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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Cooperation in Two-Person Games
with Repeated Partner Choice
or
Why Be Helpful Even If You Are Exploited
Stephan Schrader
January, 1992
WP#
3377-BPS/92
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Cooperation in Two-Person Games
with Repeated Partner Choice
or
Why Be Helpful Even If You Are Exploited
Stephan Schrader
January, 1992
WP#
3377-BPS/92
Massachusetts Institute of Technology
Alfred P. Sloan School of
Management
50 Memorial Drive, E52-553
Cambridge, Massachusetts 02139
(617 253-5219)
Games with Repeated
Cooperation in Two-Person
Partner Choice
or
Why Be Helpful Even If You Are Exploited
*>
Summary
Frequently, actors face a rep)eated choice betweer\ several dyadic relationships
in
which
to interact.
For example, firms
various partners, and employees
may
enter a series of joint ventures with
may exchange
information with a
number
of
colleagues from other firms. This paper introduces the notion of two-person
games
with repeated partner choice to conceptualize such situations. Players repeatedly face
two
decisions, (1) with
A
whom
to interact
and
computer simulation demonstrates
(2)
how
to interact in a
chosen dyad.
that in such situations a strategy of
unconditional cooperation in a chosen dyad can perform as well or even better than
retaliatory strategies like Tit-for-Tat
— as long as the players tend to at
least slightly
favor those relationships in which they gain higher payoffs.
I
propose that
many
situations resemble
more
the characteristics of two-
person games with repeated partner choice than of a traditional two-person game
viewed isolated from other
relations.
This might explain the puzzle of
cooperation can frequently be observed even
defecting player
if
defection creates an advantage for the
and the other players apparently do not
situations, unconditional cooperation
may
why
to retaliate. In
such
not only be justified as morally
appropriate but also as "good business."
*)
I gratefully acknowledge the research assistance of Ava Kuo, who skilfully develop)ed the
simulation software as part of her Undergraduate Thesis Project at the Massachusetts Institute of
Technology. I thank Lael Brainard, Dietmar Harhoff and William Riggs for their insightful
comments.
Introduction
Game
theoretical
to
sodal interactions with
components. The Prisoner's Dilemma framework, for example,
strategic
employed
to conceptualize
1988; Johnston
(Jarillo,
frameworks are frequently applied
&
is
such diverse issues as buyer-supplier relationships
Lawrence, 1988), informal information trading (von
Hippel, 1987; Schrader, 1990), marketing strategies (Hauser, 1987), inter-group
communication (Bornstein, Rapoport, Kerpel,
1986), trade policy
(Yarbrough
&
&
Katz, 1989), market fraud (Opp,
Yarbrough, 1986), and trench warfare (Axelrod,
1984).
and the threat of
Retaliation for non-cooperative behavior
elements of most game-theoretical analyses
Schelling, 1960;
&
Yarbrough
(e.g.
it
constitute core
(Axelrod, 1984; Axelrod, 1986;
Yarbrough, 1986). Bhide and Stevenson, however,
suggest that, at least in the business world, non-cooperative behavior seldom evokes
retaliation.
"Trust breakers are not only unhindered by
usually spared retaliation by parties they injure" (Bhide
bad reputations, they are
&
Stevenson, 1990,
p. 125).
Despite the absence of retaliation, the authors observe that most businesspeople are
trustworthy,
(Bhide
&
explain
i.e.
behave cooperatively. But "why be honest
Stevenson, 1990,
how
p. 121).
The authors argue
that
theory
fails to
cooperation can persist in situations in which non-cooperative
is
not used. They
dominance of cooperative behavior not with game-theoretic arguments
such as provided for example by
the result of moral choices.
so,
honesty doesn't pay?"
game
behavior creates additional payoffs and in which retaliation
interpret the
if
not because
it is
In this paper,
Stiglitz (1989)
"We keep promises because we
good business" (Bhide
I
and Fudenberg
&
&
Tirole (1989) but as
believe
Stevenson, 1990,
it is
right to
do
p. 128).
propose that the puzzle raised by Bhide and Stevenson can be
explained in game-theoretical terms
if
one conceptualizes the underlying
3-
interactions as two-person
games with repeated partner choice The novel concept
.
of
two-person games with repeated partner choice takes into account that players often
have the choice between several dyadic relationships
each chosen dyad a two-person game
select
played
which
to interact.
— frequently repeatedly
if
Within
players
each other anew.
game with
In a two-person
They have
to decide, first, with
chosen dyad. The
first
based on
repeated partner choice, actors face two decisions:
whom
decision takes
person environment and thus
actors
is
in
ii\to
to interact in a
switch interaction partners at any given point,
as a two-person
games with repeated partner
how
account that actors are part of a multi-
The second decision
their preferences.
and can be viewed
may
to interact and, second,
relates to the interaction of a pair of
game. Thus the designation two-person
choice.
Take
Situations that follow these characteristics can be observed frequently.
informal information trading as example (von Hippel, 1987; Schrader, 1990;
Schrader, 1991a).
Informally information trading refers to the exchange of valuable
information between employees working for different companies.
are
common
in
many
industries.
multi-person environment.
Information trading takes place in the context of a
Usually, several potential information trading partners
are available to a given employee. Consequently
to enter information trading relationships.
with those colleagues with
1990).
to
whom
it is
in a given dyad.
Stevenson's observation
infrequently. In
most
necessary to decide with
Employees
whom
clearly prefer to cooperate
they had beneficial contacts in the past (Schrader,
In addition to choosing trading partners, each
behave
Such exchanges
Interestingly
enough
employee has
to decide
on how
— and supporting Bhide and
— retaliation within a dyad can be observed very
cases,
employees do not
retaliate directly
if
a colleague acts
non-cooperatively in a situation in which cooperation would be expected. However,
4-
they are
now
less inclined to interact
this colleague in the future (Schrader,
with
1990).
I
as,
or
propose that non-retaliatory, cooperative strategies can be as advantageous
may even dominate
retaliatory strategies like Tit-for-Tat in
with repeated partner choice
if
two-person games
players adjust their interaction preferences for other
players based on past payoffs. In this paper,
present results from a computer
I
simulation to support this claim.
The main outcome
of the
computer simulation of two-person Prisoner's
Dilemma games with repeated partner
choice
is
surprisingly different to
from the traditional analysis of two-person games
Rapoport, Guyer,
&
Gordon, 1976; Schelling,
(Axelrod, 1984;
(e.g.
1960).
The
what
Owen,
results
1982;
strategy of unconditional
cooperation fares as well or even better than a retaliatory strategy like Tit-for-Tat
—
as long as player's preferences for specific interaction partners are at least partly
updated according
effective, players
to the payoffs received.^
do not need
For unconditional cooperation to be
to take radical measures.
It is
already sufficient
are to a small degree less inclined to rechoose another player
created a disadvantage, and slightly
more
These adjustments might be so small that
to
inclined
at
if it
if
if
they
interacting has
has proven to be beneficial.
any given point they are not discernable
an outsider.
In the next section,
I
present briefly the arguments that are put forward in
support of retaliatory strategies in two-person Prisoner's Dilemma games.
I
then
introduce the notion of two-person games with repeated partner choice. Next,
present a computer simulation of several two-person
I
games with repeated partner
The term "unconditional cooperation" refers to the interaction behavior luithin a chosen dyadic
relationship. The adjustment of interaction preferences might be interpreted as a form of
retaliation.
will show later, however, that already small adjustments that might not be
1
discernable for the partners suffice to reach the described results. Since a characteristic of effective
retaliation is supposedly its visibility to the other players, I will not use the term retaliation (or
reward) for a slight adjustment of interaction preferences based on past pay-offs.
-5-
choice.
These simulations study the effectiveness of different interaction strategies
depending on
how
sensitive players' interaction preferences are to payoffs
depending on the number of players involved.
simulation are put into context.
Finally, the results of this
suggest that unconditional cooperation might
I
have some additional advantages over more sophisticated
Retaliation
by
and thereby
Tit-for-Tat can
create a negative
and
strategies like Tit-for-Tat.
be misinterpreted as a tendency
image
The Argument for Retaliatory
to
be uncooperative
spillover.
Two-Person Games Without
Strategies in
Partner Choice
In
two person games, players
face frequently the possibility to exploit their
partners, e.g. to gain a benefit at the cost of the other.i
Repeated games provide the
partners with the opportunity to retaliate such exploitation.
punishes exploitation right after
it
exploited player (Schelling, 1960).
measures can serve as means
desirable way.
(A threat
is
— even
if it
the ex-ante
retaliatory strategy
creates additional costs for the
Retaliation or sanctions
to motivate or
in order to deter undesired actions
A strategy
occurs
A
threat of such
and the
induce other players to perform in a
announcement of the willingness
by the other
to retaliate
player.)
that does not retaliate exploitation will be exploited
by non-
cooperative strategies in the long run (Axelrod, 1984). In the context of repeated
Prisoner's
Dilemma games, Axelrod
defects after a defection
Dilemma and
defines a strategy as retaliatory
by the other player (Axelrod,
related situations, retaliation serves
futvire exploitation
by the other
player.
Second,
it
if it
immediately
1984, p. 44). In a Prisoner's
two purposes.
First,
it
prevents
might induce the other player
to
return to a cooperative strategy, especially since the temptation benefit of
^
An extensive discussion and
and Gordon
(1976).
exhaustive
list
of two-by-two
games can l>e found
in Rapoport, Guyer,
exploitation can
& Raiffa,
no longer be reaped (Luce
1957, p. 101;
Rapoport
& Guyer,
1966).
To
how
investigate
Dilemma game
to play a Prisoner's
organized a seminal computer tournament.^
invited to submit a
game
Fourteen
effectively,
Axelrod
theory experts were
computer program that embodied a decision rule on
how
to play
an interactive Prisoner's Dilemma game. Tit-for-Tat, submitted by Anotal Rapoport,
won
the tournament,
i.e.
gained the highest overall payoff. Tit-for-Tat
starts
cooperative choice and thereafter mimics the other players previous action.
with a
In a
second round of the tournament, 63 programs were submitted. Again, Tit-for-Tat
turned out to be the overall most effective strategy.
The successful
strategies identified
three characteristics in
First,
they are "nice",
Second, they are retaliatory,
cooperatively.
And
behavior.
common.
by Axelrod's tournaments have
third, they are forgiving,
i.e.
i.e.
i.e.
at least
they start out
they punish non-cooperative
after
having punished non-cooperative
behavior, these strategies provide the possibility to return to a cooperative mode.
Donninger (1986) tested the generalizability of Axelrod's findings. In a similar
computer tournament, he both varied the payoff matrix and the composition of
strategies included.
Especially
if
His results demonstrate that
the temptation payoff
the counterpart cooperates)
better.
is
(i.e.
not always efficient to be nice.
the payoff a player receives
if
she defects while
raised, non-cooperative strategies tend to
Retaliation (either direct or with
characteristic of all well
it is
some time
performing strategies
lag),
perform
however, proved to be a
.
In sum, retaliation or sanctioning undesirable behavior appears to be an
element of successful strategies in non-zero-sum repeated games (Hardin, 1982).
Retaliation
is
A detailed
employed
to
prevent exploitation and to induce the other player to act
description of the tournament can be found in (Axelrod, 1984).
-7
A
cooperatively.
strategy of unconditional cooperation,
on the other hand,
is
unstable (Taylor, 1976)1 and likely to be exploited by other strategies (Rapoport et
Although nnutual cooperation can
1976).
strategies,
"it is
result
al.,
from conditional, retaliatory
never rational for a player to use the unconditionally Cooperative
strategy" (Taylor, 1976, p. 89).
Two-Person Games With Repeated Partner Choice
The
traditional analysis of two-person analysis a specific relationship
independent of any other relationships the players might or could be engaged
However, many
which a
situations exist in
viewed isolated from other
relationships.
two-person relationships
which
in
all
For example, academicians frequently
time
is
needs
Unfortunately, the
limited.
to
It is
cross licensing, joint
Individual frequently
know many
number
these situations, an actor can or
numerous
Time and other
individuals with
constraints
whom
to write
of papers an individual can write at a given
whom
to collaborate;
A multitude of isomorphic
development
face
potential relationships are finally actuated.
necessary to choose with
be done repeatedly.
dyadic relationship cannot be
to potentially engage.
often necessitate that only a few of
joint papers.
specific
in.
projects, or the
and
this choice
situations exists, such as
exchange of birthday cards. In
must choose between
all
several potential relationships.
Within each of the selected relationships, however, an interaction occurs that can be
interpreted as a two-person game.
Thus, these situations contain clearly a multi-
person component. This component, however,
research
1
is
not addressed by the existing
on multi-person games.
Taylor defines a stable equilibrium as a situation in which "no player can obtain a larger payoff by
using a different strategy while the other players continue to use the same strategies" (Taylor, 1976:
p. 85).
-8
The
games concentrates on
analysis of multi-person
different issues.
Most
multi-person games conceptualize situations in which several actors interact
simultaneously and the action of one actor has direct consequences for the payoffs of
all
the other actors.
central topic
and
The provision
of public or collective goods, for example,
issues such as the free rider
behavioral norms are discussed
(e.g.
is
a
problem and the emergence of
(Axelrod, 1986; Hardin, 1982; Olson, 1965;
Taylor, 1976; Ullmann-Margalit, 1977).
The problems investigated
analysis of two-person
interest
is
in this article are
games nor
vdth only one other player
many
relationship occurs in the context of
different
The phenomenon of
of multi-person games.
located conceptually between two-person
A player interacts
is
covered neither by the traditional
from a two-person game
in
games and multi-person games.
at a time.
However,
potential relationships.
which a
this specific
Such a situation
specific relationship
conceptualized independent of any other relationship.
It is
dyadic
is
also different
from n-
person games that are characterized by each player simultaneously interacting with
all
other players.
At
least
two types of decisions have
played with repeated partner choice. The
to
be made
first
when two-person games
are
decision concerns the formation of
dyadic relationships. These decisions determine the pairs of interacting players. The
second type of decision relates to the behavior in each dyadic relationship. The
players have to decide
how
to act in a given dyadic.
similar to a simple two-person
game
The
structure of this decision
is
— although the appropriate strategy might be
quite different.
Axelrod (1984, pp. 158-168) discusses a situation that resembles two-person
games with repeated partner choice
dyadic relationship
Dilemma games
at the
same
insofar as each player
time.
He
is
involved in a number of
investigates strategies for Prisoner's
in a simplistic multi-person environment.
The environment
is
-9
structured so that each player has four neighbors, one to the north, one to the east,
one
to the south,
and one
to the west.
The game extends over several generations.
In each generation, a player interacts with
person games. After
measured by
more
all
direct neighbors,
plays four two-
i.e.
four interactions, they players attain a success score
all
average performance with their four neighbors.
their
If
a player faces a
successful neighbor, the player converts to this neighbor's strategy.
rules tend to perform well in such
finds that nice, retaliatory,
and forgiving
environment. However,
future payoffs are discounted strongly, a
if
Axelrod
an
community
of
ruce strategies can be invaded by non-cooperative strategies.
In Axelrod's
game, the formation of dyadic relationships
the location of the players.
players.
The formation
A
relationships
predetermined by
given player always interacts with the same four
of dyadic relationships
outcomes of the interactions and unchanged
Many
is
situations exist, however, in
unaffected by the
is static, i.e.
in the course of the
game.
which the formation of dyadic
depends on past payoffs. Consider informal information trading.
Persons tend to prefer those relationships that have proven to be beneficial
They
(Schrader, 1991b).
are less inclined to continue relationships that have been
disadvantages. Frequently,
it
contact with those
who have
not return a phone
call.
can be observed that individuals try to avoid to get in
taken advantage of them. They might, for example,
In this case, the
two players do not
interact at
contact cannot be avoided, the individual might be as helpful as ever.
Stevenson (1990) have observed
strategy
(i.e.
this
if
players interact at
the partner's
is
Yet,
if
a
Bhide and
tendency of not adjusting one's interaction
the strategy used for interaction in a chosen dyad)
cooperative even
all.
known
to take
advantage of
and
this
to
remain
— as long as the
all.
In the following, results of a computer simulation that takes into account that
players might update their preferences for specific interaction partners based on past
-10
payoffs are presented.
It
will
be seen that a strategy of unconditional cooperation
might actually be more advantageous than
Simulation of Two-Person
Computer simulation
interaction strategies for
simulation
is
it
first sight.
Games With Repeated
used here
to
Partner Choice
study the effectiveness of different
two person games with repeated partner
frequently used to investigate
is
appears on
games with complex
choice.
Computer
strategic
interactions (e.g. (Axelrod, 1984; Axelrod, 1986; Behr, 1981; Donninger, 1986; Fader
Hauser, 1987;
SchiilBler, 1986).
Simulation allows to probe into situations that do not
lend themselves to formal analysis, including
many games with
using different, sometimes varying strategies with
The Structure of the Simulation
The computer simulation
have
a recurrent choice of
relationship, the
they choose
random
elements.
investigates a situation in
which dyadic relationship
interaction partner
is
played.
from the
which several players
to enter.
Once
player, he
player
is
1).
more
In other words,
if
set of possible partners for the next
Instead he will lean
is
influenced by past
a relationship has proven to be beneficial for a
inclined to choose this relationship again.
was exploited by another
In each dyadic
the players have interacted,
round. Their preferences for specific relationships, however,
payoffs (Figure
multiple players
Game
same two-person game
anew an
&
player,
he
is
less inclined to
more towards choosing another
On
the other hand,
if
a
choose the same player.
interaction partner.
-11-
/
-12
Player B
13-
likelihood that one player chooses another player as the desired interaction partner.
These probabilities can be interpreted as a function of the expectations regarding the
usefulness of future interactions with a given partner.
In the beginning of the
simulation, no prior biases towards specific interaction partners exist,
preference matrix
uniform. ^
is
In other words, each player has the
i.e.
the
same chance
to
be
picked by another player as the preferred interaction partner for that round.
Players update their preferences based
on past
experienced that interacting with a specific partner
interact
is
more with
this
payoffs.
is
Once
a player has
beneficial he will prefer to
partner and less with the other players. Similarly,
if
a player
exploited by another player he will be less inclined to choose that player again as
an interaction partner and will be more inclined
to
choose another player.
This
assumption coincides with empirical research on the formation of exchange
networks that stipulates that those
ties
get reinforced that create benefit for the
assumed
an expectation-
network members (Schrader, 1991b). Thereby
it is
supporting behavior changes preferences
than behavior that contradicts existing
less
that
expectations.
A
sensitivity
parameter
s is
used
to characterize
how
strongly a player's
preference for a specific interaction partner (not the player's interaction strategy)
influenced by past payoffs.
A
is
large sensitivity parameter indicates that a player's
interaction preferences are strongly influenced
by past payoffs, a small
sensitivity
parameter indicates that a player considers past payoffs only slightly when choosing
interaction partners.
The following procedure was used
matrix. In the case that
A
and B
interact
and
A
to
update the probability
receives a negative payoff,
A
will
reduce the probability to choose B as an interaction partner by the negative payoff
multiplied v^dth the sensitivity parameter and increase the other probabilities
^
Such biases could be caused
for
example by physical or any other type of proximity.
14-
(The sensitivity parameter
accordingly.
probabilities result.)
If
A
restricted so that
is
receives a positive payoff,
A will
no meaningless
decrease the probability to
choose somebody other than B by the positive payoff multiplied with the sensitivity
parameter and increase the probability
to
choose B accordingly. Through
this
procedure, events that contradict existing expectations receive a stronger weight than
events that are expectation supporting.^
The following two
tables exemplify the
updating procedure. Table la depicts a
probability matrix at the beginning of a unspecified
updated probability matrix
example
it
after
A
and B have interacted
A
has been assumed that
interact
to
be
0.05,
i.e.
Table lb contains the
t.
in
round
t.
cooperates and B defects. Thus
negative payoff of -2 and B a positive payoff of
assumed
round
2.
The
sensitivity
For this
A
has received a
parameter
is
a negative payoff of -2 leads to a reduction of the probability to
by 10 percent. B reduces by the same percentage the probability of
with anybody other than
A
and increases accordingly the probability
to
interacting
choose
A
again in the future.
^
The formulas to update the interaction probabilities are as follows
players A and B if A receives a negative pay-off in round t-1.
If
A
(1)
Pt(A,B) =
(2)
Pt(A,K) = Pt.i(A-K)
Pt.i(A-B)Ml +
-
receives a piositive pay-off in
an interaction between
s*7i)
(s»n)
,
Pt-i(A,B)
p ,. „, Pt.i(A,K)
round
t-1,
the probabilities are updated as follows.
(3)
Pt(A,B) = Pt-i (A,B) + (s»Jt)»(l-Pt-i ( A,B))
(4)
Pt(A,K) = Pt-i(A,K)
* (1
for
-
s*7t),
with
,
A
The
Pt(A,K)
=
=
=
=
S
=
Sensitivity parameter, indicating
A, B
K
7C
interacting players
Another player but
A
and
and
B.
A
will
B.
pay-off to player A.
Probability that player
choose player B in round
t
as desired interaction
partner (Interaction probability).
when
how
strongly players consider past pay-offs
adjusting the interaction probabilities.
-15-
TABLE
Assumed
la
Distribution of Interaction Probabilities Before Interaction Between
B
in
Round
t
A
and
16
based on past payoffs? Here the extent of the preference adjustment will be
center of our interest.
environment
whether the
And
second,
how
does the number of players in the
effects the usefulness of different strategies?
common
at the
It
will be investigated
notion can be confirmed that cooperation
is
less likely to
evolve the greater the number of players that interact.
To generate
a
common
point of reference for the subsequent discussion, a
used as base
game
in
case.
This base case assumes that every player picks randomly one of the remaining
which players do not update
their interaction preferences is
players as desired interaction partner without preferring one player over another.
Twelve players participate
Defect, Tit-for-Tat,
The four generic
strategies (Cooperate,
and Random) are represented with equal number of
The game runs
number
in this simulation.
for 1,000 rounds.
However, since an
of rounds.
On
first sight, this
players.
appears to be a large
interaction occurs only
if
two player choose
each other simultaneously, only approximately 1.09 dyadic interactions occur per
round. Thus after 1,000 rounds, circa 1090 interactions have occured. Consider that
each of the 12 players can interact with
dyads can be formed. Thus,
11 other players.
in the base case, the
Consequently 66 unique
members
of a specific
dyad have
interacted approximately 16.5 times with each other after 1,000 rounds.
The expected payoffs and the expected number
determined
easily.
of interactions can be
Table 2 reports the expected payoffs for the
subsequent rounds.
first
and
all
17-
Strategy
-18-
not necessarily assumed that players outright refuse to interact with another player
They
that has exploited them.
rather
may
adjust their interaction preferences only
Such a small adjustment would coincide with the observation by Bhide and
slightly.
Stevenson (1990) that most people do not retaliate in any obvious
way once
they
have been exploited.
The
first
simulation reflects the case that players sparingly adjust their
interaction preferences at
any given time. Previous payoffs change
to only a
relatively small extent the players' inclination for choosing other interaction
partners.
The
sensitivity
and the partner
parameter
s
equals 0.05. That means,
a player cooperates
defects, the player reduces the probability to again
partner by 10 percent of the original probability. E.g.,
other partner
if
was 30 percent before
the interaction
it
if
choose the same
the probability to choose the
will
be 27 percent afterwards.
Such a small adjustment of interaction preferences already changes the
outcome dramatically
in
performed best
base game, turns out to be the least efficient strategy
in the
comparison
to the base case.
interaction preferences are adjusted slightly
(Figure
3).
The two winning
Defect, the strategy that
— except during the very
strategies are Tit-for-Tat
first
if
the
rounds
and Cooperate. Their average
payoffs per player per round are nearly the same. Tit-for-Tat outperforms Cooperate
in the first
one hundred rounds
better thereafter.
to a small extent
and Cooperate performs somewhat
-19
20-
interaction
way during
is faster.
Second, Cooperate outperforms Tit-for-Tat in a more significant
the later stages of the game.
--
Average
pay-off
-21
motivation results from the losses Cooperate experiences
Random. The
positive motivation
with other cooperative
losses
when
strategies.
is
due
it
for-Tat "hopes" to turn the relationship with
it is
it
limits potential
does not experience a negative
motivation for not interacting with Random. Or to say
that
from interacting
on the other hand,
Tit-for-Tat,
which naturally cannot be achieved. Thus
interacting with
to the gains that result
Random. Thus,
interacting with
when
Random
it
in simplifying
words,
into a cooperative
one
Tit-
—
does not learn as quickly as Cooperate
not beneficial to interact with Random.
Small Versus Large Number of Actors
Is
the finding that unconditional cooperation performs well in
games with repeated partner choice
The
sensitive to the
number
of participating players?
traditional theory of collective action argues that cooperation
emerge
in large
To
12 to 20.
s=0.05.
groups than
in small
investigate the effect of
It is
The
assumed
first
less likely to
group
size, the
number
of players
is
increased from
that players adjust their interaction preferences sparingly,
more time
needed
is
to the 12-player case, Tit-for-Tat
payoff for the
is
ones (Hardin, 1982; Olson, 1965).
resulting distribution of payoffs
the difference that
two person-
1000 rounds
is
similar to the
for 12 players,
to reach a stable pattern (Figure 5).
outperforms Cooperate
is 5.6
one
with
Similarly
slightly; Tit-for-Tat's
average
percent higher than Cooperate's. After 3000
rounds, however, both strategies gain the same payoffs.
i.e.
-22-
-•
Average
pay-off
-23
The surprising
result of the simulations
well as Tit-for-Tat even
if
is
that
Cooperate perfomns nearly as
the preferences are adjusted only to such a small degree
Under such
that outsiders are likely to be unable to detect this adjustment.
Retaliation appears to
circumstances, the adjustment does not constitute retaliation.
not always be necessary for cooperation to emerge as a stable behavioral pattern.
Conclusion: Unconditional Cooperation Because
This paper has introduced a
new
repeated partner choice. These games
and multi-person games. Players
class of
show
games:
Two
interact in dyadic relationships.
Although one nnight have expected
repeated partner choice
is
Pays
person games with
characteristics of both
however, they can choose between several dyads in which
it
two-person games
At any point,
to interact.
that a series of two-person
similar to a collection of two-person
games with
games played nearly
simultaneously, the possibility to choose between different interaction partners has a
profound impact on the efficiency of interaction
strategies.
Unconditional cooperation proves to be a surprisingly effective strategy
— as
long as the players adjust their preferences for interacting with other players at least
to a small extent
based on past payoffs. This adjustment of interaction preferences
could be interpreted as indirect retaliation. Effective retaliation, however, requires
that the
punished player recognizes the punishment (Bayard
«Sc
Elliott, 1991).
Yet, the
computer simulations have demonstrated that even very small adjustments are
sufficient to protect
drastic
Cooperate from being overly exploited by other
measures are necessary. Exploitation does not need
cancellation of the relationship.
Rather,
it is
strategies.
to result in
sufficient to only slightly
No
an outright
reduce the
chance of again choosing the exploiting player as interaction partner. These
adjustments can be so small that other players might be unable to perceive them as
-24
This
retaliation.
would help explain why Bhide and Stevenson's
(1990) believe that
frequently untrustworthy behavior remains unpunished.
If
and Cooperate perform
interaction preferences are updated, both Tit-for-Tat
approximately equally well Under some conditions Tit-for-Tat slightly outperforms
Cooperate, under other conditions Cooperate outperforms Tit-for-Tat.
strategies isolate non-cooperative players.
cooperation.
Both
punishes non-
Tit-for-Tat, in addition,
This punishment generates short term advantages over the strategy of
unconditional cooperation.
It
does not, however, create a significant benefit in the
long run.
One major
cooperation
is
not captured in the presented simulation model.
environment, unconditional cooperation
long as reputation spillover can occur.
is
In a multi-person
a less risky strategy than Tit-for-Tat, as
that
A
that
two
players,
A
and
B, interact
plays Tit-for-Tat and punishes B's
Outsiders might only observe this punishment and
defection once they meet again.
know
is
Assume
and B defects on A. In addition, assume
might not
and the strategy of unconditional
difference between Tit-for-Tat
the history of the relationship.
a non-cooperative player with
whom
They might wrongly deduce
better not to interact.
punishment bears the danger of creating
a negative external
that
A
Consequently,
image spillover
for the
punishing party. This danger does not exist for the strategy of unconditional
cooperation.
The strength of unconditional cooperation
with repeated partner choice
Cooperate
is
is
in the context of
even more astorushing
a very unsophisticated strategy.
No
if
two-person games
one considers
strategic
game plan
is
that
used.
Only
the preferences of Cooperate for interacting with other players change slightly based
on the past payoffs received.
The
and the
distinction
strategies
between choosing interaction partners based on past payoffs
used within a given dyadic relationship
is
of great empirical
25
Take informal trading of technical information
importance.
Hippel, 1987; Schrader, 1991a; Schrader, 1991b).
If
for
an example (von
some employee demonstrates
uncooperativeness by providing unreliable information, other employees do not
tend to counter by providing false information themselves.
Instead, they are
inclined to decrease the likelihood of interacting v^ith this uncooperative
again.
Sometimes, such behavior
is
even codified, as
employee
the case in the oil-
is
exploration industry, where exploration firms delegate to
some employees,
the so
called oil-scouts, the task of exchanging valuable proprietary information informally
(Schrader
&
von Hippel, forthcoming). These
and meet
professional societies
societies enforce
an elaborate
oil-scouts are organized in
at regular intervals,
set of rules of
usually once a week.
conduct such as "no member
intentionally report false information" (By-Laws of the
Association).
If
members
violate these rules, the only
The
(...)
shall
Houston Oil Scouts
punishment
is
may
that they
be expelled from one or several future association meetings. By expelling nonthe meetings, their chance to interact with other
cooperative
members from
members
greatly reduced.
is
outside the meeting, those
If,
however, they do interact with other members
members have
to follow the rules of the association.
sum, defection does not induce counter-defection, but does lead
In
to a reduction of the
interaction probability.
It is
frequently proposed that general behavioral norms like telling the truth
emerge out of dyadic
cooperation
is
relationships
(i.e.
two-person games) in which conditional
a stable strategy (Hardin, 1982; Ullmann-Margalit, 1977).
Unconditional cooperation, however,
is
argued not
person games. To be unconditionally cooperative
nearly
all
specifications of traditional two-person
paper, however, has
effective in
some
shown
to
is
be a stable strategy in twonon-rational in the context of
games
(Taylor, 1976, p. 89). This
that the strategy of unconditional cooperation can
be
situations that are easily misinterpreted as two-person games:
26
situations in
which a player has a repeated choice between
to interact. In
such situations, which
I
different
dyads
in
which
have labeled as two person games with
repeated partner choice, unconditional cooperation can perform as well as other
retaliatory strategies, as long as the players tend to at least shghtly favor those
which they gain higher payoffs.
relationships in
Several authors view retaliation of non-cooperative behavior as a prerequisite
for cooperation to evolve
the
one
who
retaliates.
and
stabilize.
Frequently, however, retaliation
This led Axelrod (1986) to introduces a meta
punishes non-punishment.
The
results presented in this paper,
that at least for
some two-person games with repeated partner
not necessary.
Even small behavioral adjustment
interest
and
that
may
is
norm
costly to
that
however, suggest
choice retaliation
is
that are in the individual's self-
not be detectable by an outsider
may be
sufficient to stabilize
cooperation.
I
person
would argue
game with
that
many
situations resemble
more
the characteristics of a two-
repeated partner choice than of a traditional two-person game.
This might explain the puzzle of
why
cooperation can frequently be observed even
though defection creates an advantage and players apparently do not
such situations, unconditional cooperation
appropriate but also as "good business."
may
to retaliate.
not only be justified as morally
In
27-
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55t+3
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