University of Innsbruck
Institute of Public Economics
Discussion Paper 2000/4
When the ‘Decision Maker’Matters:
Individual versus Team Behavior
in Experimental ‘Beauty-Contest’Games*
Martin G. Kocher and Matthias Sutter
Institute of Public Economics
University of Innsbruck
Universitaetsstrasse 15/4
A-6020 Innsbruck
Austria
e-mail: [email protected] – [email protected]
Abstract: In our beauty-contest game participants simultaneously guess a number in
[0,100], the winner being whose number is closest to two-thirds of the average number.
We compare behavior of individuals and small teams, a hitherto unexplored issue in the
context of beauty-contest games. Our findings suggest that teams have the same depth
of reasoning (number of steps of eliminating dominated strategies) as individuals in the
first round, but that teams learn faster in subsequent rounds, applying deeper depths of
reasoning. Learning direction theory describes individuals’ and teams’ behavior very
well. However, teams’behavior is even better accounted for by that theory.
JEL classification: C7, C91, C92
Keywords: beauty-contest experiments, bounded rationality, team behavior, individual
behavior, learning direction theory
*
We would like to thank Colin Camerer, Rupert Sausgruber, and, in particular, Rosemarie Nagel for
many helpful suggestions and comments. All remaining errors are ours, of course. Financial assistance by
the Royal Economic Society’s Small Budget Scheme is gratefully acknowledged.
I.
Introduction
In economics a ‘decision maker’ is usually modeled as an individual. However, in
many real-life bargaining situations the negotiators are, in fact, teams rather than
individuals, such as families, boards of directors, legislatures or committees. Think, e.g.,
of an oligopolistic market. When firms try to optimize their production decisions by
thinking through the strategic structure of the game and forming expectations about
competitors’ decisions, it should not make a difference from a theoretic point of view
whether all firms in the market are either run by a single entrepreneur or by an executive
board. Similar arguments are at stake, when one studies, e.g., jury decisions and
political or military decisions, which are mostly dealt with by small groups of experts or
politicians.
Experimental economics provides a suitable tool to study behavior of individuals
and teams under controlled conditions which differ only with respect to the ‘decision
maker’. The beauty-contest game – which was likened by Keynes (1936) to professional
investment activity – has been studied extensively, and exclusively, with individuals as
decision makers (see Rosemarie Nagel, 1995; John Duffy and Nagel, 1997; Teck-Hua
Ho, Colin Camerer and Keith Weigelt, 1998). Individual behavior shows very similar
patterns across very different subject pools (see survey results in Camerer, 1997; Nagel,
1999; Nagel et al., 2000). By extending the beauty-contest game to a team setting and
comparing individual and team behavior, we are able to explore whether the ‘decision
maker’ matters, which is a hitherto neglected issue in the context of the beauty-contest
game. The main purpose of this paper is to address two basic questions: (1) Does it
make a difference in outcomes with regard to the iterated elimination of dominated
strategies, whether individuals or small teams compete against each other? (2) Is there a
difference in dynamic learning processes between individuals and small teams?
The characteristics of the game are straightforward. Decision makers compete for a
fixed prize by simultaneously guessing a number in a given interval, the winner being
the person or team whose number is closest to a positive fraction of the average of all
chosen numbers. The game-theoretic structure makes the beauty-contest game an ideal
tool to study how many iterations of eliminating dominated strategies a ‘decision
maker’ actually applies (Camerer, 1997). Textbook rationality would require the
decision maker to iterate infinitely. Since this is only seldom observed, the beauty1
contest game enables us to study bounded rationality. By repeating the beauty-contest
game for four rounds we are also able to observe learning dynamics.
A.
Individual versus team reasoning
The widely held belief that teams reach ‘better’ decisions than individuals is far
from being confirmed by psychological literature.1 In the idealized form of the team
superiority argument teams are, e.g., considered to balance biases, catch errors and
stimulate thoughtful work (James Davis, 1992). Since the 1950s the conventional
wisdom of team superiority has been challenged by numerous experiments (among
other classics, see Irving Lorge and Herbert Solomon, 1955; Solomon Asch, 1956),
leading to the conclusion that group discussion “can attenuate, amplify, or simply
reproduce the judgmental bias of individuals” (Norbert Kerr et al., 1996, p. 693).
First, team conformity and self-censorship may lead to so-called ‘groupthink’,
which may result in dangerous symptoms like stereotyping outside people, putting
pressure on inside people who disagree and creating the illusion of invulnerability
(Kleindorfer et al., 1993). Especially teams with designated leaders are prone to
groupthink’s adverse effects, resulting in inefficient decision making.
Second, teams tend to polarize individual attitudinal judgement in many
circumstances, which is also known under the label ‘risky shift’ (James Stoner, 1968).
This effect, which is contrasting the intuitive conjecture that teams tend to moderate
extreme positions, has been proved in many different settings (Davis, 1992; Kerr et al.,
1996) and is sometimes referred to as the ‘group polarization hypothesis’ (Timothy
Cason and Vai-Lam Mui, 1997).
Even when we would presuppose that there is a good deal of intuitive arguments
that corroborates the superiority-conjecture of teams in a direct comparison with
individuals, it is far from clear that teams generally perform better to the baseline
1
The poorly defined term ‘better’ cannot be substituted by a more precise term in general. Although
there are some situations in which the ‘correctness’ of a decision may be evaluated, most decision
contexts exhibit factual as well as value issues and, therefore, lack an unambiguous evaluation criterion
(Randy Hirokawa et al., 1996). The beauty-contest game is also subject to this double-standard, which
makes it difficult to define ‘better’concisely. Building on a central economic assumption the yardstick of
measurement is provided by the degree of rationality of the decision. A faster adjustment of teams to the
2
aggregation of its members’ potential contributions (Davis, 1992). The comparative
performance of individuals or teams hinges crucially on the kind of task in question.
Basically, one can distinguish between intellective tasks (where there is a clear
evaluation criterion for the quality of performance) and judgmental tasks (where there is
no such criterion). Intellective tasks differ further with respect to their ‘demonstrability’,
i.e., to which degree the knowledge of the solution to the task is shared by team
members once it is voiced. The beauty-contest game exhibits characteristics of an
intellective task (iterated elimination of dominated strategies), but also of a judgmental
task (judgments on others’ bounded rationality), and it is partly demonstrable, since the
rationale of the game can be explained relatively easy. We believe that the intellective
characteristics outweigh the judgmental, leading to the hypothesis that teams should
outperform individuals, especially with regard to the iterated elimination of dominated
strategies. In other words, we expect teams to apply deeper levels of reasoning than
individuals in the first round.
Assessments of the differences between individual and team decisions are
surprisingly scarce in experimental economics so far.2 Gary Bornstein and Ilan Yaniv
(1998) have studied individual versus team behavior in a standard, one-shot ultimatum
game, where a fixed amount of money c is split between a proposer and a responder. If
the responder accepts the proposer’s offer x, she gets x and the proposer keeps c − x .
However, if the responder rejects the offer, both get nothing. Bornstein and Yaniv
compare two treatments, one with individuals playing against individuals and one with
teams (of three subjects each) playing against teams. Their main result is that teams are
more (game-theoretically) rational players than individuals by demanding more than
individuals in the role of proposer and accepting less in the role of responder.
theoretic Nash equilibrium might be viewed as an indicator for ‘better’ decision making. The discussion
of our reasoning and learning models will clarify the notion more precisely.
2
The team game literature (see, e.g., Bornstein et al., 1996, 1997) is related to our team treatment, but
not to our individual treatment. A team game involves two groups, or teams of players. Each player
chooses how much to contribute towards the team effort. Since contribution is assumed to be costly, but
benefits associated with winning are public goods for the members of a team, this situation corresponds to
a prisoner’s dilemma situation within teams. Payoff to a player is an increasing function of the total
contribution made by members of her own team, and a decreasing function of the total contribution made
by members of the opposing team. In contrast to the team game literature, our beauty-contest does not
exhibit this intra-team prisoner's dilemma situation, since contribution to the group discussion is not
costly (at least in monetary terms), but might raise chances of winning.
3
James Cox and Stephen Hayne (1998) have explored decision making of teams and
individuals in common value auctions, characterized by risky outcomes. Though both
teams and individuals deviate from rational bidding when they have more information,
teams are more affected by the ‘disadvantage’ of information, leading to the conclusion
that teams are less rational decision makers than individuals.
Contrary to the studies of Bornstein and Yaniv and of Cox and Hayne, in the
beauty-contest game no fairness or distribution arguments are involved and loss or risk
aversion cannot occur. These characteristics of the beauty-contest game constitute, in
our mind, a very favorable environment for testing the rationality of players.
B.
Individual versus team learning
Learning theories are naturally numerous in psychology and they have started to
find attention in economics as well. Due to the simplicity of the beauty-contest game
and the possibility to control for many disturbance variables in individual decision
making processes, several learning theories have been tested for the individual setting.
Nagel (1995) tests a simple ‘directional’ learning model, based on previous work of
Reinhard Selten and Rolf Stoecker (1986; see also Selten and Joachim Buchta, 1994;
Selten, 1998). Learning direction theory is related to reinforcement learning models by
implying that subjects change unsuccessful behavior in the direction of behavior which
would have been successful in the past. The reinforcement learning model by Alvin
Roth and Ido Erev (1995) captures the basic insights from psychology that choices
leading to good outcomes in the past are more frequently repeated in the future (law of
effect; Edward Thorndike, 1898) and that learning curves are relatively steep in early
periods, but flattening out afterwards (power law of practice; Julian Blackburn, 1936).
Using Nagel’s (1995) data, Dale Stahl (1996) finds the best fit to the data by combining
reinforcement and directional learning. He rejects, however, a Bayesian reasoning
model. Camerer and Ho (1999) integrate reinforcement learning models and beliefbased models into a single experience-weighted attraction (EWA) learning model
which, using data of Ho et al., performs better than the reinforcement or belief-based
learning model alone. Recently, Camerer et al. (2000) have estimated an EWA-learning
model with data from experienced subjects who participated twice in a beauty contest.
Camerer et al. find that experienced players are more likely to be sophisticated, i.e., to
anticipate how others learn.
4
Our experimental design allows to compare learning of individuals with learning of
teams. So far, no learning model has been used for such a comparison. We take a first
step in this direction by applying a relatively simple learning direction theory as a
yardstick for comparing individual and team learning.
There might be several reasons for expecting teams and individuals to behave
differently in a dynamic setting. One of them stems from social psychology research,
stating that teams are more competitive or aggressive than individuals (Bornstein and
Yaniv, 1998). But it might also be that teams are more rational in a sense that team
discussion helps to understand the strategic situation of a game faster than individuals
are able to do on their own and this superiority is revealed gradually through a team
learning process. More (and possibly competing) explanations of a strategic game might
be voiced in team discussions than individuals can (or are willing to) think through in
their mind; dominated solutions may be ruled out faster by good arguments. Therefore,
we expect teams to learn faster in the course of the game. Put differently, they should
converge faster to the game theoretic solution and apply deeper levels of reasoning
immediately after the first round of experience with the game.
The remainder of the paper is organized as follows. Section II introduces the
beauty-contest game and its game-theoretic solution. Section III reports on our
experimental design. In Section IV we present results for individuals and teams by
distinguishing between first round and consecutive rounds behavior. The learning
direction theory is confronted with the data in Section V. Finally, Section VI concludes
the paper.
II.
The Beauty-Contest Game and its Game-Theoretic Solution
In our experiment N individuals or small teams (of three subjects each)
simultaneously choose a real number from the closed interval I ≡ [0,100]. The mean of
all choices for round t is denoted xt . The winner is the individual or team whose
5
number is closest to a number x* , being defined as p ⋅xt , where p ∈ (0,1) is fixed for
all rounds and announced at the beginning of the game. We chose p = 2/3.3
This game is dominance solvable. The process of iterated elimination of dominated
strategies leads to the game’s unique equilibrium at which all players choose zero.4
Nevertheless, it has been shown by Nagel, Stahl, Duffy and Nagel and Ho et al. that a
model of iterated best reply describes subjects’ behavior better than the equilibrium
obtained by iterated elimination of dominated strategies. Classifying subjects according
to the number of steps of their reasoning, in the first round we have level-0 players
choosing arbitrarily in the given interval I, with the mean being 50, whereas level-1
players give best replies to level-0 players by choosing 50 ⋅p = 33.3&. A level-2 player
chooses 50 ⋅p² and so on. Only players with infinite steps of reasoning will choose the
equilibrium number zero.
III.
Experimental Design
In our experiments we had an ‘individual’ treatment, where individuals compete in
the beauty-contest, and a ‘team’ treatment, where teams of three subjects each compete
against each other. Experiments were conducted at the University of Innsbruck. We ran
two parallel sessions on May 11th (with 17 individuals and 17 teams, respectively) and
on June 6th 2000 (with group sizes5 of 18)6. In total, 140 subjects participated in our
experiments, providing us with 35 observations per treatment and round.7
3
Nagel (1995) has shown that players are systematically influenced by the parameter p of the game.
Since we are interested in differences between individuals and teams, but not in the influence of p, we
restrict ourselves to the single parameter p = 2/3.
4
Rational players will exclude the interval [100·p,100] because any number in this interval is
dominated by 100·p. If a rational player believes all others to be rational as well (by also excluding the
interval [100·p,100]), she will exclude [100·p²,100], and so on. Choosing zero remains the only nonexcluded strategy, given common knowledge of rationality.
5
Henceforth, we denote the number of observations per session with ‘group’ size (i.e., the sample
size) and the groups in the experimental treatment (consisting of three individuals) with ‘teams’.
6
We have chosen a relatively large group size, like in Nagel (1995), because convergence of behavior
towards the game-theoretic solution is faster in relatively large groups, since single players have less
influence on group outcome. Therefore, if we will observe systematic differences between teams and
individuals in a relatively large group, where convergence has been shown to be rather quick and stable,
6
Subjects were participants of three parallel undergraduate courses in public
economics (May) and two parallel undergraduate courses in microeconomics (June).8
Students had not been confronted with game theory in any class before participating in
the experiment. Participants in parallel courses were assigned randomly to experimental
treatments (individuals versus teams) and to teams in the team treatments.
The winner of each round in the individual treatment was paid 140 Austrian
Schillings (about 10.5 Euro or 10 US$ at that time), whereas winning teams were paid
three times the individual amount (420 Austrian Schillings). Hence, we keep the persubject monetary incentives constant across the individual and team treatments. In case
of a tie, the amount was split equally between individuals or teams. Winners were paid
privately in cash at the end of the experiment, all others received nothing.
Each session lasted at most 40 minutes and was conducted as follows. Subjects got
written instructions9, which were read aloud, offering subjects the opportunity to ask
private questions. In each session there were four rounds. In each round subjects wrote
their guesses on a separate response card. These cards were collected after each round
and numbers were read aloud and written on an overhead projector without identifying
individuals or teams. Then we calculated and announced the total sum, the average,
two-thirds of the average, and the winning number. Once this information had been
revealed, the next round was started.
Subjects in the individual treatment were isolated from one another and were not
allowed to communicate with each other. They were given up to five minutes time per
round to decide on their number.
Teams in the team treatment gathered in the Aula of the faculty, which gives room
for 500 persons. Each team sat at a separate table. The minimum distance to the next
team (table) was about 5 meters. Teams had five minutes time10 to discuss face-to-face
we can exclude differences to arise from comparatively erratic behavior in very small groups (see Ho et
al., 1998, who had groups of three and seven, respectively).
7
The raw data are available upon request from the authors.
8
Due to a specific course schedule at Innsbruck University we had no subject attending both types of
courses. Therefore, it was guaranteed that each subject could only participate in one session.
9
The one-page instructions, originally in German, are included in a translated version in an appendix
to this paper.
10
The time limit was not strictly enforced in either of the treatments. Discussions in teams were
finished in less than five minutes in most cases in round 1 and in all cases in subsequent rounds.
7
and agree on a single number to be written down on the ‘team card’ for a given round.11
Team members were requested to speak with as low voice as possible and were strictly
forbidden to speak to members of other teams.
IV.
A.
Experimental Results
First Round Behavior
The mean and median of first round chosen numbers are 34.9 and 32 for individuals
as well as 30.8 and 29.05 for teams. The mean and median for individuals are close to
36.73 and 33 reported in Nagel (1995) for first round choices. The cumulative
frequencies of guesses in round 1 are plotted in panel A of Figure 1. Individual guesses
are more evenly spread than those of teams. However, we cannot reject the null
hypothesis that both samples are drawn from the same population ( p = 0.61 , MannWhitney U-test; p = 0.32 , Kolmogorov-Smirnov test).
INSERT
FIGURE 1. CUMULATIVE FREQUENCIES OF GUESSES
Like in previous studies of the beauty-contest game, first-round choices are far
from equilibrium, and 0.01 was the smallest number chosen. Numbers below 10 are also
infrequent (6% of teams and 12% of individuals). Dominated choices (those larger than
100 ⋅p ) rarely occur; we have only one observation for teams (3%) and four
observations for individuals (11%). Contrary to most previous studies we have a
considerable number of non-integer numbers already in the first round, namely 11 and
14 for individuals and teams, respectively.
Given this evidence we have to reject our hypothesis that teams outperform
individuals with regard to the iterated elimination of dominated strategies. They do not
apply deeper levels of reasoning in the first round.
11
Team cards with more than one number on it would have been invalid. Yet, there was no such case.
8
B.
Behavior in Rounds 2, 3, and 4
In Figures 2 and 3 we plot the transitions from round t to round t + 1 for t = 1, 2,
and 3. Observations below the diagonal indicate that the chosen number in round t + 1 is
smaller than the number in round t. As can be seen, chosen numbers decline
significantly over time in both the individual and the team treatment ( p < 0.025 and
p < 0.001 for each transition in the individual and team treatment, respectively;
Wilcoxon signed-ranks test). Only 18, respectively 11, out of 105 observations lie above
the diagonal in the individual and team treatment.
INSERT
FIGURE 2. TRANSITIONS FROM ROUND T TO ROUND T + 1 – INDIVIDUALS
INSERT
FIGURE 3. TRANSITIONS FROM ROUND T TO ROUND T + 1 – TEAMS
Comparing both treatments with respect to chosen numbers in rounds 2 to 4 we find
that teams choose systematically lower numbers ( p < 0.001 in any round; MannWhitney U-test). This is also immediately clear from looking at panels B, C and D in
Figure 1, where cumulative frequencies of team guesses are systematically to the left of
individual guesses. Given the fact that there was no statistical difference in chosen
numbers in the first round, the results seem to be an indication for team learning to be
faster than individual learning.
However, the mean as well as the median of chosen numbers were already lower
for teams than for individuals in the first round. Therefore, the reason for systematic
differences in chosen numbers in rounds 2 to 4 might be due to the lower reference
point (mean of round 1) in the team treatment. To check for that possibility, we recalculated chosen numbers in rounds 2 to 4 as a fraction of the corresponding previous
round’s mean and tested whether percentages were different between individuals and
teams. Actually, teams choose systematically lower fractions ( p < 0.001 in any round;
Mann-Whitney U-test), corroborating our hypothesis that teams converge much faster
towards the equilibrium level than individuals do.
The same pattern can be detected in the percentage changes of median guesses from
round to round, which are shown in Table 1. Percentage changes of medians are
considerably larger for teams than for individuals until round 3. From round 3 to round
4 there seems to be no difference in the percentage changes of the median guess
between individuals and teams. However, this is mainly due to the fact that the medians
9
in team sessions were already very low in round 3 (2.63 and 3.74, respectively, but 9.71
and 15.74 for individuals).
TABLE 1 – MEANS AND MEDIANS OF ROUNDS 1 – 4
A. Individuals
Session 1
round
1
2
3
4
mean
39.66
21.86
12.59
6.34
median
28.40
16.50
9.71
5.30
Session 2
median(t)/
median(t-1)
0.58
0.59
0.55
mean
30.32
27.50
16.99
7.830
Median
33.00
22.00
15.74
7.60
median(t)/
median(t-1)
0.67
0.72
0.48
B. Teams
Session 1
round
1
2
3
4
mean
30.71
11.39
6.13
7.56
median
30.32
9.51
2.63
1.70
Session 2
median(t)/
median(t-1)
0.31
0.28
0.65
mean
30.86
13.94
6.24
7.18
Median
28.52
12.35
3.74
1.74
median(t)/
median(t-1)
0.43
0.30
0.47
According to the definition of step reasoning, after round 1 level-0 players will
choose, on average, the mean of the previous round, level-1 players will choose p ⋅mt − 1 ,
with mt − 1 as the previous round’s mean, and so on.
Denoting player i’s guess in round t by xi ,t , then player i’s depth of reasoning in
round t (indicating her iterated best reply) is defined as the value of d that solves
xi ,t = p d ⋅mt − 1 .
We group the continuous d values into discrete categories (d = 0, 1, 2, 3, 4) by
defining
neighborhood
intervals
for
guesses
in
round
t
with
boundaries
[ p d + 1 2 ⋅mt − 1 , p d − 1 2 ⋅mt − 1 ], the right-hand boundary for d = 0 being mt − 1 . All guesses
xi ,t > mt − 1 are aggregated into a single category with d < 0 . For t = 0, we set m0 = 50 ,
which has been shown to be a reasonable assumption in this type of beauty-contest (see
Duffy and Nagel, 1997; Ho et al., 1998).
Table 2 reports (pooled) relative frequencies of individual’s or team’s depth of
reasoning. The underlined figures represent the modal values of d. For individuals we
have either d = 1 or d = 2 as modal values, which was also the case in Nagel (1995).
This is in marked contrast to teams, for which we also have d = 3 (round 4) and even
d = 4 (round 3) as modal values. A Mann-Whitney U-test confirms the impression
10
arising from Table 2 that teams apply deeper levels of reasoning (higher d’s) than
individuals for rounds 2 to 4 ( p < 0.003 in any case). Note that the difference was not
significant for the first round ( p > 0.6 ).
TABLE 2 – RELATIVE FREQUENCIES OF DEPTHS OF REASONING IN ROUNDS 1 – 4
depth
d<0
d=0
d=1
d=2
d=3
d=4
d>4
Round 1
0.11
0.14
0.37
0.14
0.06
0.09
0.09
A. Individuals
Round 2
0.11
0.09
0.26
0.34
0.17
0.03
0.00
Round 3
0.11
0.03
0.34
0.34
0.17
0.00
0.00
Round 4
0.03
0.06
0.09
0.66
0.09
0.06
0.03
depth
d<0
d=0
d=1
d=2
d=3
d=4
d>4
Round 1
0.09
0.06
0.40
0.31
0.09
0.03
0.03
B. Teams
Round 2
0.00
0.09
0.06
0.40
0.34
0.09
0.03
Round 3
0.06
0.03
0.03
0.17
0.23
0.37
0.11
Round 4
0.06
0.00
0.03
0.23
0.29
0.20
0.20
Contrary to Nagel (1995) and Duffy and Nagel (1997), who did not find any
significant evidence that subjects employ increasing depths of reasoning over the first
four rounds of the beauty-contest game, we provide clear evidence that teams apply
increasing depths of reasoning in the transitions from round 1 to round 2 ( p < 0.01 , sign
test on whether team i's depth of reasoning increased, decreased or remained unchanged
from round t to round t + 1 ) and from round 2 to round 3 ( p < 0.01 , sign test). There is
no further increase in depth of reasoning between rounds 3 and 4. For individuals we
find no statistically significant increase in depth of reasoning from round 1 to round 3,
but in the transition from round 3 to round 4 we observe an increase ( p = 0.043 , sign
test).
Comparing individual patterns of learning with the one of teams leads to the
conclusion that teams learn faster and adapt faster to a competitive environment than
individuals do. We trace this back to the possibility of discussing the structure of the
beauty-contest game in teams. Yet, our result that even individuals increase their depths
of reasoning in the final round might be an indication that more experience with the
game can serve as a substitute for team discussion. We can, therefore, confirm our
11
hypothesis that teams learn faster than individuals in the context of the beauty-contest
game.
Nagel et. al. (2000) provide evidence that once subjects reach the second, or third
reasoning level, they often jump all the (infinite) steps towards the Nash equilibrium:
one, two, (three), infinity. In our sessions we never had any individual or team choosing
zero.12 This is a quite reasonable behavior since zero could pay off only in case all other
competitors would choose the Nash equilibrium as well. In other words, the Nash
solution is not trembling-hand proof. What is usually taken for rational behavior
(choosing Nash) represents, in fact, a boundedly rational ignorance of other players’
bounded rationality.13 In our experiments, teams – as well as individuals – correctly
predict that other competitors do not go all the way long to infinite reasoning. However,
teams proceed systematically further to the theoretical rationality threshold than
individuals immediately after the first round of experience with the game.
V.
Learning Direction Theory
Learning direction theory (proposed initially by Selten and Stoecker, 1986)
suggests that players adjust their guesses from round to round by applying an ex-post
reasoning process, taking into account the previous period’s outcome. The theory
captures the idea of reducing errors in chosen strategies by switching strategies in the
direction of higher payoffs.
We call the relation of player i’s guess in round t (xi,t) to the mean in the previous
period ( mt − 1 ) her adjustment factor ai ,t . (Remember that we set m0 = 50 .) From an expost point of view it is clear that player i would have won if she had chosen the optimal
adjustment factor aopt ,t , which is given by (see Nagel, 1995)
12
In Nagel (1995) only three subjects choose zero in the fourth round in treatment p = 0.5 and none in
p = 2/3. Therefore, our results are very similar to hers in this respect.
13
This is related to the false consensus-phenomenon in psychology, which implies that people assume
others to reason as themselves. See, e.g., Robin Dawes (1990).
12
aopt ,t =
xopt ,t p ⋅mt
=
50
50
for t = 1
xopt ,t p ⋅mt
=
mt − 1
mt − 1
for t = 2 ,3,4.
Learning direction theory then states that if ai ,t > aopt ,t ⇒ ai ,t + 1 < ai ,t , or if
ai ,t < aopt ,t ⇒ ai ,t + 1 > ai ,t . This means that a player adapts ex post her adjustment factor in
the direction of the optimal adjustment factor. Similar kinds of adjustment processes
have been successfully used to explain behavior in a wide variety of different games
(see, e.g., Michael Mitzkewitz and Nagel, 1993).
Table 3 shows the relative frequencies of changes in adjustment factors due to
experience in the preceding rounds. Contrary to Ho et al.'s (1998) rejection of learning
direction theory in their ten-round versions using considerably smaller group sizes (of
three and seven), we find that learning direction theory is a very good predictor for
changes in adjustment factors in our experiment.14 Using aggregated data for each
treatment, between 69% and 83% of individuals and between 86% and 89% of teams
adapt their adjustment ratio in accordance with learning direction theory.15,16
In any transition from one round to the next teams adapt more frequently in the
direction predicted by learning direction theory than individuals. From round 2 to round
3 this difference (31 adjustments consistent with learning direction theory for teams, 24
for individuals) is statistically significant ( χ ² = 3.05, p < 0.05 ; one-tailed).
14
One explanation for this difference is offered by Duffy and Nagel (1997) who state that learning
direction theory might only predict well when aggregate behavior adheres to a certain established time
trend, which is more likely to evolve with group sizes of 15-18 (as in Nagel, 1995) or 17-18, as in our
case, but not necessarily with group sizes of three or seven (as in Ho et al., 1998), where erratic behavior
of single players influences the time path of means much stronger.
15
This might also be an indication that participants took the experiment very serious, although it was
run in class without paying show-up fees.
16
In each of the three round-to-round transitions and in each of the four sessions the relative
frequencies of correct adjustment are greater than 0.5 (p = 0.062 for any transition; one-sided binomial
test; N = 4).
13
TABLE 3 – RELATIVE FREQUENCIES OF CHANGES IN ADJUSTMENT FACTORS DUE TO
EXPERIENCE IN PREVIOUS ROUNDS
A. Individuals
ai,t > aopt,t
ai,t < aopt,t
adjustment ratio is
decreased
increased
increased
decreased
consistent with
L.D.T.*
rounds 1 – 2
0.46
0.20
0.29
0.06
rounds 2 – 3
0.43
0.20
0.26
0.11
rounds 3 – 4
0.66
0.14
0.17
0.03
0.74
0.69
0.83
rounds 1 – 2
0.69
0.06
0.17
0.09
rounds 2 – 3
0.77
0.09
0.11
0.03
rounds 3 – 4
0.26
0.06
0.60
0.09
B. Teams
ai,t > aopt,t
ai,t < aopt,t
adjustment ratio is
decreased
increased
increased
decreased
consistent with
0.86
0.89
0.86
L.D.T.*
* Sum of the frequency numbers in boldface type, which are consistent with the predictions of learning
direction theory (L.D.T.)
We were also interested whether the absolute difference between the adjustment
factor ai ,t and the optimal adjustment factor of the previous period aopt ,t − 1 is smaller for
teams than for individuals. If that were the case it might be an indication that teams
adjust more precisely to the competitive outcome of a previous round. Indeed, absolute
differences between adjustment factors in round 2 and the optimal adjustment factor in
round 1 are significantly smaller for teams than for individuals ( p < 0.01 , MannWhitney U-test). However, for rounds 3 and 4 we find no significant differences, which
is another indication for the intuition that teams learn faster but team discussion can be
substituted by individuals,when they are more experienced with the game.17
17
We also tested whether the relative deviation δi,t of chosen numbers from the target number of the
same round ( δi,t = xi ,t xt* ) is different between both treatments. δi,t can be interpreted as a measure of
competitiveness. If, e.g., relative deviation was significantly smaller in the team treatment, that would be
evidence for the conjecture that competition for the prize is tougher between teams than between
individuals. However, we do not find any significance for that.
14
VI.
Conclusion
In psychology, the literature is ambiguous on whether teams are more rational or
whether they can learn faster than individuals. Our paper has addressed the importance
of the ‘decision maker’ and the capabilities of teams versus individuals with respect to
rationality and learning in an experimental beauty-contest game.
So far, the beauty-contest game has been studied extensively with individuals as
decision makers. Our data for individuals are in line with most previous studies.
Drawing subjects from the same subject pool we have shown that depth of reasoning
and learning is different if either individuals or teams compete against each other: (1)
Our findings do not lend support to the view that teams are more rational players in the
sense that they have deeper levels of reasoning than individuals per se. First round
behavior with respect to rationality is uniform across the individual and the team
treatments. (2) However, in the course of the experiment, teams increase their depth of
reasoning. They learn faster than individuals which might be due to the opportunity to
discuss the game before making their team decision. Another explanation for teams
learning faster might be that teams anticipate other teams to learn faster than they would
anticipate individuals to learn. (3) The learning direction theory of Selten and Stoecker
(1986) is a good predictor for both individual and team behavior. However, teams’
behavior is even better accounted for by the theory than individuals’behavior.
15
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18
APPENDIX
Instructions for the team beauty-contest game (originally in German)
Instructions for individuals are analogous and available upon request (the winning
individual received 140 Austrian Schillings per round).
Each team (consisting of three persons) has to write down a number xi in the interval [0,100] on its
response card, which contains an anonymous team code. Zero and one hundred are also possible choices.
The number need not be integer.
The winning team is the one whose number is closest to x , defined as:
n
2 ∑
x = ⋅ i =1
3 n
xi
I.e., winner is whose number is closest to two thirds of the average of all chosen numbers. [It was either
announced that n = 17 or n = 18.]
The winning team receives 420.-- Austrian Schillings. If there are more teams equally close to x , then
the prize is split equally among those teams.
After all teams have written down their number, response cards will be collected. Chosen numbers will be
announced and written down on a transparency sheet, without revealing team codes. Then we will
calculate the total sum, the average, two-thirds of the average (= x ), and we will encircle the winning
number.
There will be four rounds, so that each team has to make four separate decisions on xi . After each round
you will be informed about the decisions of all other teams and the winning number, before the next
round starts.
In each round, a team has a maximum of 5 minutes time to discuss and agree on a single number xi . If
there is more than one number on a response card, the card is invalid. We would be grateful if you could
write down your motives for choosing a certain xi on the explanation sheet at your table.
In discussions, please, speak with as low a voice as possible! You are not allowed to speak to members of
other teams!
If you have any further questions, please raise your hand and the instructor will come to you.
FIGURE 1: CUMULATIVE FREQUENCIES OF GUESSES
Cumulative frequency
A. Round 1
100
90
80
70
60
50
40
30
20
10
0
Individuals
0
10
20
30
40
50
60
70
Teams
80
90
100
Guessing range
Cumulative frequency
B. Round 2
100
90
80
70
60
50
40
30
20
10
0
Individuals
0
10
20
30
40
50
60
Guessing range
70
Teams
80
90
100
FIGURE 1: CONTINUED
Cumulative frequency
C. Round 3
100
90
80
70
60
50
40
30
20
10
0
Individuals
0
10
20
30
40
50
60
70
Teams
80
90
100
Guessing range
Cumulative frequency
D. Round 4
100
90
80
70
60
50
40
30
20
10
0
Individuals
0
10
20
30
40
50
60
Guessing range
70
Teams
80
90
100
FIGURE 2: TRANSITIONS FROM PERIOD T TO PERIOD T+1 – INDIVIDUALS
A.
Individuals
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Choices in First Round
Individuals
B.
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
70
80
90
100
Choices in Second Round
C.
Individuals
100
90
Choices in Fourth Round
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
Choices in Third Round
FIGURE 3: TRANSITIONS FROM PERIOD T TO PERIOD T+1 – TEAMS
A.
T eam s
100
90
Choices in Second Round
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Choices in First Round
B.
T eam s
100
90
Choices in Third Round
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
Choices in Second Round
C.
T eam s
100
90
Choices in Fourth Round
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
Choices in Third Round
70
80
90
100