ERIC EJ862905: Delay, Probability, and Social Discounting in a Public Goods Game

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by  ERIC

JOURNALOFTHEEXPERIMENTALANALYSISOFBEHAVIOR 2009,91,61–73 NUMBER1(JANUARY)
DELAY, PROBABILITY, AND SOCIAL DISCOUNTING IN A PUBLIC GOODS GAME
BRYAN A. JONES AND HOWARD RACHLIN
STONYBROOKUNIVERSITY
Ahumansocialdiscountfunctionmeasuresthevaluetoapersonofarewardtoanotherpersonata
given social distance. Just as delay discounting is a hyperbolic function of delay, and probability
discountingisahyperbolicfunctionofodds-against,socialdiscountingisahyperbolicfunctionofsocial
distance. Experiment 1 obtained individual social, delay, and probability discount functions for a
hypothetical$75reward;participantsalsoindicatedhowmuchofaninitial$100endowmenttheywould
contribute to a common investment in a public good. Steepness of discounting correlated, across
participants, among all three discount dimensions. However, only social and probability discounting
werecorrelatedwiththepublic-goodcontribution;highpublic-goodcontributorsweremorealtruistic
andalsolessriskaversethanlowcontributors.Experiment2obtainedsocialdiscountfunctionswith
hypothetical $75 rewards and delay discount functions with hypothetical $1,000 rewards, as well as
public-goodcontributions.TheresultsreplicatedthoseofExperiment1;steepnessofthetwoformsof
discountingcorrelatedwitheachotheracrossparticipantsbutonlysocialdiscountingcorrelatedwith
the public-good contribution. Most participants in Experiment 2 predicted that the average
contributionwouldbelowerthantheirowncontribution.
Key words: altruism, self-control, risk-taking, social discounting, delay discounting, probability
discounting,publicgoodsgame,cooperation,humans
_______________________________________________________________________________
TheeconomistJulianSimon(1995)claimed discounting by pigeons (Rachlin, 1989; Ra-
that people allocate available resources on chlin, Logue, Gibbon, & Frankel, 1986).
threedimensions:(a)currentconsumptionby Social discounting also takes this same
the person, (b) consumption by the same hyperbolic form. Prior studies in our labora-
person at later times (delay discounting), and tory (Jones & Rachlin, 2006; Rachlin & Jones,
(c) consumption by other people (social 2008) have obtained social discount functions
discounting). Simon said: ‘‘Instead of a one- as follows. Participants were asked to imagine
dimensional maximizing entity, or even the that they had made a list of the 100 people
two-dimensional individual who allocates in- closest to them (but not to actually make the
tertem 6 porally, this model envisages a three- list). Order on the list (N, ranging from 1 to
dimensional surface with an interpersonal 100)wastaken asameasure ofsocial distance.
‘distance’ dimension replacing the concept Then participants were asked to choose hypo-
of altruism’’ (p. 367). Simon did not list thetically between a varying amount of money
probabilistic discounting as an additional forthemselvesandafixedamountforaperson
form, perhaps because he considered expect- at a given social distance. For example, a
ed value (probability times amount) to be an participant might prefer $75 for himself to
obvious rational adjustment to current con- $75forthe10thpersononhislist(N510)but
sumption. However, in practice, probability prefer$75forthatpersonto$5forhimself.At
discounting does not simply track expected
some‘‘crossover’’amount(v)between$75and
value (Kahneman & Tversky, 1979). When
$5forhimself,hemustbeindifferentbetween
probability (0 # p # 1) is converted to odds-
an amount for himself and a (usually) lesser
against (h 5 (12p)/p) which, like delay, amount for the 10th person on his list. In our
increases indefinitely from zero, human prob-
studies, as N increased, crossover amount
ability discounting takes the same hyperbolic
decreased. In other words, as social distance
form found byMazur (1987)todescribedelay
increased,participantswerewillingtoforgoless
money for themselves in order to give a fixed
ThisresearchwassupportedbyagrantR01MH044049
fromtheNationalInstituteofMentalHealth. amount to another person. The function thus
Correspondence concerning this article should be obtained,relatingsocialdistancetoamountof
addressed to Howard Rachlin, Psychology Department, moneyforgone,isasocialdiscountfunction.
Stony Brook University, Stony Brook, NY, 11794 (e-mail:
In Rachlin and Jones’s (2008) experiment
[email protected]).
doi:10.1901/jeab.2009.91-61 the (hypothetical) alternatives were $75 for
61
62 BRYAN A. JONES and HOWARD RACHLIN
person-Nandnothingfortheparticipantversus one defects, the cooperator earns a very low
alesseramountfortheparticipantandnothing reward while the defector earns a very high
for person-N (as in the example above). In reward. As in the PGG, defection in the PDG
Jones and Rachlin’s (2006) experiment the maximizes the reward for the individual while
hypotheticalalternativeswereanevenlyshared cooperation increases reward for the group.
$150 ($75 for the participant and $75 for Since the PGG was played only once by
person-N) versus an amount (between $75 participants in these experiments, and all
and $150) for the participant and nothing for choices were made simultaneously and anon-
person-N.Thesocialdiscountfunctions(medi- ymously, no player’s contribution could have
an amounts of money forgone in the two influenced that of any other player. With 100
experiments as functions of social distance) players, each $1.00 contributed would earn a
werevirtuallyoverlapping;theywerewellfitted return of $0.02 for the contributor ($1.00
bythefollowinghyperbolicequation: doubled and divided among 100 players)
V resulting in a loss of $0.98, for that dollar,
v ~ 1zk N ð1Þ regardless of others’ contributions. But each
social dollar contributed would also result in a gain
where v and V are discounted and undis- of $0.02 for each of the other 99 players. The
counted money amounts, N is social distance, net loss of $0.98 for the individual may be
and k is a constant that differed among balanced by the total gain of $1.98 for the
social
individuals; the greater is k the steeper is other players in the group. Thus there are
social,
socialdiscounting. structural similarities between the PGG and
The present experiments measured social, the social discounting task; in both, partici-
delay, and probability discounting of partici- pants may forgo smaller rewards for them-
pants who also played a public goods game selves in exchange for larger rewards to other
(PGG) where they chose the fraction of an people. The main purpose of both experi-
initial endowment to contribute to a common ments was to determine the extent to which
fund that would be doubled and distributed behavior in one of these tasks predicts
equally to all players. The PGG is a laboratory behavior in the other.
model of such real-world activities as contrib- Like social discounting, delay and probabil-
uting to charity or public television, voting, itydiscountingarewelldescribedbyhyperbol-
recycling, and other acts of social cooperation ic functions in the form of Equation 1 (Green
(Camerer,2003).InaPGGeachplayerhasthe & Myerson, 2004). It has been speculated that
option of contributing some fraction of an one of these discounting processes is funda-
initial endowment to a common investment. mentalwhileanother(orothers)arederivable
The investment has a positive return and is from it or that all three, or two of the three,
distributed equally to all players regardless of depend on some common mechanism (Ra-
the amount of their contribution. As an chlin, 1989, 2002b). If this were the case,
illustration, consider a PGG with 100 players, Simon’s three dimensions of utility would not
each of whom is given (a hypothetical) $100 vary independently of each other; the steep-
endowment. A player may contribute any ness of the three discount functions would be
fraction of the $100. The total amount expectedtocorrelateacrossindividualsandto
contributed is then doubled by the experi- vary together as discounting conditions vary.
menter and distributed equally among the Such common variation has not been
players regardless of their contribution. The consistently obtained with delay and probabil-
amount earned by a player would then be the ity discounting (Green & Myerson, 2004).
amountthatplayeroriginallykeptplus1/50of Moreover, Rachlin and Jones (2008) found
the total contributed. that,likeprobabilitydiscounting,butopposite
The PGG is structurally the same as a to delay discounting, social discounting was
prisoner’s dilemma game (PDG: Rapoport, steeper with higher undiscounted amounts
1965). In the two-player version of the PDG, (Vs).Asecondpurposeoftheexperimentswas
either player may ‘‘cooperate’’ or ‘‘defect.’’ If to further examine relationships among the
bothplayerscooperate,bothearnamoderate- three forms of discounting as well as the
ly high reward; if both defect, both earn a relationship of each with performance in the
moderately low reward; if one cooperates and PGG.
DISCOUNTING IN A PUBLIC GOODS GAME 63
EXPERIMENT 1 For the delay discounting measure, five
METHOD delays (D 5 1 day, 1 week, 1 month, 1 year,
Participants and 5 years) in 10 increments of money
offered immediately were presented. Each
Participantswere103StonyBrookUniversity
delay was presented on its own page, with
businessschoolstudents(51male,47female,5
page order randomized with the following
unreported, Mdn age 5 24), enrolled in two
instructions:
advanced undergraduate classes and onegrad-
Please choose which amount of money you would
uate class. They completed a pencil-and-paper
rather have for each line.
PGG as well as social, delay, and probability
A. $75 for you right now. B. $75 for you after
discounting tests for partial course credit.
[D].
Participants in each class completed the ques-
A. $70 for you right now. B. $75 for you after
tionnairesatthesametimeinalecturehall.
[D].
--------Down To-------
Materials and Procedures
A.$5foryourightnow.B.$75foryouafter[D].
The PGG instructions read as follows:
For the probability discount measure, five
Imagine the following situation (purely hypothet- probabilities, expressed as percentages (p 5
ical we regret to say): 90%, 70%, 50%, 30%, and 10%), were
1. The experimenter gives you $100. presented in 10 increments of money guaran-
2. A box is passed around to each person in this teed. Each probability was presented on its
room. own page, with page order randomized:
3.Eachpersonmayputalloranypartornoneof Please choose which amount of money you would
the $100 into the box. No one else will know how rather have for each line:
much money anyone puts into the box. A. $75 guaranteed or B. A [p]% chance of
4. After the box goes around the room, the winning $75.
experimenter doubles whatever is in the box and A. $70 guaranteed or B. A [p]% chance of
distributes it equally to each person in the room winning $75.
regardless of how much money they put into the box. -------Down To-------
Eachperson willthengo homewithwhateverthey A. $5 guaranteed or B. A [p]% chance of
kept plus what they received from the box. winning $75.
Notethatyouwillmaximizethemoneyyoureceive Social discounting instructions were as
by not putting any money in the box. Then you will follows:
take home the original $100 you kept plus what the The following experimentasks you to imagine that
experimenter distributes after doubling whatever youhavemadealistofthe100peopleclosesttoyouin
money was in the box. theworldrangingfromyourdearestfriendorrelative
HOWEVER: If everybody kept all $100, nothing atposition#1toamereacquaintanceat#100.The
would be in the box and each person would take personatnumberonewouldbesomeoneyouknowwell
home $100. and is your closest friend or relative. The person at
Whereas, if everybody put all $100 in the box, #100mightbesomeoneyourecognizeandencounter
each person would take home $200 after the money butperhapsyoumaynotevenknowtheirname.
in the box was doubled and distributed. You do not have to physically create the list- just
Please indicate below how much of the $100, if imagine that you have done so.
any,youwouldputintothebox.Pleasetrytoanswer Six social distances [N9s] were presented:
the question as if the money were real: #1,#5,#10,#20,#50,#100,eachonitsown
I would put the following amount into the box: page, with order of pages randomized. The
$______ amountofmoneytoforgoinordertogive$75
I would keep the following amount: $______ to another person was presented in 10
Sum must equal $100 incrementsrangingfrom$85to$0.Eachpage
Immediately after the PGG each participant contained the following instructions:
completed social, delay, and probability dis- Imagineyoumadealistofthe100peopleclosestto
counting tests in a counterbalanced order. youintheworldrangingfromyourdearestfriendor
Each discounting test consisted of a page of relative at #1 to a mere acquaintance at #100.
instructions followed by five or six pages of Now imagine the following choices between an
questions. amount of money for you and an amount for the
64 BRYAN A. JONES and HOWARD RACHLIN
V
#[N] person on the list. Circle A or B to indicate v ~ ð3Þ
which you would choose in EACH line. 1zkprobhs
A. $85 for you alone or B. $75 for the #[N]
whereD5delay indays;h5(12p)/p5odds
person on the list.
A. $75 for you alone or B. $75 for the #[N] against.
person on the list.
-------Down To------- RESULTS
A.$0foryoualoneorB.$75forthe#[N]person Figure 1showsmediancrossoverpointsand
on the list. crossover points of individual participants at
ThemaximumamountinColumn-Awasset the 20th and 80th percentiles for the three
at $85 rather than $75 because in previous types of discounting. The exponent, s, repre-
research many participants preferred $75 for sents sensitivity to the discounting variable
the 1st or 2nd person on their list to $75 for (Green & Myerson, 2004). With social dis-
themselves (Jones & Rachlin, 2006). counting (Equation 1), obtained values of s
For half of the discount tests the money have not been significantly different from
amountsinColumnAdecreased(D)asshown unity and were therefore set at unity in fitting
above; for the other half, the amounts in Equation 1 to the median social crossover
Column A increased (I). The eight orders of points. (But s was allowed to vary in fitting
increase and decrease across the three tests Equations 2 and 3 to the median delay and
(III,IID, IDD, IDI, DDD,DDI, DII, DID) were probability crossover points.) As in previous
social-discountingtests,manyparticipantspre-
counterbalanced across participants.
ferred $75 for people at close social distances
Data Analysis (N 5 1, 2) to $75 for themselves. Crossover
points above $75 were obtained at these close
Thedataofanyparticipantwhocrossedover
social distances. The undiscounted value, V,
morethanonceonanypageofadiscounttest
was therefore allowed to vary in fitting
wereeliminatedfromallcalculationsinvolving
Equation 1 to the median social crossover
that test. Also, the data of any participant
points. Parameters and fits of Equations 1, 2,
whose PGG answers did not add to 100% of
and3tomediancrossoverpointsareshownin
the total were eliminated from all calculations
the left side of Table 1. In addition, the three
involving the PGG. A maximum of 11 partic-
equations were fitted to individual partici-
ipants were removed from the PGG analysis
pants’ crossover points. For the individual fits,
and a maximum of 10 participants were
all parameters other than k were held con-
removed from the analyses comparing pairs
stant. A k value was then determined for each
of discounting measures. The number of
participant for each of the three discount
participants removed for each analysis is
functions. The right side of Table 1 shows the
reflected by the number of participants re-
mean of individual ks and mean of individual
ported on each table in the results section.
fits to Equations 1, 2, and 3.
The point (v) at which each of the Area under the curve (0 # AUC # 1) is a
remaining participants crossed over from
single-parametermeasureofdiscountratethat
preference for the Column-A amount to the does not depend on functional form (Myer-
Column-B amount (or vice versa) was deter- son,Green,&Warusawitharana,2001).AUCis
minedateachsocialdistance(N),delay(D)in the sum of the areas of the trapezoids formed
days,andprobability(p).Equations 1,2,and3 by adjacent crossover points and the abscissa
(see Rachlin, 2006 for discussion of this form (normalized with respect to the maximum).
of the equation) were fitted to the median The closer the AUC is to unity, the shallower
crossover point at each social distance, delay, the discounting. Table 2 shows that partici-
and probability. In addition, individual social, pants with shallower social or probability
delay, and probability discount functions were discount functions (higher AUC) tended to
determined for each participant by fitting contribute more in the PGG than did partic-
individual crossover points (vs) to Equation 1, ipants with steeper probability discount func-
2 or 3: tions. However, there was no significant
V correlation between steepness of delay dis-
v ~ 1zk Ds ð2Þ counting and PGG contribution. Analyses
delay
DISCOUNTING IN A PUBLIC GOODS GAME 65
reported in Tables 2 and 3 were performed
usingbothAUCandlogk.Thesignificanceof
theresultswasthesameforbothmeasures.We
report AUC because this measure involves no
assumptionoffunctionalform.Figure 2shows
individual-participant delay AUC’s (upper
graph) and probability AUC’s (lower graph),
each plotted against individual-participant
social AUC’s.
PGG contributions were not normally dis-
tributed across all participants. In order to
account for non-normal distributions, we
divided donations into those who contributed
less than the median donation ($0–$25) and
those who contributed more than the median
($26–$100) to create a binary variable. Four
outliers of individual AUC social scores (all
above .6) were removed from this analysis.
Significancesofallcorrelationsdidnotchange
when the outliers were eliminated. The corre-
lation between social discounting (AUC) and
the binary PGG contribution was significant,
r(90) 5 .222, p 5 .036. The correlation
between AUC probability and the binary PGG
contributionwasalsosignificant,r(92)5.371,
p , .001. Table 3 shows that the AUC’s of all
three discount functions were significantly
correlated with each other.
The average stated PGG contribution was
$31.Thedistributionofcontributionsisshown
in Figure 3 and will be discussed along with
that of PGG contributions in Experiment 2.
DISCUSSION
Participants who contributed higher per-
centages of their $100 endowment to the
common good in the PGG were more gener-
ous to others at various social distances
(shallowersocialdiscountfunctions)andwere
less risk-averse (shallower probability discount
functions)thanwereparticipantswhocontrib-
Fig. 1. Upper graph: Social discount function: uted less in the PGG. This is some evidence
Amount of money for the participant equivalent to $75 that social and probability discount functions
forapersonatagivensocialdistance(N).Thesolidcircles
may be meaningful measures of individual
aremediancrossoverpoints.ThelineisEquation1fitto
the medians. Open squares are crossover points of the altruism and social cooperativeness.
participantatthe80thpercentile(k5.04);opentriangles
are crossover points of the participant at the 20th
percentile (k 5 .27). Middle graph: Delay discount r
function:Amountofmoneynowequivalentto$75delayed
by D days. The solid circles are median crossover points. for sure equivalent to $75 at a given odds against (h 5
The line is Equation2 fit to the medians. Open squares (12p)/p). The solid circles are median crossover points.
are crossover points of the participant at the 80th The line is Equation3 fit to the medians. Open squares
percentile (k 5 .01); open triangles are crossover points are crossover points of the participant at the 80th
oftheparticipantatthe20thpercentile(k5.09).Lower percentile (k 5 2.8); open triangles are crossover points
graph: Probability discount function: Amount of money oftheparticipantatthe20thpercentile(k59.0).
66 BRYAN A. JONES and HOWARD RACHLIN
Table1
Experiment1.Bestfittomedianindifferencepointsandmeanofindividualparticipantfitsto
Equations1,2and3.
BestfittingEquationtomedian MeanoffittoEquationforindividuals
Typeofdiscounting V k s R2 V Meank SD s R2
Social(Eq.1) $77 0.067 1.00f .97 $77f 0.081 0.272 1.00f .80
Delay(Eq.2) $75f 0.029 0.35 .99 $75f 0.018 0.092 0.35f .91
Probability(Eq.3) $75f 3.270 0.57 .97 $75f 4.570 2.949 0.57f .89
ffixedatthisvalue.
However, no significant correlation was In prior experiments, participants discount-
found between delay discounting and PGG ed higher delayed amounts (Vs) more than
contribution.Thatis,aparticipant’sgenerosity lower delayed amounts (the ‘‘amount effect’’)
to her future self, as measured by her but they discounted higher probabilistic and
individual delay discount function, did not socially distant amounts less than lower prob-
predict her generosity to a group of fellow abilistic and socially distant amounts (the
students, as measured by her PGG contribu- ‘‘reverse amount effect’’). These effects are
tion. Steepness of delay discounting has been summarized for probability and delay by
repeatedly found to correlate with failure of Green and Myerson (2004) and for social
self-control or addiction in everyday life; discounting by Rachlin and Jones (2008).
individual delay discount functions have been Experiment 1 adds to the evidence that social
obtained for alcoholics, cigarette smokers, and probability discounting share similar
gamblers, cocaine addicts, and heroin addicts; propertieswhiledelaydiscountingdiffersfrom
the delay discount functions of addicts were both.
significantly greater than those of nonaddicts As Table 3 shows, despite the differing
(Bickel & Marsch, 2001). Steepness of delay properties of social discounting together with
discounting has also been found to decrease probability discounting on the one hand and
with age of the discounter (Green, Fry, & delay discounting on the other, all three
Myerson, 1994). The high predictive power of discountfunctionscorrelatedsignificantlywith
the delay discounting measure in these cases each other. The lack of transitivity (delay
together with the lack of correlation between discounting uncorrelated with PGG contribu-
delay discounting and social cooperation in tion, social and probability discounting corre-
thePGGinExperiment1isthereforeevidence lated with PGG contribution, but delay dis-
against Rachlin’s (2002b) argument that self- counting correlated with both social and
control(asmeasuredbydelaydiscounting)on probability discounting) is possible due to
the one hand, and social cooperation (as the looseness of the correlations involved
measured by social discounting) on the other, (significant r-values ranging from .22 to .32).
are ultimately the same process. For example, even though social and delay
Table2
Experiment 1. Best fit to median indifference points and Area Under the Curve (AUC)
correlatedwithPGGcontribution.
FittoEquation AUCvs.PGG
Typeofdiscounting V k s R2 AUC r p n
Social(Eq.1) $77 0.067 1.00f .97 .256 .245* .016 96
Delay(Eq.2) $75f 0.029 0.35 .99 .391 .009 .935 92
Probability(Eq.3) $75f 3.270 0.57 .97 .248 .269** .009 93
n:numberofparticipants.
ffixedatthisvalue.
*p,.05.
**p,.01.
DISCOUNTING IN A PUBLIC GOODS GAME 67
Table3 school seniors and graduate students of
Experiments 1 and 2: Correlations between AUCs for Experiment 1 were atypical. Accordingly,
differentformsofdiscounting. Experiment2repeatedthePGGandthesocial
and delay discounting tests (but not the
r p n
probabilitytest)ofExperiment1withahigher
Social-ProbabilityExperiment1 .334** .001 94 undiscounted amount ($1,000) in the delay
Probability-DelayExperiment1 .205* .048 94
tests, with a larger number of participants,
Social-DelayExperiment1 .282** .006 93
Social-DelayExperiment2 .249** .001 160 and with less sophisticated participants
(mostly freshmen in an introductory psychol-
n:numberofparticipants.
ogy class).
*p,.05.
**p,.01.
EXPERIMENT 2
discountingcorrelatedsignificantly,manypar-
METHOD
ticipantsdifferedstronglyinsteepnessofsocial
Participants
anddelaydiscounting.Somehadshallowdelay
discount functions but steep social discount Participantswere196students(85male,111
functions (‘‘Scrooges’’) while some had shal- female, Mdn age 5 19) enrolled in an
low social discount functions but steep delay undergraduate Stony Brook University psy-
discountfunctions(‘‘generousspendthrifts’’). chology class. They completed a pencil-and-
Intransitivitiesmayoccurwhen,asinthiscase, paper PGG, social discount, and delay dis-
multiple factors underlie a given measure count tests simultaneously in a lecture hall.
(Tversky, 1969).
Materials and Procedures
It is conceivable that the three-way correla-
tion found between the three discount mea- A single-page PGG was presented to each
sures is an artifact of the very similar choice participant as in Experiment 1 with the
proceduresused to determinethem(contrast- addition of the following instructions:
ed with the judgment procedure used to Howmuchdoyouthinktheaveragepersoninthe
measure PGG contribution) but we believe class will put in? $______
this is unlikely. First, any tendency to cross ImmediatelyafterthePGG,eachparticipant
over at the upper or lower parts of the answer completed delay and social discount tests in a
sheet irrespective of the discounting variable counterbalanced order. Each discount test
would have been cancelled out by the coun- consisted of a page of instructions followed
terbalanced up–down orders of the A-col- by four pages. Four delays (1 day, 1 week,
umns. Second, probabilistic alternatives were 1 month,1 year)were testedin11increments
expressed intheanswersheetsaspercentages, ranging from $1,000 to $0. Participants were
whereastheAUCmeasuresofprobabilitywere presented with a page for each delay in a
determined after conversion to odds-against. randomized order that included the following
Odds-against ((12p)/p) varies inversely with instructions:
probability.Ifparticipantstendedtocrossover Please choose which amount of money you would
at high or low values of the variable being rather have for each line.
measured,probabilityAUCshouldhavevaried A. $1,000 for you right now. B. $1,000 for you
inversely, not directly with both social and after [D].
delay AUCs. A. $900 for you right now. B. $1,000 for you
However, it is possible that the long delays after [D].
tested in Experiment 1 and typically used in A. $800 for you right now. B. $1,000 for you
delay discounting studies with humans (run- after [D].
ningto5 years)dwarfedthe$75undiscounted ------Down To------
reward and created a‘‘peanuts’’effect, where- A. $ 0 for you right now. B. $1,000 for you after
in the delay discounting measure was some- [D].
how distorted. Or perhaps the similar undis- Social discount instructions were identical
counted amounts of the discounting variables to those of Experiment 1 except four social
created an ‘‘atmosphere effect’’ wherein the distances (1, 10, 50, and 100) were tested.
two measures were confused. Another possi- For half of the discount tests the money
bility is that the ‘‘sophisticated’’ business amountsinColumnAdecreased(D)asshown
68 BRYAN A. JONES and HOWARD RACHLIN
Fig. 2. Upper graph: Social AUC versus delay AUC of individual participants. Lower graph: Social AUC versus
probabilityAUCofindividualparticipants.Thedifferentnumbersofpointsinthetwographsatthesameabscissavalue
areduetoeliminationfromanalysisofdelayorprobabilitydataofparticipantswhocrossedovermorethanonceonthat
measure.Thesolidlinesareregressionlines.Thedottedlinesaremedians.
DISCOUNTING IN A PUBLIC GOODS GAME 69
tions into below median contribution ($0–
$40) and above median contribution ($41–
100) correlated withsocialAUC,r(196) 5.29,
p , .001. Correlations using individual-partic-
ipant log ks corresponded to those obtained
with individual-participant AUC’s.
Figure 3 shows the distribution of PGG
contributions in both experiments and the
distributionoftheparticipants’predictionsfor
the average contributions in Experiment 2. As
is typical of PGG contributions (Camerer,
2003), the distributions are not unimodal;
there are large modes at $0 and $50, and a
smalloneat$100.ThemeanandmedianPGG
contributions were $40 and $38, respectively.
Fig. 3. Distributions of stated PGG contributions in
The mean and median PGG predictions were
Experiment 1 and Experiment 2 and predictions of
averagecontributioninExperiment2. $20 and $30. The mean difference across
participants (contribution minus prediction)
above; for the other half, the amounts in was $8, significantly greater than zero, t(285)
Column A increased (I). The four orders of 5 4.95, p , .001. That is, on average,
increase and decrease for the two tests (II, ID, participants claimed to be more generous
DD, DI) were counterbalanced across partici- than their classmates.
pants. Figure 4 shows individual participants’ pre-
dicted average contributions plotted against
RESULTS their own stated contributions. The solid line
is the best-fitting straight line through the
The same participant elimination criteria
points, r(364) 5 .60, p , .001.
wereusedinExperiment2asinExperiment1;
data from 36 participants were removed from
analysis. Table 4 shows the fit of Equations 1 DISCUSSION
and 2 to the data. As in Experiment 1, both The results of Experiment 2 followed the
social and delay discounting were well de- pattern of Experiment 1 with respect to social
scribedbyhyperbolicfunctions.Table 5shows and delay discounting and the PGG. As in
the mean individual fits of social and delay Experiment 1, in the social discounting test,
data to Equations 1 and 2. participants who indicated that they would
Again, as in Experiment 1, the correlation forgo more money for themselves in order to
betweensocialAUCandPGGcontributionwas give $75 to other people also indicated in the
statistically significant while the correlation PGG that they would contribute more to a
betweendelayAUCandPGGcontributionwas common good. But, as in Experiment 1, the
not significant; again, as Table 3 shows, social same did not hold for delay discounting;
and delay AUCs were significantly correlated participants who indicated that they would
with each other. Binary divisions of contribu- forgo more money now in order to obtain
Table4
Experiment 2. Fits of data to Equations1 and 2; Area Under Curve (AUC) correlated with
PGGcontribution.
FittoEquation AUCvs.PGG
Typeofdiscounting V k s R2 AUC R p n
Social(Eq.1) $82 0.048 1.00f 0.97 0.369 .350** 0.001 160
Delay(Eq.2) $1,000f 0.005 0.37 0.96 0.755 0.117 0.139 160
n:numberofparticipants.
fparameterfixedatthisvalue.
*p,.05.
**p,.01.
70 BRYAN A. JONES and HOWARD RACHLIN
Table5
Experiment2.MeanofindividualparticipantfitstoEquations1and2.
FittoEquation
TypeofDiscounting V Meank SD s R2
Social(Eq.1) $82f 0.119 .292 1.00f .86
Delay(Eq.2) $1000f 0.002 .023 0.35f .72
ffixedatthisvalue
$1,000 in the future contributed no more was the much greater steepness of delay
money in the PGG than did those who said discounting in Experiment 1. The delay
they would forgo less money now in order to discount constant, k, for median crossover
obtain $1,000 in the future. Despite this points,was0.029inExperiment1and0.005in
difference between social and delay discount- this experiment—more than a fivefold differ-
ing as regards the PGG, steepness of discount- ence. This is an example of the well estab-
ing was significantly correlated across the two lished ‘‘amount effect’’ in delay discounting
measures—participants who sacrificed present wherein larger amounts ($1,000 in this exper-
good for the benefit of others also tended to iment) are discounted less than smaller
sacrifice present good for the benefit of their amounts ($75 in Experiment 1; Raineri &
futureselves.Butthereweremanyexceptions; Rachlin, 1993).
in both experiments some participants were Unlike undiscounted delayed amounts, the
Scrooge-like, with steep social and shallow undiscounted social amounts were set at the
delay discount functions, while others were same value ($75) in the two experiments. Yet,
generous spendthrifts, with steep delay and the introductory psychology students in this
shallow social discount functions. experiment were (marginally) less selfish than
A notable difference between the results of the senior and graduate business-school stu-
this experiment and those of Experiment 1 dents of Experiment 1. The mean PGG
Fig. 4. TheactualcontributionversuspredictedcontributioninExperiment2.Thesolidlineisthebestlinearfit
betweencontributionandprediction.