Bargaining by Children
William T. Harbaugh
University of Oregon
and N.B.E.R.
Kate Krause
University of New Mexico
Steven G. Liday, Jr.
University of Oregon (Student)
1
Why study bargaining in children?
– Bargaining behavior is presumably learned.
– Optimal decisions depend on an understanding of how
others will respond.
– Beliefs about fairness, and about what others believe to
be fair will matter for decisions.
– Very little work has been done on when or how these
beliefs and knowledge are acquired.
– Adults exhibit wide variations in bargaining behavior,
both within and across cultures.
– Studying the development of bargaining behavior may
lead to more knowledge about the sources of this
heterogeneity.
2
310 children, grades 2-12.
– Recruited in public school classrooms in Coquille and
Myrtle Point, Oregon.
– Sample is more representative of local population than are
usual college students samples.
– But local pop is not representative of U.S.
– Play with kids from same classroom.
– Groups are homogenous w.r.t. age.
– Kids know each other very well. (But decisions, partners
are secret.)
3
Protocol
Ultimatum and Dictator Games:
In the Ultimatum Game proposer makes a proposed split,
responder accepts or rejects. (Then switch roles, w/ new
partners.)
– Subgame perfect Nash is (1,A) or (0,A)
Dictator game, responder just takes what he gets.
Younger kids bargain over tokens, which they can use to buy
toys, school supplies from our “experiment store” at the
end. Older kids bargain for quarters.
10 tokens/quarters per game, we pay for every game, before
the next is played.
4
– Two versions of the experiment, one starting with the
dictator game, one with the ultimatum.
– Within each version, two treatments, depending on
whether you started as the proposer or the responder.
Version A:
Treatment 1: DP, DR, UP, UR.
Treatment 2: DR, DP, UR, UP.
Version B:
Treatment 3: UP, UR, DP, DR.
Treatment 4: UR, UP, DR, DP.
– Note incomplete design - No one started as proposer
in one game and then as responder in the other.
5
Plan:
• Lit review
• Examine how bargaining behavior develops
with age.
• Do a within subjects comparison of
behavior in the ultimatum and dictator
games.
• Look at history and order effects.
• Differences by sex, other characteristics.
6
Econ Literature:
• Ultimatum game devised by Guth et al (1982), review in Roth
(1995)
– Proposals have mode at 50%, mean of 40%, smaller ones are rejected,
even with large sums. (Cameron, 1999)
• Not the subgame perfect prediction, but models of fairness can
explain this. (Rabin, 1993 or Fehr and Schmidt, 1999)
• Cultural differences exist:
– Roth et al, proposals average 36% in Israel, 47% in U.S., and Israelis
do accept lower proposals.
– Machiguenga propose 26%, UCLA anthro grad students 48%.
• Mixed evidence for gender differences:
– Eckel and Grossman (1998) find women make larger proposals,
Solnick finds no difference. Both find women accept less.
– Andreoni and Vesterlund (2001) find women’s dictator proposals are
less price elastic.
7
Developmental Psychology Literature:
• Deals w/ prosocial behavior and social cognition.
– Generally done with parent surveys, playground observation,
interviews and simulations of real-life situations.
• Kohlberg (1976) argues for stages of moral development.
– Away from rigid rules, towards “obligations to others”
– Meta-analysis by Fabes and Eisenberg (1996) finds mixed empirical
support for stages, but general increase with age.
• Gender differences in prosocial behavior
– Fabes and Eisenberg report girls are better at reasoning, but not so
different in behavior
• Josef Perner, others
– report development of a basic “theory of mind” even in very young
children. But very young children (3) are bad at deception.
8
Results:
Table I shows dictator and ultimatum proposals
by grade:
– Obvious differences across games, for every grade.
– On average, older children make larger proposals than the
second graders in both games.
– Dictator proposals for 2nd, 4th/5th, 9th, and 12th graders
average 0.35, 1.4, 1.2 and 2.1 respectively.
– Ultimatum proposals average 3.5, 4.1, and 4.5 and 4.3
– So, general upward trend to both, much of the increase is
from 2nd to 4th/5th.
9
Table I: Dictator and ultimatum proposals and rejection rates, by grade.
Dictator
proposal
Ultimatum
proposal
Grade
n
Rejection rate
2
74
0.35
(1.0)
3.5
(1.7)
0.11
(0.31)
4
106
1.4
(2.1)
4.1
(1.6)
0.14
(0.35)
9
90
1.2
(1.8)
4.5
(0.89)
0.033
(0.18)
12
40
2.1
(1.2)
4.3
(0.69)
0.10
(0.30)
All
310
1.2
(1.2)
4.1
(1.4)
0.10
(0.30)
Note: Means, with standard deviations in parentheses.
10
Table A.II shows significance tests.
• Within the same grade, the distributions of
ultimatum and dictator game proposals are
always different, and the ultimatum game
proposals are always larger.
• In the dictator and ultimatum games, proposals
by 2nd graders are significantly lower than those
of any other grade.
• In both the dictator and ultimatum games,
proposals by 12th graders are generally higher
than those of youngest grades.
11
Table A.II: Non-parametric tests of differences in proposals by grade and game.
Dictator
2
4/5
9
Grade
4/5
1.07***,+++
9
0.83***,+++
-0.25
12
1.77***,+++
0.70**
Ultimatum
Grade
2
4/5
0.58***,+++
9
0.97***,+++
0.39+++
12
0.81***,+++
0.23+++
4/5
0.95**,++
9
-0.16*,+++
Notes: The tables show the mean for the grade in the row minus the mean for the grade in the column.
*, **, and *** indicate the difference is significant at the 0.1, 0.05, or 0.01 level respectively using the Mann12
Whitney test.
+, ++, and ++ indicate the same using the Epps-Singleton test.
Within subjects comparison across games:
• 38% make the modal combination: 0 in the
dictator game and 5 in the ultimatum.
• 14% of the children make identical proposals in
both games
• Only 7% make a dictator proposal that is larger
than their ultimatum proposal.
• 78% act strategically, in the sense that they
make a strictly higher ultimatum proposal than
dictator proposal – 92% of second graders do
this.
• Suggests they understand the game.
13
Fourth/fifth graders
Second graders
0.50
0.50
0.40
0.40
0.30
0.30
0.20
0.20
0.10
0.00
0
1
2
3
0
4
5
1
2
4
3
>=6
5
Ultimatum
proposal
0.10
4
3
0.00
2
1
0
>=6
1
2
3
>=6
Ultimatum
proposal
0
4
5
Dictator proposal
Dictator proposal
5
>=6
Twelfth graders
Ninth graders
0.50
0.50
0.40
0.40
0.30
0.30
0.20
0.20
0.10
0.00
0
1
1
2
3
Dictator proposal
0
4
5
2
3
4
5
>=6
Ultimatum
proposal
0.10
0.00
2
0
1
1
2
3
>=6
3
4
5
>=6
Ultimatum
proposal
0
4
5
>=6
Dictator proposal
14
History and Order effects
• Behavior in one game may be affected by
general effects from previous games (order) or
specific effects (history.
• To control for order effects in our analyses of
differences in behavior, we look at behavior by
order.
• To see if individual specific history matters, we
ran regressions on each decision in each
treatment, with independent variables
representing outcomes in previous play.
15
Figure II: Proposals by game, grade, and order.
5
4
U2
U4/5
3
Proposal
U9
U12
D2
D4/5
2
D9
D12
1
0
1
2
3
4
Round
16
Order:
• The difference between ultimatum and dictator proposals is large
and it persists across rounds.
• There is a general decrease in dictator proposals when the dictator
game is conducted in the later rounds of the experiment.
– The decrease is drastic with the 12th graders.
– Consistent with Anderson et al. (2000) aversion story.
• Not much of a consistent trend with the ultimatum proposals.
• In the dictator game, in every round but the last 2nd graders make
smaller proposals than anyone else.
• In the ultimatum game, 2nd graders make smaller proposals than
the 4th/5th graders in every round but the first.
17
Dictator proposal
Ultimatum proposal
Ultimatum rejection
Round:
2
3
4
2
3
4
2
3
4
Model:
Tobit
Tobit
Tobit
OLS
OLS
OLS
OLS+
Probit
Probit
Partner’s
0.354
-0.056
0.153
0.090
0.097
0.078
(0.219)
(0.898)
(0.095)
(0.059)
(0.110)
(0.148)
dictator proposal
Partner’s
3.600*
-31.577
-0.057
0.529
ultimatum
rejection
(1.827)
(0.000)
(0.084)
(0.639)
-
0.6
03*
0.5
70*
**
**
Partner’s
-0.531
-0.802
0.138
0.085
-0.110***
ultimatum
proposal
(0.530)
(0.831)
(0.092)
(0.078)
(0.024)
(0.148)
(0.203)
Adjusted or
pseudo
r-squared
0.0105
0.0269
0.0277
0.0161
0.0209
0.0213
0.21144
0.4075
0.3323
Observations
77
79
78
78
76
77
79
77
18
76
• History doesn’t matter much:
– Out of 14 coefficient estimates only one was
significant – slightly more than would be expected by
chance.
– Suggests they understand random matching.
• How about Age, gender, size?
– Look at regressions which control for order effects:
19
Dictator proposal
order
Ultimatum proposal
Ultimatum rejection
-0.2567
-0.2563
-0.0012
0.0038
-0.0014
-0.0018
(0.176)
(0.173)
(0.0696)
(0.0695)
(0.0077)
(0.0071)
0.2314***
0.2283***
0.0932***
0.0927***
0.0060**
0.0060**
(0.0607)
(0.0594)
(0.0226)
(0.0225)
(0.0026)
(0.0025)
order &
-0.0529**
-0.0524**
grade
interaction
(0.0242)
(0.0237)
boy
-0.4121**
-0.2671
-0.0226
0.0473
-0.0102
-0.0152
(0.181)
(0.185)
(0.1569)
(0.1636)
(0.0170)
(0.0171)
grade
Height
-0.0354**
-0.0181
0.0016
(0.0143)
(0.0123)
(0.0012)
ultimatum
proposal
r-squared
0.0938
0.101
0.0436
0.0473
-0.0441***
-0.0411***
(0.0135)
(0.0138)
0.444
0.453
20
Proposals:
• Decrease with order
• For Dictator proposals, at a rate that increases with age.
• Kids in higher grades make larger proposals.
• In the dictator game the average proposal is 1.2 tokens,
and each year of age is associated with a proposal of
about 0.23 more tokens.
• For the ultimatum game the overall average proposal is
4.1 tokens, and each year of age is associated with about
a 0.1 larger proposal.
21
Gender:
• Boys tend to make smaller dictator and
ultimatum proposals, though only the dictator
differences are statistically significant.
• Very similar to the differences cited in Eckel
and Grossman for undergraduate men and
women.
• Gender is not significant in the ultimatum
response regression.
22
Relative height:
• Measured as the percentage deviation of the
participant’s height from the mean of all those
in the experiment with them (i.e., their
classroom.)
• Has a large effect: Children who are one
standard deviation taller than the mean
propose about half a token less than children
who are one standard deviation shorter.
23
Dictator Proposals by Age, Sex and Size:
Proposals by height and sex, 2nd and 4th/5th graders.
Dictator Proposal
1.5
1
Girls
Boys
0.5
0
Short
Medium
Tall
All
Height relative to class and sex
24
Proposals by height and sex, 9th and 12th graders.
3
Dictator Proposal
2.5
2
Girls
1.5
Boys
1
0.5
0
Short
Medium
Tall
All
Height relative to class and sex
25
Table A.III shows non-parametric tests for differences in dictator
proposals, according to gender and height.
• Robust support for gender differences in proposals.
• For differences across the relative height groups the support is
mixed:
– The tests generally support the existence of differences by
height in proposals by girls. For boys, the support is weak.
– None of the tests can reject the hypothesis that the short
boy/tall girl groups have the same contributions.
– Using rank correlations of height and dictator proposals,
Kendall’s test rejects the null of independence between
relative height and dictator proposals for all participants.
– Same for ultimatum.
• But can’t reject if we use the relative height within gender
measures.
26
• If we compare across sexes while holding height constant, the sex
difference actually switches sign in the tall kids, with tall boys
giving more than tall girls.
– Across sex and height, short girls give the most, but tall girls
give the least.
– This holds for young kids and all kids, not for old kids.
• Why are both the sex and the height effects larger in the dictator
than the ultimatum game?
– More scope for pure preferences.
• Interestingly, it seems to be relative rather than absolute height
that matters. The 9th graders are more than half again as tall as
the 2nd graders, but when playing against other 9th graders, they
actually make larger dictator and ultimatum proposals.
27
Responses:
• Only 30 of the 310 proposals are rejected.
• A one token increase in the proposal decreases the
probability of rejection by about 4%, sig. at 1%.
• Run probit, find that proposals of 2.0, 3.5, 3.5, and 4.0
maximize the predicted returns for grades 2, 4/5, 9, and
12 respectively.
• For every grade, average proposals are higher than the
estimated optimal proposals. Risk aversion, altruism,
and upward bias in children’s estimation of the rejection
probabilities are all possible explanations for this.
• The differences, both in terms of the deviation from the
optimal proposal, and in terms of the loss in the
expected return, are largest for the second graders.
28
Table IV: Average returns by ultimatum proposal and grade.
Grade
Proposal
Second
Fourth / fifth
Ninth
Twelfth
1.5
6.65
4.40
1.86
0.00
2.0
7.28
5.07
3.28
0.14
2.5
7.28
5.53
4.69
0.88
3.0
6.95
5.77
5.65
2.79
3.5
6.49 *
5.78
5.99
4.87
4.0
6.00
5.62 *
5.85
5.67
4.5
5.50
5.31
5.47 *
5.47 *
5.0
5.00
4.91
4.99
5.00
Average proposal
3.5
4.1
4.5
4.3
Rejections / Obs.
8 / 74
15 / 106
3 / 90
4 / 40
Pseudo R-sq.
0.542
0.456
0.404
0.444
Bold face indicates the proposal that maximizes the estimated return for that grade.
* indicates the average proposal for that grade, rounded to the nearest 0.5.
29
Figure III: Actual and stated rejection rates
1
0.9
0.8
0.7
0.6
Proportion
rejected
0.5
0.4
0.3
0.2
0.1
0
0 (n=11)
1 (n=21)
2 (n=9)
3 (n=28)
4 (n=68)
>=5 (n=173)
Proposal
Actual
Stated
30
Stated versus actual accept/reject decisions:
• We asked “What is the smallest offer that you would accept?” in
a survey immediately after the experiment.
• Most of the discrepancy between actual and stated behavior
comes from boys.
• On average, boys say that the lowest offer they will accept is
3.50 tokens, while for girls it is 3.06.
• But, if anything, boys are actually less likely to reject a given
ultimatum proposal.
• One interpretation of children’s tendency to exaggerate their
toughness as bargainers is that it is further evidence of their
strategic abilities. This result also shows that it can be very
important to use real rather than hypothetical payments with
children.
31
Conclusion:
• By age 7, children are good bargainers.
– They have preferences about their payoffs and payoffs of others.
– They understand that others do too, and they use this knowledge
strategically.
– Proposals differ drastically across games, and ultimatum proposals
are approximately optimal.
• We found differences in behavior, across age, gender, and height.
– The youngest children made considerably smaller dictator and
ultimatum proposals than adults. They also accepted smaller
proposals. Girls made larger proposals than boys, and shorter kids
made larger proposals than tall ones, particularly in the dictator
games.
– The height effect is most noticeable in young kids, the gender effect
in older ones.
– Children’s self-reported propensity to reject “unfair” ultimatum
offers was substantially exaggerated, particularly by boys.
– These results show that even within a culture individual differences
in behavior can be explained, and that the development of economic
behaviors can be described.
32
Applications:
• Research says something about the development of
behaviors that are important to adults, when they are
adults.
– In most bargaining situations optimal decisions depend on an
understanding of how others will respond. Beliefs about
fairness, and about what others believe to be acceptable, are
important considerations.
– We think it’s likely that people learn how to bargain at the
same time they learn so much else - when they are children.
– We know that adults exhibit wide variations in bargaining
behavior, both within and across cultures. It seems likely that
these variations lead to different outcomes in such matters as
wages, job searches, and housing and auto purchases. Studying
the development of bargaining behavior will lead to more
knowledge about the sources of this heterogeneity in both
behavior and in economic outcomes.
33
Another set of reasons arise because many important
decisions that children make as children involve
bargaining with others.
• For example, parents and children bargain over
children’s human capital investments.
– Who will make the sacrifices needed to procure education,
and what kind of capital will be accumulated.
• Another important example involves bargaining between
children.
– Research on classrooms consistently shows the importance
of peer effects. Modeling how peer effects work requires
an understanding of how children with different
preferences influence each other’s decisions, which is in
part a bargaining question.
• The outcome of these bargaining processes will
obviously depend in part on the bargaining abilities of
the parties involved. An understanding of how these
strategies change with variables such as age and sex may
also lead to a better modeling of human capital
accumulation.
34