
in node n about the two players™ expected payoffs. Since Qj °n; si ; sj Þ is
always positive, the sign of the kindness term, that is, whether an
action is considered as kind or unkind, is determined by the sign of
Dj °n; si ; sj Þ. The term Dj °n; si ; sj Þ is positive if player i thinks he gets
more than j. It is negative if player i thinks he gets less than j.
This de¬nition captures motive (i). We use equity as a reference
standard”that is, kindness equals zero, if both players get the same
payoff.
The answer to question (ii) showed that a purely outcome-oriented
model is not in line with many experimental ¬ndings. This fact is
implemented in the model with the intention factor Qj °n; si ; sj Þ. It mea-
sures to what extent there is a reasonable alternative for player j. This
factor is 1 if there is a reasonable alternative”that is, in a situation
where player i can conclude that the move of player j was intentionally
kind or unkind. If there were no reasonable alternative, for instance
if there were no alternative at all, this factor is positive but smaller
than 1. The detailed de¬nition of the y term incorporates the results
that followed from our discussion of table 6.1 and ¬gure 6.2. Since the
de¬nition is a little bit tedious, the interested reader should refer to
Falk and Fischbacher (1999).

De¬nition: Let strategies be given. We de¬ne the kindness term
j°n; si ; sj Þ in a node n A Ni as:

jj °n; si ; sj Þ ¼ Qj °n; si ; sj ÞDj °n; si ; sj Þ °3Þ

From (3) it immediately follows that a given outcome is perceived
as more kind or unkind depending on the size of Qj °n; si ; sj Þ. Put differ-
ently, if player j has, for example, no alternative to choose another out-
come or if he was forced to choose a particular outcome, perceived
kindness is smaller compared to a situation where Qj °n; si ; sj Þ ¼ 1
(that is, where j had a reasonable alternative). The condition that
208 Falk and Fischbacher



Qj °n; si ; sj Þ > 0 captures the fact that even if intentions are absent,
players experience the outcome per se as either kind or unkind: If we
would allow Qj °n; si ; sj Þ ¼ 0, this would imply that in the absence of
intentions, there would be no reciprocal actions anymore. As our dis-
cussion about the games in ¬gure 6.2 and table 6.1 have shown, how-
ever, people reward and punish even in these situations.

6.3.3 The Reciprocation Term
The second ingredient of our model concerns the formalization of
reciprocation. Let us ¬x an end node f that follows node n. Then we
denote v°n; f Þ, as the unique node directly following the node n on the
path leading from n to f .

De¬nition: Let strategies be as given in the previous section. Let i and
j be the two players and n and f be de¬ned as in the previous para-
graph. Then we de¬ne
si °n; f ; si ; sj Þ :¼ pj °v°n; f Þ; si ; sj Þ À pj °n; si ; sj Þ °4Þ

as the reciprocation term of player i in node n.

The reciprocation term expresses the response to the experienced
kindness”that is, it measures how much player i alters the payoff of
player j with his move in node n. The reciprocal impact of this action
is represented as the alteration of player j™s payoff from pj °n; si ; sj Þ to
pj °v°n; f Þ; si ; sj Þ.
With this approach, we take into account ¬nding (iii). Players do not
attempt to reduce inequity. Instead, players in our model gain utility
from punishing unkind behavior (or rewarding kind behavior)”that
is, by lowering or increasing their opponent™s payoff.

6.3.4 Utility and Equilibrium Concept

Notation: Let n1 and n2 be nodes. If node n2 follows node n1 in the
game tree (directly or indirectly), we denote this by n1 ! n2 .

Having de¬ned the kindness and reciprocation term, we can now
derive the players™ utility of the transformed ˜˜reciprocity game™™:

De¬nition: Let player i and j be the two players of the game. Let f be
an end node of the game. We de¬ne the utility in the transformed reci-
procity game as:
Modeling Strong Reciprocity 209


X
Ui ° f ; si ; sj Þ ¼ pi ° f Þ þ ri jj °n; si ; sj Þsi °n; f ; si ; sj Þ °5Þ
n!f
n A Ni


For ¬xed °si ; sj Þ, this utility function de¬nes a new game G°si ; sj Þ. If
°si ; sj Þ is a subgame perfect Nash equilibrium in G°si ; sj Þ, we call °si ; sj Þ
a reciprocity equilibrium.
The strategies si and sj in the game G°si ; sj Þ can be interpreted as the
beliefs of the players. For instance, player i believes player j will use
strategy sj , and he thinks player j expects him to use strategy si . Given
this belief, player i chooses an optimal strategy. A reciprocity equilib-
rium can then be considered as a combination of strategies and beliefs
in which the strategies are optimal and consistent with the beliefs. The
presentation of our theory in this form (without beliefs) follows an idea
of Gintis (2000).

6.3.5 Games with More than Two Players
There seem to be two ways for generalizing the model to more than
N > 2 players. We could de¬ne aggregated kindness and recipro-
cation terms, for instance, in the form jÀi °n; sÞ ¼ Qj °n; sÞ°pi °n; sÞ À
P
1
j0i pi °n; sÞÞ. Or we could sum up the reciprocity utility towards
NÀ1
each of the other players. From our analysis of question (iv), we know
that the second way is the right way to go, because people perform
individual comparisons instead of making comparisons to the group
average. This is also how the model is generalized in Falk and Fisch-
bacher (1999). We de¬ne jj!i °nÞ as player j™s kindness towards player
i and si!j °n; f Þ as player i™s reciprocation towards player j. The utility
of player i is then de¬ned as
XX
Ui ° f Þ ¼ pi ° f Þ þ ri jj!i °nÞsi!j °n; f Þ °6Þ
j0i n!f
n A Ni

The model just outlined explains the relevant stylized facts of a wide
range of experimental games (see Falk and Fischbacher 1999). Among
them are the ultimatum game, the gift exchange game, the reduced
best-shot game, the dictator game, the Prisoner™s Dilemma, and pub-
lic goods games. Furthermore, the theory explains why the same
consequences trigger different reciprocal responses in different envi-
ronments. Finally, the theory explains why in bilateral interactions
outcomes tend to be ˜˜fair,™™ whereas in competitive markets even ex-
tremely unfair distributions may arise.
210 Falk and Fischbacher



6.4 Other Approaches

Several other theoretical models have been developed to account for
observed reciprocal behavior. The models make very different assump-
tions about how people evaluate the fairness of an action and how they
respond to perceived fairness or unfairness. In this section, we brie¬‚y
discuss the most important models and relate their main assump-
tions to the four motives discussed in section 6.2.
Two well-known models rest on the premise that reciprocal actions
are driven by the desire to reduce inequity (Fehr and Schmidt 1999
(henceforth FS), and Bolton and Ockenfels 2000 (henceforth BO)). In
the speci¬cation of FS, it is assumed that in general, subjects suffer
more from inequity that is to their material disadvantage than from in-
equity that is to their material advantage. Formally, consider a set of N
players indexed by i A f1; . . . ; Ng and let p ¼ °p1 ; . . . ; pn Þ denote the
vector of monetary payoffs. In the FS model, the utility function of
player i is given by

ai X bi X
Ui ¼ pi À °pj À pi Þ À °pi À pj Þ °7Þ
N À 1 j; p >p N À 1 j; p >p
j i i j


with
ai b b i b 0 and bi < 1:

The ¬rst term in (7), pi , is the material payoff of player i. The second
term in (7) measures the utility loss from disadvantageous inequality,
while the third term measures the loss from advantageous inequality.
Given his own monetary payoff pi , player i™s utility function obtains a
maximum at pj ¼ pi . The utility loss from disadvantageous inequality
°pj < pi Þ is larger than the utility loss if player i is better off than player
j °pj < pi Þ. In a similar spirit, BO model inequity aversion. According
to model BO™s speci¬cation, a player i™s utility is weakly increasing
and concave in player i™s material payoff and”for a given material
payoff”utility is strictly concave in player i™s share of total income
and maximal if the share equals 1/N. Both models, FS and BO, are
able to correctly predict experimental outcomes in a wide variety of
experimental games. This includes two-person bargaining games
where outcomes tend to be ˜˜fair,™™ as well as market games where the
model (correctly) predicts that very unfair outcomes can emerge.
Moreover, both models are quite tractable and are therefore well-
Modeling Strong Reciprocity 211



suited as predictive tools if fairness issues have to be analyzed in
complex environments.
In light of the presented evidence, it seems to us that the FS model
has two major advantages over the BO model. First, according to the
FS model, inequity is evaluated towards each individual with whom a
player is interacting (see equation (7)). The BO model on the other
hand, measures inequity considerations of a person towards some ag-
gregate measure. As a consequence, the BO model predicts that in the
public goods game outlined in section 6.2.4, cooperative players pun-
ish defectors. However, the data shows that it is just the other way
round. Another drawback of the BO model is the measurement of in-
equity in terms of relative share. Even though this is appropriate for
some games, it seems that the FS approach, which relies on the differ-
ence of payoffs, does a bit better in general (see section 6.2.4).
The strongest objections against both approaches concern the miss-
ing account of intentions and the fact that the strongest motives for
punishments are retaliation motives and not the desire to reduce in-
equity. Both models take a consequentialistic perspective”for example,
they predict the exact same rejection rates of the (8/2)-offer across all
games in ¬gure 6.2. While distributive consequences clearly matter,
we have seen that the attribution of fairness intentions also plays a ma-
jor role. The fact that much of the observed punishment (in UG games
as well as in public goods games with punishment) are incompatible
with the desire to reduce inequity further limits the validity of the ineq-
uity aversion approach.
Another class of models assumes that intentions are important and
that reciprocal responses are not driven by the desire to reduce in-
equity aversion, but by the desire to retaliate and to reward. These
so-called ˜˜reciprocity models™™ include Rabin (1993), Dufwenberg and
Kirchsteiger (2004, henceforth DK), Levine (1998), Charness and Rabin
(2002, henceforth CR), and Falk and Fischbacher (1999, henceforth FF),
which we have sketched in section 6.4. Common to these approaches is
a strong emphasis on the concept of reciprocated kindness. All models
point out the importance of intentions for the evaluation of kindness.
According to Rabin, DK, and FF, intentions depend on the different
alternatives available to players. As we have outlined in our discussion
on table 6.1 and the mini-ultimatum games in ¬gure 6.2, players infer
different intentions by looking at the available alternatives. An impor-
tant difference between the models of FF, on one hand, and Rabin and
DK on the other concerns the interaction of outcomes and intentions.
212 Falk and Fischbacher



While DK and Rabin model kindness as solely determined by inten-
tions, the FF approach combines distributive concerns with the im-
portance of intentions. In the light of the experimental evidence (see
section 6.2.2) this is important because many people care about both
outcomes and intentions.
A completely different approach for measuring kindness is sug-
gested by Levine (1998). As opposed to the reciprocity model ex-
plained earlier, the players in Levine™s model do not reward or punish
kind or unkind actions. They reward or punish kind or unkind types.
They reward altruistic types and punish spiteful types. Levine assumes
that players differ with respect to their other-regarding preference.
This preference is described by a parameter ai . It measures the relative
importance of the payoff of another person compared to one™s own
payoff. If ai > 0, player i has some altruistic preference. If ai < 0, he is
spiteful. Secondly, players like to reward players with high aj and pun-
ish players with a low (negative) aj .
P
The utility in Levine™s model is given by Ui ¼ pi þ j0i °°ai þ raj Þ=
°1 þ rÞÞpj . The parameter r is a universal reciprocity parameter”that
is, all players are assumed to have the same reciprocity parameter. The
model is an incomplete information model, since people have an initial
prior about the type of their opponent. After observing their oppo-
nent™s action, players update their beliefs. If the action was ˜˜friendly,™™
the belief that the person is altruistic gets larger, which implies a
friendly response and vice versa.
This approach is very elegant and offers interesting insights. How-
ever, it has also some serious limitations: Since the reciprocity parame-
ter is universal, there are no sel¬sh players in this model. This is not
only theoretically unsatisfactory, but also empirically wrong. A further
problem stems from the fact that in this model, the equilibria are dif¬-
cult to ¬nd. Moreover, the use of an incomplete information approach
implies the existence of many equilibria. This limits its use as a predic-
tive tool.
The CR model combines a consequentialistic model of positive reci-
procity with a type-based model of negative reciprocity. In this model,
players care, in addition to the own payoff pi , about the social bene¬t
P
(modeled as pj ), and about the payoffs of ˜˜those who need it™™
(modeled as minfpj g). This part captures a new motive”the search
for ef¬ciency”a motive that is neglected in all previous models. In
this part of the model, all payoffs are weighted positively, meaning
this part of the model accounts only for positive reciprocity. How is
Modeling Strong Reciprocity 213



negative reciprocity modeled? In CR, the weight for social welfare in
a player™s utility function is expressed as a number. If this number is
lower than a certain threshold, then the weight of this player™s payoff
in the utility function of the other players is reduced. (It can even be
reduced to a negative number.)
The CR model is very rich and captures much of the experimental
data. It does this at the cost of many parameters (six) and at the
cost of a high complexity. In particular, the reciprocity part of the
model is hard to solve for a particular game. Furthermore, this model
shares with Levine and DK models the problem of multiple equilibria.
However, the CR model is particularly interesting because it models
positive and negative reciprocity in a different way. While positive rec-
iprocity is modeled in a consequentialistic way, negative reciprocity is
modeled in a purely intentional way. If a player does not deserve posi-
tive reciprocity, positive reciprocity is reduced or even negative reci-
procity applies.
Taken together, all models have certain advantages and disadvan-
tages. As it is usually the case, more realistic features imply a higher
degree of complexity. As the predictive power and the psychological
richness of a model increases, the tractability suffers. Therefore, a re-
searcher™s purpose will determine the model he or she uses.

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7 The Evolution of Altruistic
Punishment

Robert Boyd, Herbert Gintis,
Samuel Bowles, and Peter J.
Richerson




7.1 Introduction

Cooperation among nonkin is commonly explained by one of two
mechanisms: repeated interactions (Axelrod and Hamilton 1981;
Trivers 1971; Clutton-Brock and Parker 1995) or group selection (Sober
and Wilson 1998). Neither allows the evolution of altruistic coopera-
tion in large groups of unrelated individuals. While repeated inter-
actions may support cooperation through the use of tit-for-tat and
related strategies in dyadic relations, this mechanism is ineffective un-
less the number of individuals interacting strategically is very small
(Boyd and Richerson 1988). Group selection can lead to the evolution
of altruism only when groups are small and migration infrequent
(Eshel 1972; Aoki 1982; Rogers 1990). A third recently proposed mech-
anism (Hauert et al. 2002) requires that asocial, solitary types outcom-
pete individuals living in uncooperative social groups, an implausible
assumption for humans.
Altruistic punishment provides one solution to this puzzle. As we
have seen in previous chapters of this volume, in laboratory experi-
ments, people punish noncooperators at a cost to themselves even in
one-shot interactions, and ethnographic data suggest that altruistic
punishment helps to sustain cooperation in human societies (Boehm
1993). It might seem that invoking altruistic punishment simply cre-
ates a new evolutionary puzzle: Why do people incur costs to punish
others and provide bene¬ts to nonrelatives? However, in this chapter
we show group selection can lead to the evolution of altruistic punish-
ment in larger groups because the problem of deterring free-riders
in the case of altruistic cooperation is fundamentally different from
the problem deterring free-riders in the case of altruistic punishment.
This asymmetry arises because the payoff disadvantage of altruistic
216 Boyd, Gintis, Bowles, and Richerson



cooperators relative to defectors is independent of the frequency
of defectors in the population, while the cost disadvantage of those
engaged in altruistic punishment declines as defectors become rare,
because acts of punishment become very infrequent (Sethi and Soma-
nathan 1996). Thus, when altruistic punishers are common, selection
operating against them is weak.
To see why this is the case, consider a model in which a large popu-
lation is divided into groups of size n. There are two behavioral types,
contributors and defectors. Contributors incur a cost c to produce a
total bene¬t b that is shared equally among group members. Defectors
incur no costs and produce no bene¬ts. If the fraction of contributors in
the group is x, the expected payoff for contributors is bx À c and for
defectors the expected payoff is bx”so the payoff disadvantage of the
contributors is a constant c independent of the distribution of types
in the population. Now add a third type, altruistic punishers, who co-
operate and then punish each defector in their group, reducing each
defector™s payoff by p at a cost k to the punisher. If the frequency of al-
truistic punishers is y, the expected payoffs become b°x þ yÞ À c to con-
tributors, b°x þ yÞ À py to defectors, and b°x þ yÞ À c À k°1 À x À yÞ to
altruistic punishers. Contributors have higher ¬tness than defectors if
altruistic punishers are suf¬ciently common that the cost of being pun-
ished exceeds the cost of cooperating ° py > cÞ. Altruistic punishers suf-
fer a ¬tness disadvantage of k°1 À x À yÞ compared to nonpunishing
contributors. Thus, punishment is altruistic and mere contributors are
˜˜second-order free-riders.™™ Note, however, that the payoff disadvan-
tage of altruistic punishers relative to contributors approaches zero as
defectors become rare because there is no need for punishment.
In a more realistic model (like the one later in the chapter), the costs
of monitoring or punishing occasional mistaken defections would
mean that altruistic punishers have slightly lower ¬tness levels than
contributors, and that defection is the only one of these three strategies
that is an evolutionarily stable strategy in a single isolated population.
That is, a population of defectors cannot be successfully invaded by a
small number of cooperators or altruistic punishers, whereas a popula-
tion of cooperators can be successfully invaded by a few defectors, and
a population of altruistic punishers can be invaded by a small number
of cooperators, assuming there is a positive level of punishment due,
for instance, to error in perception.
However, the fact that altruistic punishers experience only a small
disadvantage when defectors are rare means that weak within-group
evolutionary forces”such as mutation (Sethi and Somanathan 1996)
The Evolution of Altruistic Punishment 217



or a conformist tendency (Henrich and Boyd 2001; Bowles 2001; Gintis
2003)”can stabilize punishment and allow cooperation to persist. But
neither produces a systematic tendency to evolve toward a cooperative
outcome. Here we explore the possibility that selection among groups
leads to the evolution of altruistic punishment when selection is too
weak to ensure the emergence of altruistic cooperation.
Suppose that more cooperative groups are less prone to extinction.
This may be because more cooperative groups are more effective in
warfare, more successful in co-insuring, more adept at managing com-
mons resources, or other similar reasons. All other things being equal,
group selection will tend to increase the frequency of cooperation in
the population. Because groups with more punishers will tend to
exhibit a greater frequency of cooperative behaviors (by both contribu-
tors and altruistic punishers), the frequency of punishing and coopera-
tive behaviors will be positively correlated across groups. As a result,
punishment will increase as a ˜˜correlated response™™ to group selection
that favors more cooperative groups. Because selection within groups
against altruistic punishers is weak when punishment is common, this
process might support the evolution of substantial levels of punish-
ment and maintain punishment once it is common.
To evaluate the plausibility of this argument, we studied the fol-
lowing more realistic model using simulation methods. There are N
groups. Local density-dependent competition maintains each group at
a constant population size n. Individuals interact in a two-stage game.
During the ¬rst stage, contributors and altruistic punishers cooper-
ate with probability 1 À e and defect with probability e. Cooperation
reduces the payoff of cooperators by an amount c and increases the
ability of the group to compete with other groups. For simplicity, we
begin by assuming that cooperation has no effect on the individual
payoffs of others but does reduce the probability of group extinction.
We also assume that defectors always defect. During the second stage,
altruistic punishers punish each individual who defected during the
¬rst stage. After the second stage, individuals encounter another indi-
vidual from their own group with probability 1 À m and an individual
from another randomly chosen group with probability m. An individ-
ual i who encounters an individual j, imitates j with probability
Wj =°Wj þ Wi Þ where Wx is the payoff of individual x in the game,
including the costs of any punishment received or delivered.
As a result, behaviors (like defection) that lead to higher payoffs will
tend to spread within groups and diffuse from one group to another at
a rate proportional to m. Group selection occurs through intergroup
218 Boyd, Gintis, Bowles, and Richerson



con¬‚ict (Bowles 2001). Because cooperation has no individual level
effects, there is no tendency for group bene¬cial behaviors to spread
by imitation of more successful neighbors. Each time period, groups
are paired at random, and, with probability e, intergroup con¬‚ict
results in one group defeating and replacing the other group. The
probability that group i defeats group j is °1 þ °dj À d i ÞÞ=2 where dq is
the frequency of defectors in group q. This means that the group with
more defectors is more likely to lose a con¬‚ict. Finally, with probability
m individuals of each type spontaneously change into one of the two
other types. The presence of mutation and erroneous defection insure
that punishers will incur some punishment costs, even when they are
common, thus placing them at a disadvantage with respect to the
contributors.

7.2 Methods

Two simulation programs implementing the model just described were
independently developed, one by Boyd in Visual Basic and a second
by Gintis in Pascal. (These programs are available on request.) Results
from the two programs are very similar. In all simulations, there were
128 groups. Initially, one group consisted of all altruistic punishers,
and the other 127 groups were all defectors. Simulations were run for
2,000 consecutive time periods. The ˜˜steady state™™ results plotted in ¬g-
ures 7.1, 7.2, and 7.3 represent the average of frequencies over the last
1,000 time periods of ten simulations.
Base case parameters were chosen to represent cultural evolution in
small-scale societies. The cost of cooperation, c, determines the time
scale of adaptive change. With c ¼ 0:2 and k ¼ p ¼ e ¼ 0, ˜˜defection™™
becomes a simple individually advantageous trait that spreads from
low to high frequency in about ¬fty time periods. Since individually
bene¬cial cultural traits, such as technical innovations, diffuse through
populations in 10 to 100 years (Rogers 1983; Henrich 2001) setting
c ¼ 0:2 means that the simulation time period can be interpreted as ap-
proximately one year. The mutation rate was set to 0.01, so the steady
state value of such a simple individually advantageous trait was about
0.9. This means that considerable variation is maintained, but not so
much as to overwhelm adaptive forces. The probability that contribu-
tors and altruistic punishers mistakenly defect, e, was set to 0.02. In the
base case k ¼ 0:2, so that the cost of altruistic cooperation and altruistic
punishment are equivalent.
The Evolution of Altruistic Punishment 219



We set p ¼ 0:8 to capture the intuition that in human societies
punishment is more costly to the punishee than the punisher. With
e ¼ 0:015, the expected waiting time to a group extinction is twenty
years, which is close to a recent estimate of cultural extinction rates in
small-scale societies (Soltis, Boyd, and Richerson 1995). With m ¼ 0:02,
passive diffusion (i.e., c ¼ p ¼ k ¼ e ¼ 0) will cause initially maximally
different neighboring groups to achieve the same trait frequencies
in approximately ¬fty time periods. Results of simulations using this
model indicate that group selection can maintain altruistic punishment
and altruistic cooperation over a wider range of parameter values than
group selection will sustain altruistic cooperation without altruistic
punishment.

7.3 Results

Our simulations indicate that group selection can maintain altruistic
cooperation over a much wider range of conditions than group selec-
tion will maintain cooperation alone. Figures 7.1, 7.2, and 7.3 compare
the steady state levels of cooperation with and without punishment for
a range of parameter values. If there is no punishment, our simulations
replicate the standard result”group selection can support high fre-
quencies of cooperative behavior only if groups are quite small. How-
ever, adding punishment sustains substantial amounts of cooperation
in much larger groups. As one would expect, increasing the rate of ex-
tinction increases the steady state amount of cooperation (¬gure 7.1).
In the model described in the last few paragraphs, group selection
leads to the evolution of cooperation only if migration is suf¬ciently
limited to sustain substantial between-group differences in the fre-
quency of defectors. Figure 7.2 shows that when the migration rate
increases, levels of cooperation fall precipitously. When altruistic pun-
ishers are common, defectors do badly, but when altruistic punishers
are rare, defectors do well. Thus, the imitation of high-payoff individu-
als creates a selection-like adaptive force that acts to maintain variation
between groups in the frequency of defectors. However, if there is too
much migration, this process cannot maintain enough variation be-
tween groups for group selection to be effective. This means that the
process modeled here is likely to be much less important for genetic
evolution than for cultural evolution”because genetic adaptation by
natural selection is likely to be weaker compared to migration than is
cultural adaptation by biased imitation, and thus less able to maintain
variation.
220 Boyd, Gintis, Bowles, and Richerson


a.




b.




Figure 7.1
The evolution of cooperation is strongly affected by the presence of punishment. Part (a)
plots the long run average frequency of cooperation (i.e., the sum of the frequencies of
contributors and punishers) as a function of group size when there is no punishment
( p ¼ k ¼ 0) for three different con¬‚ict rates. Group selection is ineffective unless groups
are quite small. Part (b) shows that when there is punishment ( p ¼ 0:8, k ¼ 0:2), group
selection can maintain cooperation in substantially larger groups.
The Evolution of Altruistic Punishment 221


a.




Frequency
of

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