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;
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.
R
••
qq,
TESTING GAME-THEORETIC MODELS OF FREE RIDING:
NEW EVIDENCE ON PROBABILITY BIAS AND LEARNING
Thomas R. Palfrey
Howard Rosenthal
No.
549
April 1990
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
TESTING GAME-THEORETIC MODELS OF FREE RIDING:
NEW EVIDENCE ON PROBABILITY BIAS AND LEARNING
Thomas R. Palfrey
Howard Rosenthal
No.
549
April 1990
/
/
Introduction
I
The
rider problem
free
organizational
bear
activists
transactions
costs
for
benefit
the
with
little
risk
affecting
of
Congressmen can avoid taking an unpopular
election.
(say,
costs
political processes.
to
congressional
raising
policing
activities
professors
salaries)
who
of
avoid
of
a
large
At
committee
onerous
outcome
the
stand on
other
an
of
an
issue
congressmen
are
lesser powers can benefit from the
mighty.
the
enough
if
,
In international politics,
willing to.
Political
Nonvoters can avoid bearing informational costs
(free-riding) membership.
and
endemic
is
quite
not
teaching
and
grand
so
a
level,
assignments benefit
from these activities because some colleagues agree to serve.
All of these problems have in common the following features.
group,
all
whose
of
members
stand
.
both theory and experiments
al,1983,
1984,
about
In this paper, be present
particularly simple class
a
voluntary contribution threshold games
Rapoport,1985, Lipnowski and Maital,
1988,
generous
the
They have been studied extensively both theoretically and
experimentally in most fields of social science.
good problems:
from
They are all examples of public
contributions of some subset of the group.
good problems
benefit
to
There is a
1989).
a group
In these games,
w of N players "contribute"
1983,
t^
as
N)
a
(Van de
Kragt et
Palfrey and Rosenthal,
benefit is produced if at least
An individual obtains the
their endowments.
highest possible payoff (when w
of public
successful free rider- -the good
is produced without the individual having contributed.
Models of this kind of public goods problem that could eventually apply
to natural
settings should,
information.
certain
we argue,
incorporate some element of private
Each individual will generally be incompletely informed about
characteristics
of
other
individuals.
uncertainty as pertaining to endowments.
own endowment,
individual.
values
are
we
model
is
randomly
private information to the
and
independently
assigned
according to a probability distribution that is common knowledge.
formal
model
of
decision-making,
this
How much an individual values his
relative to the public good,
Endowments
Here
subjects
are
assumed
to
In the
be
risk
.
neutral. The predicted outcomes under the assumption that players maximize
expected earnings are provided by game theory.
Game theory makes "rational expectations" predictions about behavior. In
behavior reduces to a single binary decision
our case,
endowment or keep it. In order to make this decision,
likely behavior of the other players.
the
assessment
of
is
Players
contribute.
likelihood
the
given
rational
expectations
setting,
relevant
the
players
other
the
these
hypothesis:
expectations
will
about
the
optimization
player's
each
expectations
probabilistic
their
given
of
each player assesses
An equilibrium of the game imposes the
behavior of the other players.
following
each
that
optimize
then
In our
contribute the
--
generates
probability
the
distribution over behavior that everyone had been anticipating.
This
is
called a Bavesian Nash Equilibrium of the game of incomplete information.
Corresponding to any equilibrium, there will be a dual prediction about (1)
probability
the
distribution
over
and
outcomes,
the
(2)
decision
rules
adopted by the players
This
presents
paper
experiments
and
compares
number
these
in
laboratory
of
experiments
with
the
We report three major findings.
the equilibrium predictions from game theory are very accurate,
First,
at least at an aggregate
On a qualitative level,
level.
as we vary
the
the observed outcomes always move in the direction the theory
treatments,
would predict.
provision
theory.
large
a
behavior
the
predictions of the theory.
good
from
findings
On a quantitative
almost
Furthermore,
exactly
level,
mirrors
the
the
efficiency of actual public
numerical
predictions
of
the
the level of voluntary contribution is usually very
close to the predicted level of contribution.
Moreover, once subjects gain
familiarity with
patently
the
irrational
rules
of
behavior
observe
game,
we
do
(violation
of
dominated
the
not
the
kind of
strategies)
often
observed in other public goods experiments.
Second,
to
predictions,
the extent that we
observe variations
from the
theoretical
there is a relatively simple model with is very successful in
accounting for the error.
Specifically,
for
some
treatments we
overcontribution, while for others we observe undercontribution.
most
part,
these
deviations
are
consistent
with
subjects
observe
For the
behaving
but
optimally,
approximately
others
that
other
that
players
are
more
In other words individuals underestimate the
"civic-minded" than they are.
probability
assuming
free
We
ride.
compare
explanation of the
this
variations from the theory to a variety of other possibilities, based on
incomplete experimental control over utility functions.
These alternative
explanations fare poorly in comparison to the probability bias model.
Third, we find little evidence that individuals update their inaccurate
inclinations to free ride.
beliefs about the other subjects'
While we are
unable to cleanly reject the hypothesis that individuals are learning about
other
individuals'
behavior,
learning hypothesis
rigidity
find
the
paucity of
the
Nevertheless,
surprising.
beliefs,
of
we
data
still
are
in
support
spite
remarkably
of
such
for
a
apparent
the
supportive
of
the
equilibrium predictions from noncooperative game theory.
While we thus want to emphasize that we found mostly small deviations
the deviations do require further explanations.
from the theory,
to
pursue
two
alternatives
within
the
basic
framework
We chose
of
our
decision-making model.
One possibility is that the observations may have been measured badly,
which in our case means that some variables we thought we were controlling
perfectly
in
the
laboratory
setting
were
in
not
fact
perfectly
controlled. The most obvious possibility would be that the players were not
risk neutral
with
respect
to
the
dollar
payoffs,
but
had a non
linear
utility function over the outcomes, that might even depend upon the payoffs
to the other subjects.
2
A second possibility
be wrong.
expected
is
that the rational expectations hypothesis might
This would not necessarily mean that players
utility,
but
that
their
subjective
did not maximize
assessments
of
players' behavior was inaccurate or even systematically biased.
one particular hypothesis about a systematic bias
in players'
the
We propose
expectations
about the behavior of the other players:
H:
other
Individuals overestimate the probability that others contribute.
Equivalently
individuals underestimate the probability that others do
,
not contribute.
individuals underestimate the probability
In looser terms,
that others will be tempted to either free ride or to avoid the potential
Here the hypothesis has been stated verbally.
loss of their endowment.
A
precise specification of the hypothesis that will allow us to investigate
its validity with our experimental data is provided later.
H leads directly to predictions about how behavior will systematically
deviate from the predictions of (risk-neutral)
if only
game theory.
For example,
of N individuals need contribute to produce the public benefit,
1
overestimation will lead to a reduced level of contribution.
that, in any
1
of N game,
The reason is
raising the probability others contribute raises
the expected utility of not contributing but leaves the expected utility of
contributing unchanged.
There
The
reversed in the N of N game.
reasoning is
overestimation increases
expected utility
the
contributing but
of
leaves the expected utility of not-contributing unchanged.
in
some
overestimation
games,
predicted
is
More generally,
individual
increase
to
contribution; inother games, the prediction is reversed.
We specifically chose experimental parameter values- -in terms of w,
and the distribution of endowments
relative
value
the
to
of
the
N,
public
benefit- -that lead to a prediction of increased contribution and those that
In section 3, we show that
lead to a prediction of decreased contribution.
the
predictions developed from overestimation of probabilities
replicated by non-linear utility models
by models
or
of
cannot be
cooperative
or
altruistic behavior.
We
were
in
fact
overestimation rather
led
to
develop
than non- linear
hypothesis
the
utility on
the
probability
of
basis
of
a
set
of
experiments conducted at the California Institute of Technology in 1987-88
and
Carnegie-Mellon University
reasonably
contribution
successful
games
in
(without
in
early
accounting
While
1989.
for
communication)
the
,
outcomes
there
were
theory
game
voluntary
of
systematic
some
departures from the predicted equilibrium rates of contribution.
on
the
experimental
parameters,
we
found
that
sometimes
was
Depending
there
was
significantly higher contribution rates than predicted and sometimes there
was
significantly
less
than
predicted.
After
developing H,
we
designed
several
experiments
"critical"
We
original experiments.
In
conducted
too
in others,
predicted;
at
the
relative
game
to
theory,
was
The results of these new experiments,
little.
California
the
than
replicated the original parameter values.
also
overcontribution,
new experiments,
some
with different parameter values
Institute
Technology
of
in
the
summer
of
1989, also strongly support H.
most adaptive expectations models or Bayesian learning models
Finally,
would predict that if an experiment
environment
with
expectations
about
in
other
light
of
their
information from
in
Forsythe,
This has
the
experimental literature
and Plott
Palfrey,
implications
the
for
data,
[1982],
(e.g.
Camerer
as well.
For
in contrast to the non- linear utility explanations for deviations
example,
from
in
update
will
In fact there is a great deal of substantial
favor of this elsewhere
[1988]).
subjects
behavior
players'
McKelvey and Palfrey [1989],
and Weigelt
then
groupings,
random
early play in the experiment.
evidence
conducted repeatedly in a stationary
is
our
predictions,
adaptive behavior,
probability
the
bias
together
hypothesis,
predicts that behavior will be closer to
the
equilibrium predictions as the number of replications increases.
some evidence supporting this, but it is very weak.
with
Bayesian
We find
Overall, there appears
to be a great deal of persistence to the biased beliefs.
In Section II
Section
some
III
we
of
outline
alternatives
Section
IV,
experiments.
we
the
to
paper,
the
present
model
the
model
the
develop
we
the
game -theoretic model.
of overestimation
of
risk
experimental
neutral
design
of probabilities
selfish
and
Concluding remarks appear in Section V.
the
behavior.
results
of
In
and
In
the
.
II
.
The Equilibrium Model of Voluntary Contribution
A group project requires
A group consists of N persons.
units of input.
least w
at
Each group member is endowed with one indivisible unit of
input, which may be either "consumed" by the individual or "contributed" to
The voluntary contributions game consists of a single
the group project.
simultaneous
move
which
in
individual's
each
choice
set
pair
the
is
{contribute, not contribute).
The project succeeds
if and only
if at
least w units are contributed.
The value of the project to any individual is normalized to equal
The
1.
private value of the endowed unit of input to an individual is denoted
c
.
i
Each person knows his or her own c
but only knows that the other players'
c's are independent random draws from some common probability distribution
with CDF F(').
Assume F is continuous and strictly increasing on
c>0, with F(0) =
and F(c) =
The payoff for player
1.
with endowment (cost)
i
[0,c],
is given by:
c
1+c
if
i
does not contribute and at least w others contribute
c
if
i
does not contribute and fewer than w others contribute
1
if
i
contributes and at least w-1 others contribute
if
i
contributes and fewer than w-1 others contribute
3
No side payments are permitted.
Individual
for
rationality
4
contribution.
(complete
Thus,
information)
equilibrium
is
and
makes
if
than
to
1
costs
all
fewer
everyone
for
<
c
w
not
a
necessary
condition
public
information
were
players
had
c
contribute.
<
1,
the
Indeed,
only
total
non-contribution is always an equilibrium unless w=l
If l^w players had c<
be
of
1
in a game of complete information,
there would
pure strategy equilibria where exactly individuals contribute.
these,
where
the
w lowest cost
"efficient" equilibrium.
individuals
contribute,
(If side payments were permitted,
would be
One
the
it would always
efficient
be
There
be
also
are
w
the
for
lowest
multitude
a
individuals
cost
of
contribute
to
equilibria where
some
or
if
all
wc<N.)
of
the
players used mixed strategies (Palfrey and Rosenthal [1984]).
Because communication among the players is ruled out in our environment,
there is no direct way in which players can coordinate or correlate their
arrive
strategies
or
efficient
outcome.
players- -that
outcome
an
at
Moreover,
similar
asymmetries
the
since
complete
the
to
between
their different endowments- -remain private
is,
information
the
information,
asymmetries cannot be used, even tacitly, as a coordinating mechanism.
It is natural then to assume that all individuals will use an identical
when
rule
make
they
simultaneously.
shown
As
contribution
their
Palfrey
in
decisions
and Rosenthal
independently
[1988],
a
and
symmetric
Bayesian equilibrium to this game always has a particularly simple form.
For any beliefs
contribute,
rule
.
that player
there
Therefore,
symmetric
a
has
unique best
a
is
i
threshold cost level, call it
c
,
one equilibrium value of c
guarantee
existence
of
,
decisions
strategy which is a cutpoint
response
simply
is
characterized by
such that contribution is optimal if
c
>
c
to
c
a
<
Vhile there may be more than
.
the mild regularity conditions imposed on F(')
least
at
other players'
the
equilibrium
and noncontribution is optimal if
c
about
one
such
value.
The
equilibrium cutpoints is the set of all solutions (in c
set
)
of
all
such
to the following
equation:
*
c
-
'V^h*r^
1-F(c*)
N-w
Prob(k-=w-l)
(1)
where k is the number of contributors other than
i.
The interpretation of equation (1) is that a person with a private value
of
c
to
be
faces an opportunity cost of contributing equal to c
an equilibrium,
opportunity costs
it must
sufficiently
given others are using the
greater than
c
c
In order for c
be that everyone with a cost below c
low
so
as
to
be
has
better off contributing,
decision rule and everyone with private costs
have too high an opportunity cost.
with a cost of exactly
contributing.
c
.
Therefore,
individuals
must be indifferent between contributing and not
Since the value of the group benefit was normalized to one.
that
implies
indifference
contribution
of
cost
the
must
equal
the
probability of being pivotal to the success of the group project.
probability is Prob(k-=w-l)
poses
potential problem
a
possibility of multiple
The
.
in
evaluating
data
the
However, for the experiments that we conduct,
This
solutions
from
to
(1)
experiments.
the
it turns out that there is a
natural concept of stability of equilibrium that nearly always generates a
unique equilibrium prediction with our range of parameters. We say that a
Bayesian equilibrium is Expectationally Stable if the following tatonnement
process converges to the equilibrium c*.
Let c_ be
cutpoint in the open interval
initial
some
everyone started out using c^ as their cutpoint rule.
players will observe a frequency of contribution
on average,
Then,
F(Cp,)
q^"=
Suppose
(0,c).
Next suppose
.
that this results in each of the players having expectations q
about the
likelihood that a randomly selected opponent will contribute.
One
imagine this,
for example as the outcome of a learning process after many
repetitions with opponents using c_.
Then the best response under these
expectations (q^) is to use a cutpoint
c^= G(Cq)=
We
exists
an
(c„-G(c
)
.w-lT „,
Jn-w'
F(Cq)
I-F(Cq)
w-1
an Expectationally Stable Equilibrium
is
open
set
>
(c^-c*)
C
0.
E
a Bayesian Nash
C
(c*)
[0,c
such
]
for
that,
all
c^
there
if
(ESE)
-
G
C
E
(c*)
,
define c* as being a Globally Expectationally
We
Stable Equilibrium (GESE)
Thus,
satisfying:
c.
fN-ll („,
that c*
say
could
if it
is
an ESE relative to the open set (0,c).
equilibrium to our game is GESE if the adjustment
process moves in the direction of the equilibrium from any initial cutpoint
that is above
or below the maximum possible cost, c.
In the experiments we report on here F is always uniformly distributed
between
and
exception,
there
and c <
c
there
so
is
always
is
In that case both c
an unstable equilibrium between
and
point of the adjustment process G(c
c^> c
.
When w=N and
c>l,
c
*
=0
and
with
one
-
c
c.
For c^ < c
while
=Oisa
ESE,
= c are ESE and there is also
and
*
)
one
The exception occurs when w=N
always a unique GESE.
*
1.
least
at
c
= c
*
o
,
is
unique GESE.
c
=
the
is
the limit
limit point if
In all other cases
with w >
one with c -
there are exactly two Bayesian equilibria,
1,
the other with c
and
*
*
It is easily shown that c
> 0.
If w -
other equilibrium is a GESE.
1
-
is unstable but the
then there
is
a
unique globally
stable Bayesian equilibrium.
parameters,
which
also
are
There
c,
w and N.
particular
a
some
equilibria
asymmetric
subset
(say
player
contribute regardless of their cost (i.e. c
numbers
<
1
.
occur
1
through
w)
- c)
.
always
This is possible as
We do not consider these (or other) asymmetries.
for nearly all of our experimental parameters we will have a
*
stable symmetric Bayesian equilibrium, c
that provides a strong
,
prediction about individual behavior:
PI.
some
- c for these players) and the
Therefore,
unique
for
—
other members of the group never contribute (c
c
may
An example of such an equilibrium would be one in
*
long as
which
Given
c
(N,w,c), predict
1
contributes if and only if
c
<c
.
Ill
Explaining deviations from the theory
.
While
Pi
roughly
is
will
we
supported,
many
that
individual
The model might nonetheless be
decisions are not in accord with the rule.
This suggests the following hypothesis:
correct in the aggregate.
P2
see
*
*
The aggregate probability of contribution is q »=F(c ).
.
A natural estimator of
is
q
q,
the observed frequency of contribution
Standard tests can then be used to see if this observed
in an experiment.
frequency differs significantly from q
.
We will in fact find significant deviations, even in the aggregate, from
the
theoretical probabilities.
section we discuss two different
In this
approaches to "rationalizing" these deviations,
one based on relaxing the
rational expectations restriction of the theory and one based on modifying
the utility hypotheses of our equilibrium predictions.
A.
The biased probabilities hypothesis
In
consistently
proposed
we
introduction,
the
underestimate
hypothesis
the
probability
the
that
individuals
that
others
but
freeride,
otherwise behave roughly in accordance with expected payoff maximization.
What does this hypothesis,
predict about these deviations?
H,
There are
several alternative ways to operationalize H to apply it to our data. One
interpretation
is
simply
act according to
others
that
some
H
c
implies
> c
That
.
individuals
that
believe
expectations
is
that
about
the
likelihood others will contribute exceed the equllihrlum likelihood others
will contribute.
Using these beliefs,
they then use a outpoint
represents the optimal decision rule given beliefs
To study how
c
> c will
affect the
dT
>
<
n
°
.^
'^
"^
< w-1
>
tn—
(observed)
decision rule c
(2)
10
which
c
suffices (for small amounts of bias) to differentiate (1):
dc
c
,
it
-
Let q
estimator
consistent
of contribution.
the actual probability
+
of
q
relationship
The
.)
P3a.
If w>=l, q
P3b.
If w=N and
P3c.
If
*
< q
is
q
at
a
*
q
H:
7
.
"k
St
-4-
0<q <l,q >q^
+
0<q,q>q
< w < N and
I
that
evaluated
(2),
immediately leads to the following predictions under
+
(Note
it
ifq
Sip
< (w-l)/(N-l) and q
+
< q
^
St
if q
(w-l)/(N-l).^
(
design consists
Our experimental
of
some
of
1
experiments where we
3
always expect undercontribution relative to the game -theoretic predictions,
some
3
of
and
3
of
3
experiments where we always expect overcontribution, and some
of 4 where the prediction depends on the value of c
2
2
We varied
.
to produce the appropriate contrasts.
c
*
For our design, except in the unanimity game w-3
unique.
*
9
solution
In unanimity games,
*
when
c-=l
a second solution c
>
c
*
> 0,
if c
,
1,
When
.
c
<
we have one solution with c
1,
c
is
and a unique
.
.
c =1
N-3
when
>
there is no q
,
*
-2
= c
and
= c.
Combining H with the auxiliary hypothesis that the bias is always with
respect
strong
to
the
unique
restrictions
on
expectationally
stable
observations.
When
equilibrium
multiple
thus
leads
to
equilibria
stable
=0.
exists, we will assume that the bias is with respect to c
A second way to operationalize H is that players expectations about the
likelihood others will contribute exceeds the empirical likelihood rather
than exceeding the equilibrium likelihood.
respect
to
actual
contribution rates.
contribution
Again,
In other words the bias is with
rates,
not
the measurement problem
data on contribution decisions, not on beliefs.
rates of contribution, q
infer
what
players
,
q
beliefs
would
have
,
,
we then state
P4:
equilibrium
that we only have
is
However,
for any obser\'ed
we can estimate a cutpoint decision rule, c
optimizing relative to those beliefs.
e
necessarily
q^
>
q"^
11
to
have
been
if
they
had
and
been
Denoting these inferred beliefs by
As we will discuss in the next section, this operationalization, while more
appealing that P3 in some ways, has some drawbacks.
B.
Alternative Models of Non-Linear Utility and Cooperation
The purpose of this section is
developed on the basis
of
the
to
demonstrate that P3
probability bias
generated from plausible alternative models.
,
the prediction
hypothesis
cannot be
H,
We first deal with non- linear
utility and next with cooperation.
Non-Linear Utility
Since our experimental subjects were paid in dollars,
rule out utility being non- linear in money as
it is important to
an alternative explanation
Without loss of generality, we assume that each player's
for our results.
utility for the outcomes is given by the function u with:
u(0) = 0; u(c) = c + q(c); u(1) = 1; u(l+c) = l+c+ 5(c)(3)
The case of risk neutrality is q =
= 0.
5
all players have the same utility function
We continue to assume that
and that the functions a and
are each either always weakly positive or weakly negative.
analysis
focuses
attention to
<
on
c
<
With (3), equation
c
local
changes
in
the
equilibrium.
This
(1)
-
Q-Prob(k<w)
-
This requires q >
games,
the
1+c
impossible in unanimity games.
utility
of
our
directs
our
generalizes to:
5
•Prob(k>w) (4)
Let us begin by considering the common form of utility
is
Again,
1.
= Prob(k=w-l)
global risk aversion.
5
last term in (4) must be 0.
non-contribution
is
and
5
That is,
Therefore,
raised
<
0.
But
in
economics,
in unanimity
the outcome with payoff
q >
relative
to
implies
the
that the
utility
contribution, in turn implying less contribution, contradicting P3b.
12
of
Having dealt with risk aversion, we turn to models where, in spirit with
prospect theory in psychology,
and partly risk-acceptant
the utility function is partly risk-averse
These possibilities are covered by either:
.
>
and
5
> 0, with at least one inequality strict,
(b) a <
and
5
<
(a) a
In case
(a)
,
with at least one inequality strict.
the utility of non-contribution is raised relative to
,
contribution outcomes
two
the
between
lotteries
interpreted as
with respect to
(0
non- contribution
the
the
the
The utility function is risk-averse for lotteries
utility of contribution.
between
or
utility function
and
outcomes.
Case
can
(a)
individuals who are
for
anchor formed by
the natural
risk-acceptant
but
1)
loss
for
be
averse
They are
their endowment.
risk averse for lotteries over the alternatives near the anchor point where
there
possibility
a
is
lotteries
of
guarantee
that
being
worse
they will
be
off
no
but
worse
off
than
Since the utility of non-contribution always increases,
predicts
contribution
that
always
falls
relative
acceptant
risk
are
endowment.
the
however,
case
(a)
neutrality.
risk
to
for
Therefore, case (a) is inconsistent with P3b and P3c.
In Case
(b)
the situation of case
,
Now the utility of
is reversed.
(a)
contributing is always increased and the individual is risk-acceptant for
lotteries over the contribution outcomes.
as
ascribing
completion
of
a
non-monetary
project.
the
bonus
One can think of the individual
effect
prize
or
Alternatively,
case
the utility of contribution always
successful
captures
(b)
including the model of Palfrey and Rosenthal
effects,
the
to
altruism
But since
(1988).
increases relative to risk neutrality,
case (b) is inconsistent with P3a and P3c.
The remaining possibility is q <
utility
function
is
globally weakly
situation as a priori unacceptable.
consider
1
of N games.
and
risk
6
>
If
0.
5
acceptant.
We
On the other hand,
In these games,
the outcome
>
is
if
-q;c/(1-c)
rule
5
out
,
the
this
< -qc/(1-c)
impossible.
,
The
inequality condition then implies that the utility of contribution has been
raised
relative
to
that
of
not -contributing.
should increase, contradicting P3a.
13
Therefore,
contribution
.
Cooperation
Another alternative view of behavior is provided by various models of
cooperative
behavior.
An
bound
upper
could
what
to
achieved
be
by
cooperation is the outcome that could be achieved by the players if they
This would have the w players with the
had full information about costs.
lowest costs contribute if the sum of these lowest costs
In other words,
total benefit.
good
the
provided unless
is
total
its
less
is
cost
than N.
exceeds
the
Another model of full information cooperation is for the w
players with the lowest costs to provide the good as long as none of these
costs exceeded 1.0,
the value
of the public
good.
Given that costs are
private information, these highly coordinated behaviors cannot be achieved.
Moreover, they lead to predicted contribution rates that are just very much
higher than anything we observe in the data.
A more plausible model of cooperation is for players to use (perhaps as
tacit collusion against the experimenter)
cutpomt
that maximizes
*
m
expected group payoff.
This always implies a contribution rate q
> q
*
m
where q is an ESE. As we will see in the Section V, q also substantially
a
a
c
,
exceeds observed contribution rates.
In summary,
models
we have established that models of non- linear utility
altruism
like
that
can
interpreted
be
in
terms
of
(and
non- linear
utility) cannot be used to generate predictions equivalent to those of P3
To
see
if
H
is
supported
as
a
behavioral
hypothesis,
presentation of our experimental design and results.
14
we
turn
to
a
.
Description of the Experiments
IV.
A
specification of
detailed
experiment
each
appears
Table
as
The
1.
experiments at Carnegie-Mellon University used 72 undergraduates.
The 1987
and 1988 experiments at Caltech used subjects who were undergraduates
Caltech or Pasadena Community college.
used
1989
school
high
students
The
experiments
participating
Caltech. The subjects were mainly male.
in
one
three
or
parameters.
experiments,
In
None of the participants had prior
matched pairs.
for
For
the paired
order of experiments.
the
were
there
No
12
program
identical
except
These varied slightly between
Experiments run in 1987 and
1988 used a computer program written at Caltech.
computer
of
important order effects were observed.
CMU and Caltech and within Caltech sessions.
a
set
involving three
sessions
six
everything was
sessions,
Instructions are included as an appendix.
used
Each experimental
an experiment constituting a different
CMU setting,
the
at
A given experimental session consisted of
or 12 subjects.
9
summer of
program
summer
a
experience in the tasks required in these experiments.
session used
in the
at
written
at
Experiments run in 1989
with
Carnegie -Mellon,
minor
modifications made at Caltech. None of the factors mentioned to this point
appeared to have a notable influence on results.
All experiments were designed to have 20, 25, or 30 rounds.
of rounds was
made known to
the
subjects before
the
The number
experiment started.
Three experiments were inadvertently curtailed shortly before the planned
end because of computer crashes
At the beginning of each experiment,
value,
in "francs",
subjects were told w,
of the public benefit.
cents they would receive,
N,
and the
They were also told how many
at the conclusion of the session.
These values
were held constant throughout an experiment.
In each
(endowment)
204
were
round,
.
subjects were
each
given a single
Token values in franc increments between
independently
drawn
with
replacement
distributions and randomly assigned to subjects.
the value
indivisible
of his
other subjects.
or her
13
from
1
and either 90 or
identical
uniform
Each subject was told
token but not told the values
of
the
tokens
Subjects were then asked to enter their decisions
15
"token"
of
(spend
or
not
subject
spend the
received
token).
a
nxomber
least w of the N subjects
If
at
of
francs
equal
to
value
the
of
spent,
the
each
public
benefit if he was a "contributor". If he or she was not a contributor, the
payoff was this number of francs plus the token value.
In table 1, c shows
the rescaling of the distribution of token values such that the benefit is
rescaled to
14
1.
Each
round subjects were
assigned
to
a
new
group
in
a
rotation sequence which minimized the number of times any two subjects were
paired together in the same group.
The reason for doing this was to limit
reputation and supergame effects which can occur with repeated play.
effects appear to be very limited.
Such
In the ensuing analysis, we treat each
decision as an independent observation.
16
V.
Results
Testing the Bavesian-Nash predictions
A.
1
Efficiency predictions
.
We begin
discussion of
the
results by noting
that
the
Bayesian Nash
Equilibrium is remarkably successful as a predictor of aggregate outcomes
of the experiments.
For each group
in each round of each experiment,
Using the actual token values
computed the actual earnings of the group.
drawn by
group,
the
subjects,
we
also
computed the
then computed average earnings per subject,
to
for
averaged out.
from
variations
individual
c
the
We
.
normalized so that the public
We then compared predicted earnings
actual earnings for each experiment.
where
predicted earnings
assuming that play conformed to the theoretical outpoints,
good had value 1.0 in each experiment.
we
These are aggregate comparisons
round- to -round
and
group -to -group
are
(We maintain disaggregation across replicate experiments.)
The results offer remarkably strong support for the
For data
theory.
from all rounds, the regression equation is:
Actual earnings = -0.046 +
estimated constant
The
1
.
does
040*Predicted earnings,
not
differ
R
significantly
2
=0.95,
from
zero
estimated intercept does not differ significantly from 1.0.
data points are plotted in Figure
Since
there
is
substantial
the
success
variation
in
endowments
2
the
The
actual
(via
c)
across
of 0.40), we checked
of the model was not largely driven by
between earnings and endowments.
and
1.
experiments (endowments and actual earnings have an R
that
N=33
the
correlation
Therefore, we also examined the increase
in per subject earnings over per subject endowments.
The results, plotted
in Figure 2, continue to be striking:
Actual Increase = 0.012 +
.
932*Predicted Increase,
17
R
2
=0.93,
N=33
conventional
at
Again,
levels,
significantly from zero and the
not
differ
significantly from
differ
did not
slope
did
intercept
the
unity.
from
data,
the
from theoretical
show systematic deviations
viewpoints,
though
even
occur
results
impressive
These
other
predictions.
We
discuss these deviations next with a series of tables where the rows show
value
the
of
and
c
designate
columns
the
combinations used in the design.
results
experiment
each
for
In each cell
matches
that
different
four
the
of the
the
order
The
experiments within cells always matches the order given in Table
lines in Table
2
1
N
we present
table,
cell.
w,
of
Solid
1.
separate cells in the later tables.
Individual Behavior
.
Table
2
individual
shows
risk-neutral
that
corresponding
level
classification using the
game -theory
to
There
PI.
theoretical
For w=N=3
outpoint was used where it existed.
,
many
are
outpoints.
succeed at
not
does
w<N,
(For
errors
the
of
non-zero
the
the zero cutpoint was used.
More on this later.)
On the other hand, the prediction is qualitatively correct. Contribution
rates
are
decreasing
reported here.
For
in
all
contribute/not
table
cost,
as
shown by probit
experiments,
contribute
cutpoint/endowment below cutpoint,
the
vs.
analysis
standard x
endowment
2
test
above
is
not
the
2x2
equal
to
that
for
or
-3
is
significant at p < 10
.
In the 19
experiments with a non-zero cutpoint where subjects had endowments above or
at
the
benefit
level,
significant at p <
= .0436
Thus,
lO'''
excluding
after
except
in the CIT 7/31/89
of 4,
2
p=.0882
for
and =
subjects,
such
in
.0002
the
CIT
in the
the
7/26/89
CIT 8/8/89
test
2
2
is
of
4,
of
3.
the prediction is qualitatively correct even if we exclude subjects
who have weakly dominant strategies of not-contributing.
When a subject has a dominant strategy,
in
accordance with the model,
occurs in early rounds of a
2
as
of
3
involving the high school students.
the subject almost always acts
shown in Table
experiment and a
The
3.
3
of
3
only exception
experiment, both
The classification errors of the model
18
are
almost entirely from the behavior of subjects without dominant
thus
strategies.
3
15
Aggregate behavior
.
The game- theoretic model fares somewhat better at the aggregate level of
predicting the frequency of contribution,
noted
be
game- theoretic
comparative
important
an
that
model
other hand,
c,
for
c
contribution
fixed,
claim
in the
,
contribution
that
should
of
the
N
fixed,
As contribution is cheaper on
.
On the
falls.
c
increasing
not
is
w/N,
in
which
A naive view of volunteering
table.
should
It
For w,
data.
contribution occurs more frequently as
increases from left- to-right
might
c
4.
prediction
static
strongly supported by the
is
contribution is strongly decreasing in
average for low
shown in Table
as
increase
w/N
as
increases
since
larger fraction of the group is needed to produce the public benefit.
a
This
naive view is not supported by the data.
If we examine each cell in the table, we find results that are sometimes
in
"ballpark"
the
significant
of
deviations,
but
theory,
the
leading
Of 25
rates
that
experiments,
are
with
P3
.
of
of
the
Experiments
0.
19
consistent
with w=N=3
Only
(Relative to the
with
H.
On
the
observations
can not
equilibrium,
other
hand,
be
However,
.
one
are
evaluated
of
.05
the
six
level.
statistically
inconsistent
In contrast,
significant.
statistically because
any positive contribution is
there
contribution rates for two of the cells.
these
them have observed contribution
observations is statistically significant at the
15
P2
statistically
Consider first experiments with w
H.
all but six of
consistent
many
find
also
rejection
to
deviations are mostly consistent with
< N.
we
are
also
equilibria
q
—
consistent
with
100%
The observed contribution rates
are obviously below those given theoretically and so,
relative to the 100%
contribution equilibrium are inconsistent with H.)
Results similar to those presented in Table 4 are obtained when we basis
the analysis only on those subjects with dominant strategies.
appear in Table
"altruism"
5.
The results
Two alternatives to H, non- linear utility that leads to
and cooperative behavior always predict contribution in excess
19
.
of
the
Bayesian Nash levels.
probabilities
outpoints c
that
q
Table
In
would
result
if
6,
display
we
followed
players
contribution
the
cooperative
the
(To provide readers with a sense of how parametric variation
.
affects the probabilities, the probabilities are also shown for cells where
no experiments were run.)
6
enables us to
3
and c = 2.25,
The information in Tables 4 and
When w = N =
make comparisons across the 33 experiments.
both the cooperative and self-interested models predict zero contribution.
(In this extreme case,
benefits.)
closer
to
In
the
the
the expected cost of providing the good exceeds the
other
cases,
31
Bayesian predictions
observed
the
than
except for the two observations where w =N
3,
we
always
represent
made
the
strong evidence
-=
cooperative
the
to
On
are
predictions
whole,
the
self-interested behavior
always
(For w = N
and c = 3/4.
3
Bayesian prediction.)
for
frequencies
the
data
comparison
in
-=
to
full cooperation.
B
An alternative test of the probability bias hypothesis
.
To this point,
with
bias
we have only examined the hypothesis
reference
to
local
properties
the
about probability
predicted
equilibrium
probability that a randomly selected individual will contribute.
we began by considering a unique stable equilibrium outpoint,
at the implied contribution frequencies
if everyone adopts
in
the
optimal
outpoint)
if
then looked
that strategy,
then calculated the derivative of the reaction function (i.e.
change
direction of
departed from
expectations
expectations" in a neighborhood of the equilibrium.
That is
"rational
If that derivative was
positive then our prediction was an observed level of contribution greater
than
the
equilibrium
level
of
contribution
and
if
the
derivative
was
negative then we predicted the opposite.
That method of evaluating H has two potential weaknesses.
probability bias is measured with respect to
the
not with respect to the objective frequencies of
doing.
First,
the
equilibrium prediction,
what players are actually
Second, the sign of the derivative evaluated at the equilibrium may
be different that the sign of the derivative evaluated elsewhere.
That is
we are making global comparisons by extrapolating from a local calculation.
20
we next examine whether the
To show that these are not serious problems,
expectations overestimated the actual frequencies of contributions.
We do
this in the following way.
Suppose that we observe an empirical frequency of contributions equal to
A
A
Ignoring individual differences,
q.
-1
F
implies a cutpoint rule c(q)
this
-
"
We may then perform the following
that the players are following.
(q)
First we maintain the assumption that players are using best
calculation.
given their beliefs,
responses,
expectation
others
that
are
so that they are all maximizing under the
with
contributing
probability
equal
to
q
A
(possibly not equal to q)
Next we ask "what value of
.
the use of the strategy c(q)?"
would rationalize
q
because it
> q then H is confirmed,
If q
A
means that q can only be justified as deriving from best response behavior
A
A
if players beliefs,
q
exceed
,
A
Care is required in comparing q and q
it is possible that q
4 experiments,
cutpoint
not
is
observations
best
a
reject
then H is contradicted.
If q < q
q.
response
to
2
of
and
3
of
2
sufficiently high that the implied
cutpoint.
any
hypothesis
any
Thus,
such
responding
that
subjects
of
3
experiments where a best
there are two values of q
that are consistent with
joint
the
optimally to biased priors.
response can be found,
is
for the
First,
.
Second,
in
2
are
A
observed
the
cutpoint.
If
both
of
these
exceed
H
q,
is
unambiguously
A
But the data are ambiguous if only one exceeds
supported.
q.
A
w = N =
and
3
>
c
we are guaranteed that q
1,
> q unless
experiments with these parameters, H cannot be put to
Finally, for
A
= 0.
q
For
a test.
A
Table
displays
7
experiments.
and
q
q
(the
latter
in
The hypothesis H is supported in all
all but one of the
1
of
3
experiments.
In the
2
parenthesis)
3
of
of
3
for
all
experiments and
3
experiments,
three
A
experiments
great
to
rejected the joint hypothesis
consistent
be
with
since
optimizing
the
value
behavior.
In
of
q
was
three
too
other
A
experiments,
of
3
one q
experiments,
experiments.
6
was below
H
was
q,
the other above.
supported.
H
was
In the remaining six
not supported in the
In three of the four such experiments,
there
is
2
2
of 4
no q^ that
A
rationalizes
above q
,
q;
in
the
remaining experiment,
the other below.
21
one of the
two
solutions is
examination
alternative
this
Summarizing,
of
hypothesis produces somewhat more mixed results
hold
most
probability
the
While
than before.
experiments,
bias
there
the
are
some
Stronger support for H is found in studying whether the priors q
move
players
have
hypothesis
seems
to
for
the
of
parameter values for which it clearly fails.
Learning
C.
the
in
direction
of
during
q*
experiment.
the
likelihood that others
systematically inaccurate prior beliefs about the
will
contribute,
then
as
play
they
the
game
If
observe
and
sample
a
of
contribution rates, one would expect them to adjust their beliefs in light
of this new information.
This will implicitly lead to a dynamic learning
process that will be reflected in changes in the observed q's.
One natural hypothesis
that this adjustment process will produce q
is
that are closer to the equilibrium q* late in the session compared to the
early
rounds
of
session.
the
rationalized in the
last
experiment and one
of
2
There
28
cases
Counting as
rounds.
10
are
a
where
success
can
q
one
2
be
of 4
experiment that could be rationalized in the last
3
10 rounds but not in the first 10, we find that 22 of 28 cases move in the
right direction.
An alternative hypothesis about learning is that the difference between
A
q
and
the
6
q
less in the last 10 rounds than in the first 10 rounds In
is
of
5
remaining experiments, this is the case indicating that even in these
Summarizing the
experiments there is evidence that subjects are learning.
evidence for the aggregate data,
the hypothesis
that players'
the direction of movement of q
priors on
q
supports
are biased upward initially,
but
they adjust these priors in the correct direction during the course of the
experiment.
Somewhat surprisingly,
even at the aggregate level, we find the amount
of movement is slight. Table
first
five
indicates
Statistical
rounds
a
with
higher
tests
for
8
compares frequencies of contribution in the
contribution
rate
of
in
the
contribution
differences
in
22
last
in
5
the
A
rounds.
first
aggregate behavior
"
five
over
the
+"
sign
rounds.
last
5
compared to the first
5
rounds
are
generally weak.
There are
33
tests,
only three of which have p- levels below 0.1.
We are thus left,
hypothesis,
for now at least,
together with
with a successful probability bias
Bayesian-Nash
equilibrium predictions,
as
the
major sources of explanation for the variation in aggregate contribution
rates.
However,
a
more
definitive
statement of the
results,
especially
regarding learning effects would requires a rigorous analysis of the data
at the individual level that is beyond the scope of this paper.
23
.
Table
1
Description of Experiments
Range
Cents
Rounds Reveal Sequence
Date
Site w
N
U/21/%9
CMU
1
3
90
0.7
2.25
12
20
No
3
5/2/89
CMU
1
3
90
0.7
2.25
12
20
No
1
7/26/89
CIT
2
4
204
0.1
2.22
12
17
No
1
7/31/89
CIT
2
4
90
0.3
2.25
12
20
No
3
8/8/89
CIT
2
4
90
0.3
2.25
12
20
Yes
3
4/27/89
CMU
2
3
90
0.7
2.25
12
20
No
1
5/2/89
CMU
2
3
90
0.7
2.25
12
20
No
3
4/27/89
CMU
3
3
90
0.7
2.25
12
20
No
2
5/2/89
CMU
3
3
90
0.7
2.25
12
20
No
2
7/13/88
CIT
3
90
0.5
1.5
12
30
No
-
2/15/89
CMU
3
90
0.3
1.5
12
20
No
2
2/16/89
CMU
3
90
0.3
1.5
12
20
No
2
7/31/89
CIT
3
90
0.3
1.5
12
20
No
2
8/8/89
CIT
3
90
0.3
1.5
12
18
Yes
2
11/22/87
CIT
2
3
90
1
1.5
9
20
No
-
12/3/87
CIT
2
3
90
1
1.5
9
20
No
-
12/20/87
CIT
2
3
90
1
1.5
9
20
No
-
2/15/89
CMU
2
3
90
0.3
1.5
12
20
No
1
2/16/89
CMU
2
3
90
0.3
1.5
12
20
No
3
7/31/89
CIT
2
3
90
0.3
1.5
12
20
No
1
8/8/89
CIT
2
3
90
0.3
1.5
12
20
Yes
1
7/21/88
CIT
3
3
90
0.5
1.5
12
30
No
-
2/15/89
CMU
3
3
90
0.3
1.5
12
20
No
3
2/16/89
CMU
3
3
90
0.3
1.5
12
20
No
1
12
25
No
1
Sub j s
c
20
«
8/3/89
CIT
3
3
204
0.1
2/21/89
CMU
1
3
90
0.2
3/4
12
20
No
1
3/1/89
CMU
1
3
90
0.2
3/4
12
20
No
3
2/21/89
CMU
2
3
90
0.2
3/4
12
20
No
2
3/1/89
CMU
2
3
90
0.2
3/4
12
20
No
2
2/21/89
CMU
3
3
90
0.2
3/4
12
20
No
3
3/1/89
CMU
3
3
90
0.2
3/4
12
20
No
1
7/26/89
CIT
2
4
204
0.1
2/3
12
25
No
2
7/26/89
CIT
2
204
0.1
2/3
12
24
No
24
.
.
Notes to Table
Francs
1.
This gives the upper end of the uniform (in integers)
-
of endowments
in
This number and c can be used to
francs.
distribution
calculate
the
franc value of the public benefit, B - Francs/c.
Cents
The number of cents paid per franc earned in the experiment.
-
Subjects
Ro\ands
The number of subjects in the experiment.
-
number
The
-
Experiments
of
20,
Experiments of 17,
18,
parameters.
duration.
(non-practice)
of
rounds
or
25,
for
ran
rounds
30
given
the
for
(w,
N,
c)
planned
the
and 24 rounds were prematurely terminated
by computer crashes
Reveal
When this parameter is "Yes",
-
Subjects could match individual token values and decisions but
each round.
could
all token values were revealed after
identify
not
Otherwise,
individuals.
the
values
token
were
not
revealed.
Sequence
Only one set of subjects was run on a given date.
-
subjects played three sets of parameters in sequence.
order
in
which
the
were
parameters
used.
The sequence shows the
the
In
subjects were used, subsequent to their play of a
3
of
experiment
3
game,
8/3/89,
for two other
subjects were run for only one set of parameters.
Other
-
which
franc
The
c
value of 2.22 was
values
had been
integer 91 resulted in
c
-
The 18 experiments at
drawn endowments
2
.
rescaled
from
in
to
90
Within each
c
C.*iU
value,
the
204.
2.25
condition in
Setting B
to
the
22
form a matched set in which the same randomly
used with
were
treatment
for a
six
all
of
sets
the order of the w-1
,
Two
subjects.
subjects were run for each of three different values
0.75).
on
When this entry
treatments in which groups were fixed rather than rotated.
is blank,
Some sets of
2,
sets
of
and
of c
(2.25,
and
experiments was
3
1.5,
permuted so that the first experiment and third experiments were reversed for
the two sets of subjects.
The
endowments
in the 1 of
date
and
on
3
in
the
CMU
1
of
3
c
experiment at CIT on 7/31/89.
8/8/89
followed
the
same
-
1.5
(The
sequence
experiments
matched
three experiments on that
but
differed
assignments and the Reveal treatment.)
Otherwise, new random draws were used in each experiment.
25
were
in
endowment
Table
2.
i
Classification Errors, Bayesian Nash Predictions
w/N
2/4
1/3
c
26/240
29/240
2i
2/3
49/204
44/240
29/240
64/360
33/240
34/240
53/240
35/216
li
2
3/3
46/240
38/340
23/240
31/240
23/180, 30/180
37/180, 31/240
39/240
112/360
47/240
60/240
40/240
68/240
132/300
20A
205
46/240
61/240
61/240
61/240
3
126/240
143/240
4
73/300
67/240
2
3
*
The
errors.
first number
for
each experiment
is
the
number
The second number is the total number of decisions.
26
of classification
.
Table
3.
Endowments at Least Equal to Benefit: Contributions/Endowments
i
w/N
c
25
2/4
1/3
6/112 (2/36)
2/139
2/139
0/141
3/141
,
2/124 (2/24)
0/77
1/77
1/77
2/69
1-
2/3
'
3/3
6/147 (1/3A)
0/147
2/138
0/138
0/61, 0/67
0/61, 1/89
0/89
0/89
12/89 (2/20)
19/127 (2/17)
0/87
4/87
The first number is the total number of contributions when the endowment was
at
least equal
to
the
benefit.
The
second number
the
is
total number of
occurrences of endowments greater or equal to the endowment.
contributions of this
except
zero,
boldface
where
type
in
the
shown
in
small
last
five
font.
rounds
Reveal
The number of
of each experiment was
experiments
shown
in
.
For the 2x2 comparison spend/not spend vs.
< benefit, the standard x
2
endowment > benefit/endowment
test was always significant at p < 10
—L
.
The same
holds true for the first 10 rounds of each experiment except two experiments
where
the
rounds
p- levels
except
for
were
two
.0003
and
experiments
.062,
with
rounds)
27
respectively and for
p- levels
.0006
and
the
.0001
last
(last
10
8
Table
4.
The Frequency of Contribution
w/N
2/4
1/3
c
2'-
0.217
0.213
(0.192) 0.309*
(0.192) 0.221^
0.200
2/3
(0.132) 0.192
(0.104) 0.159
(0.104)
3/3
(0)
(0)
0.096
0.129
(0)
(0)
;
2
0.311^
0.221
0.263^
0.333
0.292
0.300*
0.333.
0.417*
0.308^
0.308^
0.363.
0.379
(0.306)
(0.271)
(0.271)
(0.271)
(0.269)
20*
205
(0.239) 0.311
(0.222) 0.196
(0.244) 0.250
(0.238)
(0.238)
(0.238)
(0.238)
0.440
(0)
(0)
(0)
(0)
*
3
tt
0.379
0.404
0.521
0.558
(0.392)
(0.392)
0.500
(0.523) 0.580*
(0.596) 0.525
(0.596) 0.596
(0)
(0)
(0.674)
2
3
The entries in parentheses in each cell are the theoretical
contribution frequencies for symmetric Bayesian Nash equilibrium under risk
neutrality.
The frequencies were calculated using the actual token draws in
the experiment. The absence of an entry
in a cell
indicates that no
experiments were run for the parameters corresponding to the cell.
Actual
frequencies that represent deviations from the theory that are not consistent
with H are shown in boldface.
Note.
*
Departure from theoretical frequency statistically significant at 0.05
level on basis of t-test, using Normal approximation to binomial.
28
Table
5.
The Frequency of Contribution, Endowment Below Benefit Level
w/N
2i
0.525
0.485
(0.465) 0.620*
(0.465) 0.505^
0.465
0.466
0.325
0.380^
0.485
0.415
(0.466)
(0.399)
(0.399)
(0.399)
(0.395)
4
'I
2/3
2/4
1/3
c
(0.293) 0.430
(0.248) 0.409
(0.248)
0.453*
0.531^
0.630^
0.483^
0.470^
0.576^
0.523
See Table 4 for explanatory notes.
29
3/3
0.206
0.304
(0)
(0.361) 0.399
(0.354) 0.366
(0.370) 0.398
(0.378)
(0.378)
(0.378)
(0.378)
(0)
(0)
(0)
(0)
(0)
(0)
Table
6.
Theoretical Contribution Probabilities for Group Optimal Behavior
w/N
2/4
1/3
c
3/3
(0)
(0)
0.625
0.567,
0.431_
2/3
2i
4
(0.25
(0.134)
)
0.75
0.646
0.5
1
2
(0.313)
204
20S
0.712
0.567
0.610
0.834
(0.424)
(0.381)
3
(0.250)
(0.293)
(0)
1
(0.500)
0.75
0.875
(0)
1
4
(0.431)
0.627
(0.625)
(0.500)
0.765
0.889
(0)
1
2
3
(0.453)
(0.667)
(0.529)
* Note:
(0)
The lower
entries in each cell are the theoretical contribution
probabilities for symmetric Bayesian Nash equilibrium under risk neutrality.
The upper entries are the contribution probabilities generated by the
outpoint that maximizes expected group payoff.
30
Table
7
Estimated Prior Probabilities in the Experiments
Last 10 Rounds
All Roxinds
obs
Date
Site
4/27/89
5/2/89
CMU
CKU
w=l, N=3, c=2.25
17
13
7/26/89
7/31/89
8/8/89
CIT
CIT
CIT
v-2. N-A, c-2,.25
.309
.221
.200
4/27/89
5/2/89
CMU
CMU
w=2. N=3,
c-=2,.25
.192
.158
4/27/89
5/2/89
CMU
CMU
w=3. N-3, c-2,.25
7/13/88
2/15/89
2/16/89
7/31/89
8/8/89
CIT
CMU
CMU
CIT
CIT
11/22/87
12/3/87
12/20/87
2/15/89
2/16/89
7/31/89
8/8/89
obs
e
q
e
q
q
.242
.225
.263
.288
.298
.217
.192
.488
.315. .685
.232, .658
.183
.150
.291, .719
.215, .785
.096
.129
.464
.538
.075
.108
.411
.493
w»l, M=3, c=1.5
311
221
263
333
292
.317
.424
.394
.293
.338
.292
.233
.250
.342
.302
.339
.408
.387
.284
.327
CIT
CIT
CIT
CMU
CMU
CIT
CIT
.342, .638
.500
.311
.289
.467
.317
.292
.392
.358
.371, .629
.433, .567
w=2, N=3,
300
333
417
308
308
363
379
7/21/88
2/15/89
2/16/89
CIT
CMU
CMU
v=3, N=3, c=1.5
.311
.196
.250
.683
.542
.613
.258
.142
.175
.622
.461
.512
8/3/89
CIT
w=3, N=3, c=204/205
.440
.662
.325
.569
2/21/89
3/1/89
CMU
CMU
v=l, N=3,
c=3/4
.379
.404
.467
.449
.408
.408
.446
.446
2/21/89
3/1/89
CMU
CMU
v=2. N=3.
c=3/4
.521
.558
.266, .734
.298, .702
.583
.583
.323, .677
.323. .677
2/21/89
3/1/89
CMU
CMU
w=3. N=3,
c=3/4
.525
.596
.373, .627
.332, .668
.492
.575
.393, .607
.343, .657
7/26/89
CIT
v=2, N=4, c=2/3
.500
.156, .551
.458
.136, .582
7/26/89
CIT
v=2, N=3
.580
318, .682
558
390, .610
,
c=1.5
c=2/3
31
1
.302
.308
-
-
.363, .647
.363. .647
.
.
-
.388, .612
.323, .677
-
Table 8.
Differences In the Frequency of Contribution:
First Five Trials Frequency
Last Five Trials Frequency
-
w/N
2/4
1/3
c
-.03°
2i
+ .02°
4
, 1
1-
2
+ .12°
+ .10°
+ .02°
+ .08°
-.05°
+ .05°
-.02°
-.05°
+ .03°
+ .02°
+ .06°
+ .05°
-.02°, .00°
-.13°. -.05°
+ .08°
-.07°
+ .03°
+ .07°
+ .22^
.00°
+ .28^
204
205
3
4
-.27^
-.05°
+ .03°
+ .03°
+ .10°
+ .14°
2
3
Likelihood- ratio test significant at p > 10
2
3/3
2/3
Likelihood-ratio test significant at 10
32
-2
> p > 10
+ .05°
.00°
References
Camerer,
C.
Econometrica
Reputation Model,"
Cox,
J.,
B.
"Experimental Tests of a Sequential Equilibrium
and K. Weigelt,
Roberson, and V.
Auctions,"
Smith
V.L.
in
Greenwich:
JAI Press, 1982.
Cox, J., V.
Smith,
.
1988,
"Theory and Behavior or Single Object
Smith,
Research
(ed.)
and J. Walker,
R.
,
Palfrey,
I.
and
Maital,
S.
and Palfrey, T.,
R.
.
Plott,
1983,
C,
1982, 50,
.
12,
Economics
.
207-12.
Valuation
"Asset
in
an
537-67.
"Voluntary Provision of a Pure Public good as
Journal of Public Economics
the Game of Chicken,"
McKelvey
and
Econometrica
Experimental Market,"
Lipnowski,
T.
Experimental
in
"Tests of a Heterogenious Bidder's Theory
of First Price Auctions," Economics Letters
Forsythe,
1-36.
55,
.
1983,
20,
381-6.
"An Experimental Study of the Centipede Game,"
Working Paper, California Institute of Technology, 1989.
Palfrey
,
T.,
Rosenthal,
and H.
"Participation and Provision of Discrete
Public Goods: A Strategic Analysis," Journal of Public Economics
.
1984, 24,
171-93.
"Private Incentives and Social Dilenimas: The Effects
^
of Incomplete Information and Altruism," Journal of Public Economics
28,
1988,
309-32.
,
Game
.
with
Press)
Private
"Testing for Effects of Cheaptalk in a Public Goods
Information,"
Games
.
33
and
Economic
Behavior
.
1990
(in
Rapoport,
A.,
"Public Goods
Political Science Review
.
and the MCS
1985,
Van de Kragt, A., J. Orbell, and
a
79,
R.
Experimental Paradigm," American
148-55.
Dawes,
"The Minimal Contributing Set as
Solution to Public Goods Problems," American Political Science Review
1983,
77,
112-21.
34
.
:
APPENDIX
These are the instructions for the laboratory session run on 7/31/89 at
Subjects were seated in front of
the California Institute of Technology.
computer terminals that were separated by partitions.
These instructions
but were read aloud to the subjects.
were not distributed,
In addition,
the payoff tables were distributed to the subjects as indicated below.
After reading the instructions for the first experiment,
two practice
rounds were conducted to familiarize the subjects with the procedures and
computer
the
sense
that
practice
subjects were
round.
practice
The
screens.
rounds
very controlled
were
instructed exactly what actions
They were
also
shown how to
access
a
take
to
in
the
in
the
"history screen"
which summarized the past decisions made and outcomes in the games they had
played in previous rounds.
After going through these two practice rounds,
all subjects were given a quiz to make sure they understood the details of
experiment
the
how
and
would
earnings
their
be
Any
computed.
misunderstandings were clarified and the experiment commenced.
After the first experiment of the session had concluded,
subjects were
briefly informed of the new rules for the second experiment,
second experiment
layouts were
Because
commenced.
similar in all
and then the
keyboard tasks and the
the
three experiments
screen
conducted in each session,
practice rounds and quizzes were not always conducted before the second and
third experiments.
At the end of a session,
in a separate room.
subjects were paid in private
Each subject was then dismissed from the expermiment
before the next subject was paid.
Similar procedures were followed in the
other sessions.
INSTRUCTIONS
This
cash
at
the
is
an experiment
end of
the
in
decision making.
experiment.
The
amount
You will be paid in
of money you
earn will
depend upon the decisions you make and on the decisions other people make.
It
is
important
that
you
do
not
talk
at
all
or
otherwise
attempt
to
communicate with the other subjects except according to the specific rules
35
If you have a question,
of the experiment.
feel free to raise your hand.
One of us will come over to where you are sitting and answer your question
This session you are participating in is broken down into a
in private.
separate
three
sequence
of
rounds.
All money
experiment,
expeiments.
denominated
is
you will be
paid
Each
At
Francs.
in
$.30
for
experiment
every
the
will
end
of
Francs
100
last
the
20
last
you have
accumulated during the course of all three experiments.
RULES FOR EXPERIMENT #1
At the beginning of every round of every experiment,
assigned to a group with two other subjects.
you will be randomly
Each round in the experiment
you will have a single token to use in one of two ways:
Option #1:
Spend the token.
Option #2:
Keep the token.
The amount of money you earn in a round depends upon whether you keep or
spend your token that round and how many others in your group spend their
Each round, you will be told how many Francs your token is worth if
token.
you do not spend it.
This amount,
called your
token value,
will change
from round to round and will vary from person to person randomly.
more specific,
from
1
to
in each round,
Francs.
90
pattern to your
To be
this amount is equally likely to be anywhere
There is absolutely no systematic or intentional
token values
or
the
token values
of
anyone
else.
The
determination of token values across rounds and across people is entirely
random.
Therefore,
token values.
everyone
in your
group will generally have different
Further- more, these token values will change from period to
period in a random way.
You will be informed privately what your new token
value is at the beginning of each round and you are not permitted to tell
anyone what this amount is.
Specific instructions:
At the start of each round you are told your token value for that
round.
Remember
that
members
of
the
same
group
will
generally
have
different token values and these values change randomly for every one after
36
]
each round.
After being told your token value, you must wait at least 10
Your keyboard will
seconds before making your decision to keep or spend.
When everyone has made a decision, you
be frozen for this period of time.
are
told which
members
your
of
round
spent
given
a
each round of experiment
1,
you
are
their
and
token
what
This will continue for 20 rounds.
earnings were for that round.
each
group
randomly
new
token
value
Following
randomly
and
,
your
reassigned to a new group.
Payoffs:
In
members
your
in
group
decides
spenders and nonspenders)
addition,
nonspenders
in
to
spend
least
their
token,
group will
in your
your
at
if
group
also
2
the
of
every member
each earn
earn their
out
50
(both
In
Francs.
token value.
3
what
happens in your group has no effect on the payoffs to members of the other
groups and vice versa.
These
earnings.
table
are
Therefore,
shown
in
in each round,
the
following
you have three possible
table:
[Hand
out
earnings
.
Earnings Table for Experiment
You Spend
Number of Others Spending
Yes
1
Y'our
Earnings
Francs
Yes
1
60 Francs
Yes
2
60 Francs
Your Token Value
No
No
1
Your Token Value
No
2
Your Token Value + 60 Francs
37
Specific instructions for Experiment
This
is
1:
experiment
exactly the same as
1
except
that only
spender
1
is
needed for all members in that group to receive 60.
PAYOFFS
In each round,
group
decides
nonspenders)
nonspenders
to
in
spend
your
in your
if at
their
group
least
1
token,
will
each
out of the
every
earn
member
60
members
3
(both
Francs.
group also earn their token value.
in your
spenders
In
and
addition,
What happens
in
your group has no effect on the payoffs to members of the other groups and
vice versa.
Therefore,
in each round,
These are shown in the following table:
you have three possible earnings.
[Hand out new table to subjects and
collect old table.]
Earnings Table for Experiment
You Spend
Number of Others Spending
2
Your Earnings
60 Francs
Yes
Yes
1
60 Francs
Yes
2
60 Francs
Your Token Value
No
No
1
Your Token Value + 60 Francs
No
2
Your Token Value + 60 Francs
38
Specific instructions for Experiment
3:
First of all,
There are three differences in this experiment.
has 4 members instead of
spend in order
Third,
all
for
3.
Second,
members
2
each group
out of the 4 members of a group must
of that group
to
get
extra payment.
the
the extra payment is 40 instead of 50.
PAYOFFS
In each round,
group
decides
nonspenders)
nonspenders
to
in
spend
your
in your
if at
least
their
group
2
token,
will
each
out of the
every
earn
group also earn their
member
40
members
4
(both
Francs.
token value.
in your
spenders
In
and
addition,
What happens
in
your group has no effect on the payoffs to members of the other groups and
vice versa.
Therefore,
in each
round,
These are shown in the following table:
you have
three possible earnings.
[Hand out new table to subjects,
and collect old table.]
Earnings Table for Experiment
You Spend
Number of Others Spending
3
Your Earnings
Yes
Francs
Yes
1
40 Francs
Yes
2
40 Francs
Yes
3
40 Francs
Your Token Value
No
Your Token Value
No
1
No
2
Your Token Value +
40 Francs
No
3
Your Token Value +
40 Francs
39
Notes
Up
this
to
our
point,
approach
the
is
utility model of the Van de Kragt et al
.
same
Rapoport's
(1985)expected
experiments.
Weimpose game
as
(1983)
theoretic equilibrium restrictions, which Rapoport does not do.
2
See,
Palfrey and Rosenthal's
for example.
altruism model or,
(1988)
context of risk averse behavior in sealed bid auctions,
inthe
Robertson, and
Cox,
Smith (1982) and Cox, Smith and Walker (1983).
3
Variations
of
game
the
with different payoff
structures
can be
treated.
Similarly, the model can be generalized to allow for a non-zero lower support
to
F(c)
and to F(c)
non-uniform.
Palfrey and Rosenthal
See
1989).
(1987,
Here we introduce only parameters varied in the experiments reported below.
4
Since F is continuous and any specific c has zero measure, wewillfrequently
simplify matters by being imprecise about knife - edgesituations in this case
,
c=l.
See Palfrey and Rosenthal (1990) for a theoretical and experimental analysis
of the game when preplay communication is permitted.
Moreover,
the iterated map defined by
GESE in all but
two cases.
'^
c
G(c
converges globally to the
)
In both those cases, G(')
is explosive,
but there
exist monotone transformations of G that will converge.
7
8
9
Note that if w=l
*
,
for any F(
•
)
,
there exists a unique
q
>
0.
Note that for w < N, there never exists an equilibrium with q
,
Showing uniqueness is direct with N=3 since q=F(c
equation
(1)
noting that
is quadratic in q
^
.
In the w=2
—
,
=
dq
40
=1.
*
*
)
is
linear in
c
while
N=4 game, uniqueness follows from
2
=0 if q = 1/3 and
*
if q= 2/3"
This assumption can be relaxed.
In the actual experiment,
subjects know prior to play that, post play, they
will not only learn their payoffs but also the total number of contributors
It is possible that subjects attach utility to these reports
in the group.
(Is
being
the
contributing
non-contributor
sole
when
no
one
else
in
does?)
unanimity
a
but
,
worse
game
choose
we
than
ignore
to
not
this
complexity.
12
data
The
can be
obtained by mailing
a
request
and
a
3.5"
high
density
diskette to either author.
13
These
costs
were
generated
pseudo-random number generator.
in
advance
by
a
standard
computerized
In the three 1987 experiments,
token values
were assigned directly in cents. There do not seem to have been any important
effects from the manner of presenting or scaling the payoffs.
14
There do not appear to be important effects associated with either the manner
of presenting the payoffs (francs vs. cents) or scaling the payoffs.
Finding per\'asive use of dominant strategies is in fact surprising since the
same
result
does
not
occur
in
Prisoners'
Dilemma
experiments.
difference between our experiments and the Prisoners' Dilemma
is
A
key
that in our
experiments "All Contribute" does not Pareto dominate "All not Contribute".
16
^
In those
toward
q
cases where there are two solutions for q
.
41
e
,
both solutions move
!
1
o
u
017
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