Bentley College
From the SelectedWorks of Jeffrey A Livingston
2014
A Step Ahead? Experienced Play in the P-Beauty
Game
Jeffrey A Livingston
Susan Skeath, Wellesley College
Available at: http://works.bepress.com/jeffrey_livingston/13/
A Step Ahead? Experienced Play in the p-Beauty Game*
Jeffrey A. Livingston**
Bentley University
Susan Skeath
Wellesley College
Abstract: In the "p-beauty" contest, contestants choose a number between zero and 100 and the
winner is the player who selects the number that is closest to some fraction p of the average
chosen by the group. While the Nash equilibrium is for all players to choose 0, subjects
frequently display bounded rationality by choosing numbers that are substantially higher.
However, as the game is repeated, players learn from experience and play converges towards
prediction. This study explores how players learn by examining how players who are
experienced with the game behave when competing against opponents who they know have
never seen the game before. We find that experienced players outperform inexperienced players
on average, but there is a great deal of variation in their play and they respond to mistakes no
better than inexperienced players; accordingly, experienced players are not much more likely to
win than their inexperienced opponents. We argue that this results are in accordance with models
of learning that rely on simple pattern recognition rather than a more sophisticated process of
formation of beliefs about the strategies followed by opponents.
Keywords: bounded rationality, beauty contests, learning, experience
JEL codes: D01, C7
* Thanks to Michael Price, two anonymous referees and several seminar participants at the
Eastern Economic Association Conference and the Nordic Behavioral and Experimental
Economics Conference, all of whom provided many helpful comments and suggestions. Sharon
Lin and Brian Johns and provided excellent research assistance. Wellesley College and Bentley
University provided financial support. Of course, all remaining errors are my own.
**Corresponding author. Department of Economics, Bentley University, 175 Forest Street,
Waltham, MA 02138. Phone 781-891-2538, Fax 781-891-2896, Email [email protected]
1
I. Introduction
Game theoretic predictions of behavior rest upon a set of well-known assumptions. Under
these assumptions, when players have correct beliefs about the strategies of other players and
select the strategy that maximizes their expected playoff given those beliefs, their choices are
consistent with the predicted Nash equilibrium. However, empirical evidence, typically from
laboratory experiments, records subject behavior in some dominance solvable games that differs
from the equilibrium predictions.
One example is what Moulin (1986) dubs the p-beauty contest game (PBCG), where
contestants choose a number between 0 and 100 and win by being closest to some fraction, p, of
the average of all of the numbers chosen. As Moulin shows, the unique equilibrium of the game
is for all players to select zero. However, Camerer et al. (2004) point out that across many
experimental studies beginning with Nagel (1995), players usually select numbers above 0. For
example, when p = 2/3, the average number chosen by a group of subjects is typically between
20 and 35.
This behavior is often explained using a model of bounded rationality that incorporates
what Nagel (1995) and Stahl and Wilson (1995) term level-k thinking. In such a model, level-0
thinking implies no strategic thought or selecting a number randomly, while level-k thinking
represents the best response to a belief that all other players are following a level-(k-1) strategy.
For example, when p = 2/3, a level-1 thinker selects 33 in an effort to be close to 2/3 of the
anticipated average play of 50, and a level-2 thinker selects 22 in an effort to be close to 2/3 of
2
the anticipated average play of 33.1 Numerous experiments, including Ho, Camerer and Weigelt
(1998), Duffy and Nagel (1997) and Holt (1999), have been run using the PBCG to determine
the levels of reasoning used by players in such contests. Those results suggest that the modal
player uses level-1 thinking.2
Still, when the game is repeated, all players appear to learn from experience. Data from
iterated PBCGs in which subjects who have never played before choose numbers in a succession
of rounds show a convergence toward the Nash equilibrium choice of zero.3 The learning process
has been studied extensively, at least theoretically. Camerer and Ho (1999) describe the two
approaches taken by these modeling efforts as “belief-based” and “choice reinforcement” models
(see also Camerer, Ho and Chong (2003)).4 In belief-based models, players learn in a relatively
sophisticated fashion, forming beliefs about the strategies of their opponents and best responding
to those beliefs. Most recent theories of learning in games where level-k behavior is exhibited
1
Camerer et al. (2004) offer a related, more general model called the cognitive hierarchy model.
In their setting, "Step k" thinkers accurately predict the relative frequencies of players doing
fewer steps of thinking from levels 0 to k-1. Crawford et al. (2010) provide a review of these and
several other models of what they term "strategic sophistication," including equilibrium plus
noise, finitely iterated strict dominance and k-rationalizability (Bernheim (1984) and Pearce
(1984)), quantal response equilibrium (McKelvey and Palfrey (1995)), and noisy introspection
(Goeree and Holt (2004)).
2
Other studies show considerable heterogeneity in behavior. Economists and PhD grad students
tend to go one level deeper, exhibiting level-2 thinking by guessing numbers near 22 (see
Camerer (1997) in addition to Nagel (1995)). Cognitive ability is also associated with more
sophisticated play. Burnham et al. (2009) find that a one standard deviation increase in cognitive
ability (as measured by IQ score) is associated with a choice that is 7 to 10 points lower.
3
Nagel (1995), Alba-Fernandez et. al. (2006), Guth et. al. (2002), and Ho, Camerer and Weigelt
(1998) are among the papers that have presented evidence on this issue.
4
Many other interesting questions related to PBCG games are also discussed in the literature.
See Kocher and Sutter (2006), Kocher et. al. (2007), and Sutter (2005).
3
assume this type of sophisticated process.5 In contrast, choice reinforcement models assume
less-sophisticated players who do not have beliefs about the strategies followed by their
opponents and merely react to patterns that they witness in previous outcomes.6 To our
knowledge, neither of these learning models has been examined extensively, if at all, in the
experimental literature.
One exception is Slonim (2005) which lends some empirical support to belief-based
models of sophisticated learning.7 Slonim runs an experiment in which players participate in a
supergame of nine rounds of a PBCG, separated into three games of three rounds each, and play
against a new set of two opponents in each game. Players thus gain experience with the PBCG as
they move from round to round and from game to game. In one treatment, all three players start
the game with the same amount of experience; in the other, one player is experienced and the
other two have not seen the game before. He finds that experienced players choose lower
numbers in round one when facing experienced opponents than when facing inexperienced
opponents, often correctly predicting the guesses of their opponents. As a result, an experienced
5
Ho and Su (2013) study a level-k model of behavior in centipede games where players carefully
attempt to intuit the level-rule played by their opponents so they can play the best-response to the
behavior that this level-rule entails. Mohlin (2012) explores a more general setting where the
lowest type thinker merely best-responds to the average of past play, and higher type thinkers
develop their beliefs about how others learn using increasingly complex models.
6
Crawford (1995) and Stahl (1996) offer models of this flavor. In Stahl (1996), players follow
behavioral rules of thumb (e.g., always use level-1 thinking), learn the average payoffs
associated with various rules, and then decide which rule of thumb to follow in the subsequent
period based on that information.
7
The extant literature does include several studies of the level of thinking employed by subjects
who play the PBCG. Sbriglia (2008) shows that learning can lead to more advanced levels of
thinking. Over the course of six rounds, each round’s winning player leaves the game but gives
information about her thought process to the remaining players. The game converges more
quickly to the predicted equilibrium, and the levels of thinking advance more quickly, when
players are provided this information than when they are not. Costa-Gomes and Crawford
(2006) define several types of strategies that players might follow when playing a PBCG that are
based on level-k thinking, several of which are rational best responses to non-equilibrium PBCG
strategies. They utilize a series of 16 two-person PBCGs to decipher each player's type, and find
that many players can be neatly classified into their defined types.
4
player is far more likely to win, particularly in the first two rounds.8 Slonim argues that these
results are consistent with “the ‘sophisticated’ learning studied in Cooper and Kagel (2002),
Camerer and Ho (1998), Camerer et al. (2002), Stahl (2000) and others,” and further relates his
findings to sophisticated belief-based models where players anticipate the strategies of their
opponents.
However, Slonim’s set-up yields a strong correlation between experience and the number
a player chooses. The average first-round guess in sessions with all inexperienced players is 33.5
with a standard deviation of only 2.3. Thus, in his experiment, experience type is an excellent
predictor of the number a player can be expected to guess, resulting in a pattern of target
numbers that are relatively easy to predict without thinking carefully about the strategy that
might have led to the choice. His results are therefore consistent with both sophisticated beliefbased models and less-sophisticated choice reinforcement models.
To alleviate some of the ambiguity in Slonim’s results and to better understand how
people learn in these settings, we conduct an experiment in which players gain experience in a
session of six rounds of the PBCG and then play in another session against opponents who they
know have never played the game before. One key difference in our work is that groups of six
players, rather than three, compete against each other. This set-up results in a much larger
variance in guesses, making it more difficult to predict what the target value will be in each
round.9 We can then assess how experienced versus inexperienced players react to this variation
to test the learning-model hypotheses. Should learning be more sophisticated, as in belief-based
8
In the first round of games where one player is experienced and the others are inexperienced,
the experienced player wins 85 percent of the time. See Slonim (2005).
9
In our sample, the average first-round guess of inexperienced players is similar to that in
Slonim (35.27 compared to his 33.5) but the standard deviation in our sample is far larger (19.12
compared to his 2.3). Note that the average guess presented for Slonim’s sample is for plays
against other inexperienced players only, while the average for our sample is for all plays. The
overall average guess for Slonim’s sample is not presented in his paper.
5
models, we would expect experienced players to hold a substantial advantage over their
inexperienced opponents, as Slonim finds. If, however, players with experience are merely
reacting to patterns and results without carefully thinking about the strategies followed by their
opponents as choice reinforcement models would suggest, then they may hold less of an
advantage. When the new result does not conform to the old pattern, choice reinforcement
learners will make mistakes.
Our results, presented in detail below, turn out to be more consistent with the lesssophisticated choice reinforcement models of learning than with the belief-based models.
Interestingly, experienced players appear to have a clear edge if we look only at average play. In
each round, their average guess is close to the best-response to the average guess of their
inexperienced opponents and significantly closer to the target number; they also have less overall
variation in their guesses. However, because there is still substantial variation in the guesses of
both types, experienced players are only slightly (if at all) more likely to win in each round than
inexperienced players. Further, they are no more likely to guess a number inside the Nagel-style
“neighborhood interval” around the target number than are inexperienced players, and, perhaps
most importantly, they react no better to mistakes than do inexperienced players.
II. Experimental Design
Subjects were recruited from among students at Bentley University, where the
experiments were conducted. We conducted 32 sessions over two days, each consisting of a
group of six players playing a total of six consecutive rounds of a PBCG. Within each round of
play, subjects attempted to guess a number closest to 2/3 of the mean guess, choosing from
numbers between 0 and 100, inclusive. Compensation was provided in the form of a $5
participation payment, paid to all players in each session, and a $3 winner’s payment, paid to the
winner of each round in each session. The games were run by computer using the web-based
6
version of the PBCG provided by Charles Holt's Veconlab. As students arrived, they checked in
at a table on a different floor from the computer lab where the experiments took place. Each
subject was given a notecard with an ID number, and was instructed to log into the website using
this ID so that all choices would be anonymous. Subjects were also given a hard copy of the
experiment instructions at the registration table (see Appendix A). Once all of the subjects had
logged in, the instructions were read aloud. Subjects were then asked if they had any questions.
Once any questions were answered, they were instructed to begin the game.
Every 20 minutes four sessions were played concurrently. Thus, 24 players were in the
room simultaneously, minimizing the possibility that players could become aware of the identity
of their opponents (except for the first set of sessions where two sessions were played). In the
initial two sessions run on each day, all players were inexperienced. At the end of these sessions,
the players returned to the registration table to receive payment. At this point, four randomly
selected players were invited to participate in an additional session with the knowledge that he or
she would be competing against five opponents who had never seen the game before.
Thereafter, four more players were randomly selected from among the new inexperienced
players to participate in the next set of sessions. This process allowed us to generate
“experienced” players for each successive set of four sessions. Including the initial four allinexperienced-player sessions, we conducted a total of 32 sessions.10
10
There were some difficulties with some of the sessions due to subjects logging in to an
incorrect session. Three sessions ran with only five players, and two of these did not include an
experienced subject. One session ran with six inexperienced subjects. One session ran with two
experienced subjects and four inexperienced subjects. Data from these sessions is included in the
analysis below, but doing so has no impact on the qualitative results.
7
III. Data and Results
Summary statistics of the plays in each round are presented in Table 1.11 The average guesses
chosen in each round by all players together, by inexperienced players who played in only one
session, by experienced players, and by experienced players in their initial play (before they
gained experience) are presented in columns 1 through 4 respectively. The average target number
in each round is then presented in column 5. An initial look at the data seems to support the
conclusion that experienced players learn to anticipate the strategies of their inexperienced
opponents, and best-respond to those strategies. First, as found by Nagel (1995) and others,
players of all experience levels learn from the experience they gain from each successive round,
and their choices converge towards the Nash equilibrium play of zero. The average guess chosen
by all players falls from 33.85 in round one to 24.41 in round two and continues to decline until
it reaches 6.39 in round six. The average plays are consistent with first-level thinking: the
average guess in each round is close to two-thirds of the average choice in the previous round.
Second, despite the fact that the experienced players have observed the logical conclusion
of the game, they do not play the Nash equilibrium choice in the initial rounds of their second
session. Rather, their average choices in each round are close to the best-response to the average
choice of their inexperienced opponents (i.e., close to two-thirds of the average guess of the
inexperienced players), and accordingly are close to the average target number. For example, in
round one, the average choice of inexperienced players is 35.14, close to the play of 33 that we
would predict from players using first-level thinking. The choice of experienced players
11
The summary statistics presented here (and the analysis that follows) were calculated after
dropping seven observations from round 4 or later where a subject appeared to intentionally
disrupt the results by playing 100. These observations include one inexperienced player and one
experienced player from round 4, one inexperienced player and two experienced players from
round 5, and two inexperienced players from round 6. However, the qualitative results of all of
the analysis is unchanged if these observations are included in the sample.
8
however is 25.01 on average, roughly consistent with second-level thinking and only 2.44 above
the target number. This pattern extends through all six rounds of play. Even in round six, when
guesses have converged close to equilibrium, experienced players guess numbers that are, on
average, almost equal to the target number.
More crucial is how the experienced players react to the targets they personally observe,
relative to the inexperienced players. Table 2 presents summary statistics for both inexperienced
and experienced players12 for how close the guess comes to the previous round’s target, to twothirds of the previous round’s target (the best-response if the other players merely adjust their
guess to the previous round’s target), and to the current target. The raw data again make it appear
that experienced players enjoy a substantial advantage over their inexperienced opponents.
Inexperienced players, on average, choose numbers that are very close to their group’s target for
the previous round. Their average guesses are no more than 2.81 points away from the previous
round’s target in each round. Meanwhile, experienced players, on average, appear to bestrespond to this pattern by choosing numbers that are close to two-thirds of the previous round’s
target. Their average guesses are no more than 3.26 points away from two-thirds of the previous
round’s target in each round. Further, experienced players’ average guesses are far closer to the
target number in each round. For example, in round 1, inexperienced players choose numbers
that are 12.86 above the target on average, while experienced players choose numbers that are
only 2.77 above the target on average. Experienced players are similarly closer to the target
number on average in each subsequent round.
12
As before, the average guesses for our two groups of inexperienced players (those who later
play again as experienced players and those who do not) follow a similar pattern. Accordingly,
all inexperienced players are considered together as one group here and in the analysis that
follows.
9
Whether these differences are statistically significant is investigated using a series of
regressions of player guesses in a particular round controlling for player type. The regressions
also control for session fixed effects, so the estimation relies on how the choices of the
experienced player differ from the choices of the inexperienced opponents within each session.
The following equation is estimated separately for each round:
(1)
Dis = α + β1EXPis + β2SESSIONs + is ,
where Dis is the difference between the guess made by player i in session s and the round’s
target number in session s, EXPis is a dummy variable equal to one if player i in session s is
experienced, and SESSIONs is a vector of dummy variables indicating the session of the
experiment. The results are presented in Table 3.
In each round, the difference between the guesses of experienced players and the target
are statistically significantly smaller than the difference between the guesses of inexperienced
players (who do not play again) and the target by substantial margins. The magnitude of the
difference declines in each round as the players converge towards the Nash equilibrium. The
experienced players choose numbers that are 9.53 closer to the target number in round one, 7.05
closer to the target number in round two, 4.47 closer in round three, 2.72 closer in round four,
and 1.69 closer in round five. Each difference is significant at the five percent level or better.
Even in round six where guesses have converged fairly close to equilibrium (recall that the
10
average guess is 6.39), experienced players miss the target by 1.22 less than inexperienced
players; this difference is significant at the ten percent level.13
Further, experienced players exhibit less variation in their guesses than do their
inexperienced opponents. Table 1 shows that the standard deviation of guesses in each round is
smaller for experienced players than inexperienced players, and Table 4 reports results from a
series of interquartile range regressions where the only control variable is a dummy variable
indicating whether the subject is experienced.14 The results show that the interquartile range of
guesses is smaller for experienced players than inexperienced players in all six rounds. The
differences are significant at the ten percent level or better except in round four.
One would expect, as in Slonim (2005) where experienced players win 85 percent of the
games in round one, that these advantages would give experienced players a significant edge in
their chance of winning each round. Surprisingly, they do not enjoy nearly the same advantage in
our experiment. Table 5 presents the percentage of rounds that are won by the experienced
player, and tests whether this proportion is significantly different from 1/6 (the expected
probability of winning if all players have an equal probability of winning). The experienced
13
It is possible that the experienced players’ superior strategies may not be a result of their
experience, but rather due to inherent intelligence or skill at playing the game. To control for
this possibility, the experienced players’ choices are compared to the choices made in their initial
play, before they had gained any experience. The average choices they selected before gaining
experience are displayed in column 4 of Table 1. If their superior play was due to innate skill
rather than to experience, we would expect the choices made by experienced players in their
initial play to be similar to their choices the second time around. But this is not the case. The
numbers chosen by the experienced players are substantially lower than what they choose in their
initial play. Further, if the initial play of players who are invited back to participate again as
experienced players is considered as a separate category of player, the differences between the
choices of experienced players and their initial choices are statistically significant, and there is
otherwise no change in the qualitative results here or in any of the analysis that follows.
14
We do not control for session fixed effects because the goal of the regression is to examine
differences in the interquartile range of the different player types, which would not be possible if
we only made use of variation within each session across the choices of the six participating
players.
11
player wins only 24 percent of the plays in round one; this proportion is not statistically different
from 1/6. The winning percentage is significantly above 1/6 in round two and round six only, and
is never higher than 36 percent (the winning percentage for the experienced player in round two).
Further, despite the fact that experienced players play closer to the target number on
average than do inexperienced players (as shown in Table 3), and that their average guesses are
very close to the average target (as shown in Table 2), they are still not statistically significantly
more likely to play close to the target than inexperienced players. Nagel (1995) calculates
“neighborhood intervals” around the choices that are consistent with each level of thinking in
order to see whether choices are concentrated around those numbers.15 We follow a similar
approach by calculating the neighborhood interval around the target number for each round and
then estimating how often each type of player selects a guess in that interval. Probits of the
following form are estimated:
(2)
Pr(NIis = 1) = 1EXPis + is,
where NIis is a dummy variable that equals one if player i in session s guesses a number within
the neighborhood interval of the session s target number, and EXPis is as previously defined.
Standard errors are clustered by session.16 Panel A of Table 6 reports the estimated marginal
effect of experience and Panel B reports summary statistics on the percentage of plays that fall
within the target interval for both experienced and inexperienced players in each round. The
results show little evidence that experienced players are more likely to play in the neighborhood
interval of the target number than inexperienced players. In all rounds, except for round five, the
difference between experienced and inexperienced player guesses is statistically insignificant.
15
Each interval has the boundaries 50(2/3)i+1/4 and 50(2/3)i-1/4, where i represents the level of
thinking, rounded to the nearest integer.
16
Including session fixed effects would force the dropping of a large number of observations
because there are many sessions in each round in which no player’s guess was in the
neighborhood interval of the target number.
12
The means are also quite low; for example, in round one, only 12 percent of experienced players
choose a number in the neighborhood interval of the target. Thus, in round one, experienced
players often find that the pattern they follow fails to result in a win, and thereafter, they do no
better in adjusting their behavior to improve their chances of winning than do inexperienced
players.
So far we have seen that, on average, experienced players appear to best-respond to
inexperienced players. Yet they have only a small advantage, if any at all, in their chances of
winning the game. What explains this pattern? A closer look at the data shows that even though
experienced players outperform inexperienced players on average, there is still a substantial
amount of variation in the guesses of both types of players, making it more difficult to anticipate
the optimal guess. Figures 1 through 4 present the relative frequency distributions of the choices
made by both inexperienced players and experienced players, for round one through round four
respectively. Again following Nagel (1995), the choices are grouped by neighborhood intervals
around the choices that are consistent with each level of thinking. The interval [45, 50] is
assumed to represent level-0 thinking,17 the interval [30, 37] represents first-level thinking, the
interval [19, 25] represents second-level thinking, the interval [13, 16] represents third-level
thinking and the interval [9, 11] represents fourth-level thinking. Nagel (1995) calls intervals
between these neighborhood intervals "interim intervals," and uses the geometric mean to
determine the boundaries of adjacent intervals.
The figures illustrate vividly the extent of the variation in guesses of both player types.
For example, in round one, the modal guess of inexperienced players is in the level-1 interval,
but this accounts for only a little over 20 percent of the sample. Many others play guesses in the
17
The level-0 interval is bounded from the right by 50 since choices above 50 are strictly
dominated by choices below 50.
13
level-0, level-2 and level-3 intervals, as well as in the interim intervals. Likewise, the level-2
interval contains the largest percentage of the guesses of experienced players, but this accounts
for less than 40 percent of the sample. Similar patterns describe behavior in subsequent rounds.
This variation in guesses results in extensive variation in the target numbers in each
round. Figures 5 through 8 present relative frequency distributions of the target numbers in
rounds one through four, respectively. In each figure, the middle interval is the Nagel-style
neighborhood interval around the average target number for that round. Consider round one for
example. Only 37.5 percent of the target numbers (12 out of 32) fall in the neighborhood interval
of the average target, and this percentage declines in each subsequent round.
Because of this variation in target numbers, mistakes are common, as the analysis
presented in Table 6 shows. Our final analysis explores how players react to these mistakes. If
experienced players learn by discovering how to better anticipate the strategies of their
opponents, as they do in belief-based models of relatively sophisticated learning, we would
expect them to correct mistakes in the following rounds more effectively than do inexperienced
players. However, if they do not learn to anticipate strategies and merely follow patterns as in
choice reinforcement models of learning, then they are not likely to improve upon mistakes any
better than do inexperienced players.
To examine this issue, we explore how close players are to the target in the current round
as a function of how far they were from the target in the previous round. We estimate the
following equation:
(3)
where the dependent variable
,
is the absolute value of the difference between the guess of
player i in session s and the target seen by that player in the current round,
is the absolute
value of the difference between the guess player i in session s in the previous round and the
14
target seen by that player in the previous round, and the other variables are as previously
defined.18
Not surprisingly, players who are further from the target in previous rounds tend to miss
the target by larger amounts in the current round. If learning is sophisticated, however, then these
misses should be smaller for experienced players than for inexperienced players. Table 7
presents the results for rounds two through six. As expected, in round two for example, a one
point increase in the distance of a player’s guess from the target in the previous round is
correlated with a 0.21 point increase in the distance of a player’s guess from the current target. In
addition, there is little evidence that experienced players react better to mistakes than do
inexperienced players. The interaction term between the experience indicator and distance from
the previous target is insignificant in all but round 5, where the effect for experienced players is
actually larger rather than smaller. Thus, given similar-sized misses of the target in the previous
round, experienced players do not appear to play any closer to the target than do inexperienced
players in the next round. The results are consistent with experienced players learning by choice
reinforcement—responding to an initially observed pattern—and thereafter doing no better in
their attempts to anticipate the strategies of their opponents than do inexperienced players.
V. Conclusion
In many experimental studies across several types of games, players frequently act in
ways that are not in accordance with Nash equilibrium predictions. However, just as frequently,
play converges towards Nash equilibrium as players gain experience with such games. These
results are sometimes taken as evidence that behavioral anomalies found in experimental settings
are less important than they seem at first. If players learn to behave in the way that theory
18
In these regressions, we drop observations of winners since winners may not consider whether
they could have done better. However, using the entire sample yields qualitatively identical
results.
15
predicts, then the theoretical models of behavior likely need not account for how they behave
before gaining experience.
The precise type of learning evidenced by players is thus crucial to the conclusion that
the theoretical models remain valid. If players learn in the sophisticated, rational manner
assumed by belief-based models, in which they form beliefs about the strategies employed by
their opponents and best-respond accordingly, this conclusion seems justified. However, if
players learn via relatively unsophisticated processes by simply following rules of thumb and
observing patterns which happen to lead to behavior that is consistent with prediction, then
modeling efforts that account for the behavioral anomaly are likely still warranted.
We find evidence from beauty contest games that supports the latter point of view. In six
player PBCGs where one player is experienced and the others are inexperienced, on average the
experienced player appears to have learned to anticipate the strategies of the inexperienced
players and best-respond accordingly. A deeper look at the data reveals that this is not the case.
There is wide variation in the guesses of experienced players in each round, leading to frequent
mistakes and little if any increase in the chances that the experienced player wins the round.
Further, when these mistakes happen, experienced players do not improve upon their strategies
any better than do inexperienced players. This suggests that any learning that comes from
experience is not of the sophisticated, belief-based type envisioned in the literature and that
experience does not help players think more carefully about the strategies their opponents follow.
16
round:
1
Table 1: Summary statistics: guesses by round and player type
experienced,
experienced,
all players
inexperienced
second play
initial play
target number
(1)
(2)
(3)
(4)
(5)
33.85
35.14
25.01
35.98
22.57
(18.36)
(20.01)
(8.39)
(13.90)
(5.56)
2
24.41
(14.32)
25.68
(15.51)
18.08
(6.56)
24.15
(11.83)
16.28
(4.89)
3
17.09
(13.26)
18.18
(14.27)
12.17
(5.84)
16.32
(12.24)
11.39
(4.85)
4
11.39
(10.52)
11.64
(11.28)
8.67
(5.22)
12.61
(9.92)
7.59
(4.42)
5
8.11
(8.27)
7.91
(7.64)
5.68
(4.68)
11.49
(12.42)
5.41
(4.01)
4.51
(5.00)
10.65
(15.73)
4.26
(3.96)
6
6.39
5.92
(8.82)
(7.24)
Standard deviations in parentheses.
17
Table 2: Summary statistics, differences between guesses and targets
Round:
1a
Previous round’s target
Inexperienced Experienced
1.94
-8.33
(19.12)
(8.39)
Average distance of guess from:
(2/3)×previous round’s target
Inexperienced Experienced
13.05
2.79
(19.12)
(8.39)
Current target
Inexperienced Experienced
12.65
2.77
(17.09)
(9.34)
2
2.81
(13.54)
-4.15
(6.15)
10.35
(13.77)
3.26
(5.81)
9.13
(13.03)
1.97
(6.07)
3
1.58
(12.60)
-3.93
(4.31)
7.01
(12.85)
1.43
(4.38)
6.43
(11.87)
1.13
(5.60)
4
0.38
(9.59)
-2.67
(2.83)
4.19
(9.83)
1.11
(3.05)
4.17
(8.88)
1.33
(4.11)
5
0.82
(7.15)
-1.52
(3.59)
3.37
(7.39)
0.88
(3.48)
2.99
(6.35)
0.86
(2.36)
3.02
(8.09)
1.19
(3.79)
2.37
(7.23)
0.67
(2.86)
6
1.18
-0.47
(7.80)
(3.67)
a
In round 1, “previous target” is defined as 50×(2/3)
Standard deviations in parentheses
18
Table 3. Do experienced players come closer to the target number?
round one round two round three round four round five
(1)
(2)
(3)
(4)
(5)
Experienced
-9.10***
-6.88***
-4.11***
-2.53**
-1.90***
(1.88)
(1.51)
(1.47)
(1.21)
(0.67)
Constant
16.51***
(6.17)
15.25**
(6.61)
10.55***
(3.51)
7.71***
(1.84)
4.56***
(1.00)
round six
(6)
-1.51*
(0.80)
2.33***
(0.55)
Observations
188
188
188
184
186
186
R2
0.14
0.13
0.19
0.27
0.23
0.28
Dependent variable is the difference between the player's guess and the round's target number.
Regressions also control for session fixed effects
Robust standard errors in parentheses
*** significant at the 1% level; ** significant at the 5% level; * significant at the 10% level
19
Table 4. Do experienced players have less variation in their guesses?
round one round two round three round four round five
(1)
(2)
(3)
(4)
(5)
Experienced
-20.95*** -10.20***
-7.61**
-3.44
-5.01***
(4.07)
(3.05)
(3.22)
(2.74)
(1.83)
Constant
27.10***
(2.89)
17.97***
(1.56)
12.85***
(1.16)
9.99***
(1.22)
9.61***
(0.89)
round six
(6)
-3.41**
(1.47)
7.50***
(0.79)
Observations
188
188
188
184
186
186
Dependent variable is the interquartile range of the players’ guesses.
Bootstrapped standard errors in parentheses
*** significant at the 1% level; ** significant at the 5% level; * significant at the 10% level
20
Table 5. How often is the winner the experienced player?
Round:
1
2
3
4
5
6
0.24
0.36***
0.24
0.22
0.25
0.32**
(0.44)
(0.49)
(0.44)
(0.42)
(0.44)
(0.48)
*** the proportion of experienced winners is significantly different from 1/6 at the 1% level;
** the proportion of experienced winners is significantly different from 1/6 at the 5% level;
* the proportion of experienced winners is significantly different from 1/6 at the 10% level.
21
Table 6. Are experienced players more likely to guess in the neighborhood interval of the
target than inexperienced players?
Panel A. Regressions
Experienced
Observations
round one
(1)
0.04
(0.07)
round two
(2)
0.13
(0.09)
round three
(3)
0.02
(0.08)
round four
(4)
-0.03
(0.07)
round five
(5)
0.17*
(0.09)
round six
(6)
0.04
(0.08)
188
188
188
184
186
186
round two
(2)
0.27
(0.45)
round three
(3)
0.15
(0.37)
round four
(4)
0.13
(0.34)
round five
(5)
0.28
(0.46)
round six
(6)
0.19
(0.40)
Panel B. Summary statistics
Experienced
round one
(1)
0.12
(0.32)
Inexperienced
0.08
0.14
0.13
0.16
0.11
0.16
(0.27)
(0.34)
(0.34)
(0.36)
(0.31)
(0.36)
Dependent variable is the probability that player i’s guess is in the neighborhood interval of the
target.
Regressions also control for session fixed effects
Marginal effects are reported
Robust standard errors in parentheses
*** significant at the 1% level; ** significant at the 5% level; * significant at the 10% level
22
Figure 1. Relative frequency distributions of neighborhood interval choice, round 1
level 1
level 2
level 0
level 3
level 4
level 2
level 1
level 4
level 3
level 0
23
Figure 2. Relative frequency distributions of neighborhood interval choice, round 2
level 2
level 1
level 3
level 4
level 0
level 2
level 3
level 4
level 1
level 0
24
Figure 3. Relative frequency distributions of neighborhood interval choice, round 3
level 2
level 4
level 3
level 1
level 0
level 4
level 2
level 3
level 1
25
level 0
Figure 4. Relative frequency distributions of neighborhood interval choice, round 4
level 4
level 3
level 2
level 1
level 0
level 1
level 0
level 4
level 2
level 3
26
Figure 5. Relative frequency distribution of target numbers, round 1
* neighborhood interval around the average round 1 target number (22.41)
27
Figure 6. Relative frequency distribution of target numbers, round 2
0.3
Relative Frequencies
0.25
0.2
0.15
0.1
0.05
0
less than 9
9 to 12
12 to 15
15 to 18*
18 to 21
21 to 24
Target Numbers
* neighborhood interval around the average round 2 target number (16.17)
28
> 24
Figure 7. Relative frequency distribution of target numbers, round 3
* neighborhood interval around the average round 3 target number (11.37)
29
Figure 8. Relative frequency distribution of target numbers, round 4
* neighborhood interval around the average round 4 target number (7.60)
30
Table 7. Reaction to missing the target in the previous round
round two round three round four round five
(2)
(3)
(4)
(5)
distance from target, previous round
0.21***
0.29**
0.18*
0.16
(0.07)
(0.14)
(0.10)
(0.14)
round six
(6)
0.43***
(0.16)
experienced player
-5.88
(4.27)
-4.76
(3.24)
-0.94
(1.82)
-2.84*
(1.45)
-1.16
(1.19)
distance last round×experienced
0.12
(0.33)
0.35
(0.41)
-0.34
(0.59)
0.46*
(0.27)
0.24
(0.44)
14.44**
(5.90)
7.14**
(3.38)
6.51***
(2.38)
4.22***
(1.41)
0.91
(1.05)
Constant
Observations
156
153
150
152
2
R
0.20
0.33
0.36
0.30
Dependent variable is the distance of player i's guess from the target number.
Regressions also control for session fixed effects
Robust standard errors in parentheses
*** significant at the 1% level; ** significant at the 5% level; * significant at the 10% level
31
152
0.43
Appendix A. Instructions
Experiment Instructions
This is an experiment in economic decision-making. The experiment consists of a series
of six (6) rounds. You will play against a group of 5 other people in each round. The decisions
that you and the 5 other people make will determine the dollar winnings for each of you. Each
player will be paid $5 for participating.
At the start of each round, you will be asked to choose a number between 0 and 100,
inclusive. 0 and 100 are possible choices. Your number can include up to two decimal places,
such as 12.34 or 56.78. At the same time, each of the other 5 people will also choose a number
between 0 and 100. None of you will be able to see anyone else’s number until after your
decision is submitted.
The numbers selected by all 6 people in your group will be averaged, and then the
number that is two-thirds (0.67) of that average will be calculated and announced at the end of
the round.
The person whose number is closest to two-thirds of the average will win $3 for that
round. The 5 other people will earn $0.
If more than one person ties for having a number closest to two-thirds of the average,
then the payment of $3 will be divided equally among those who tied and the others will earn $0.
The website will keep track of the choices of each player in each round. It will also
calculate the target number (two-thirds of the average of the numbers chosen by the 6
participants), identify the winner or winners of each round, and keep track of each player’s
winnings over the six (6) rounds of play.
After the end of the final round, you will be required to complete a short online survey.
Upon completing the survey and logging out of the website, you will present your code card at
the table upstairs near the main door of Smith and collect your winnings. At that time, you will
also need to sign a receipt confirming the amount of the payment that is made to you.
32
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