Teaching Nash Equilibrium and Strategy Dominance

Teaching Nash Equilibrium and Strategy Dominance
Documento de trabajo
E2003/47
Teaching Nash Equilibrium
and Strategy Dominance:
A Classroom Experiment on
the Beauty Contest
Virtudes Alba Fernández
Pablo Brañas Garza
Francisca Jiménez Jiménez
Javier Rodero Cosano
Consejería de Relaciones Institucionales
centrA:
Fundación Centro de Estudios Andaluces
Documento de Trabajo
Serie Economía E2003/47
Teaching
Nash
Equilibrium
and
Strategy
Dominance: A Classroom Experiment on the Beauty
Contest*
Virtudes Alba Fernández
Pablo Brañas Garza
U. Jaén & centrA U.
Jaén, centrA & LINEEX
Francisca Jiménez Jiménez
Javier Rodero Cosano
U. Jaén, centrA & LINEEX
centrA & LINEEX
RESUMEN
El objetivo de esta investigación es mostrar cómo el uso de experimentos de clase
puede ser un buen instrumento pedagógico para la enseñanza del concepto de
Equilibrio de Nash. El juego utilizado para nuestros propósitos es una versión repetida
del juego del concurso de belleza (BCG), un simple juego de adivinanzas cuya
repetición permite que los estudiantes reaccionen a las opciones que elijan sus
contrincantes de forma que se converja hacia la solución de equilibrio. Llevamos a cabo
el experimento con estudiantes de carrera sin conocimiento previo de teoría de juegos.
Tras cuatro rondas, se observó en todos los grupos una clara tendencia decreciente en
el número medio escogido, por lo que podemos afirmar que a través de la repetición
del BCG, los estudiantes aprenden rápidamente cómo alcanzar la solución de equilibrio.
Palabras clave: Experimentos de clase, Juego de Concurso de Belleza, Enseñanza,
Equilibrio de Nash.
ABSTRACT
The aim of this investigation is to display how the use of classroom experiments may
be a good pedagogical tool to teach the Nash equilibrium (NE) concept. The basic game
for our purposes is a repeated version of the Beauty Contest Game (BCG), a simple
guessing game whose repetition lets students react to other players’ choices and to
converge iteratively to the equilibrium solution. We performed this experiment with
undergraduate students without any previous knowledge about game theory. After four
rounds, we observed in all groups a clear decreasing tendency in the average chosen
number. So, our findings prove that, by playing a repeated BCG, students quickly learn
how to reach the NE solution.
Keywords: Classroom Experiments, Beauty Contest Game, Teaching, Nash
Equilibrium.
JEL classification: A22, C99, D83
*
Authors wish to thank Rosemarie Nagel, Curro Martínez Mora and Quique Fatás for their helpful
suggestions; also, Ana & Pili López López for their contribution in the data gathering and
processing. We gratefully acknowledge financial support from University of Jaén R+D program (#
20210/148).
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Reasons for this study
Over the last years, Game Theory (GT) has become an essential part of
intermediate microeconomics theory. Obviously the key concept in Game
Theory is the Nash Equilibrium (NE). Therefore, this powerful notion is an
inevitable part of our teaching.
However, educators know that it is not easy to explain NE in class. Students usually find difficulties in understanding this idea and, consequently,
the implicit features of this concept.
Generally, the teaching of GT begins with the study of strategic interactions among players. Once they understand that each player’s payoffs depend
not only on their own actions but also on those by other players, the next step
is to explain the process of elimination of dominated strategies. For this, we
only require two basic assumptions: rationality (maximization) and common
knowledge (of rationality). Both concepts let us solve (dominance solvable)
games and also to predict some particular behavior. However, there exist
games that are not dominance solvable even if they have a NE. In this case,
NE requires an additional hypothesis: common knowledge of players’ beliefs
about rivals actions. So, the best response become the suitable mechanism
to reach the solution. Best response is the best strategy that a player may
choose given the strategies chosen by his rivals. The NE is reached when all
agents play (some of) their best response. So, NE is self-enforcing because
no player has incentives to deviate from it.
Clearly, the use of NE eliminates the circular reasoning (player 1 thinks
that player 2 thinks that player 1 thinks . . . ).
This paper proposes a pedagogical tool to simplify the teaching of NE:
successive repetitions of a dominance solvable game, the Beauty Contest
Game (BCG). This particular game is very useful to show the two prior
procedures intuitively. Concretely, we perform four repetitions of it as a
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classroom experiment, previous to a game theory teaching class.
Our findings clearly illustrate that students unexperienced (in GT) apply
at each round both iterative elimination of dominated strategies and bestreply behavior to the previous choices. On average, we observe a recursive
approximation to the Nash prediction, which we name learning effect. Therefore, using the repetition of this static game guarantees us that students learn
how to solve it in some way or the other.
Theoretical Background
Since the introduction of the concept in Nash (1951), the Nash equilibrium
is not only the standard tool for the economic scientific community but also
the basis for a systematic teaching of the discipline in the modern era.
In game theory, it is a common practice to assume that individuals “magically” choose a set of actions such that all the (infinitively recursive) predictions become true. So as to reach this hard-to-believe solution, theorists can
follow one of the two original Nash interpretations: agents are either perfectly rational, or there is an evolutionary equilibrium1 . As it is well known,
the first one relies on the assumption that all agents are able to compute
the game equilibria and they reach one of them. Instead, the second one
presupposes the existence of a big population of simple people that play the
game in an evolutionary framework: they pick up one strategy randomly,
if the outcome is good they will repeat it again (or they disappear if the
strategy is bad). After some time, the game will converge to one of the Nash
equilibriums. The BCG is a good example of the underlying convergence to
both interpretations. From a rational point of view, it is not credible that
the subjects will solve this problem (few people can at the first try!) but if
1
Although Nash only uses the rationality explanation in the published version due to
space limitations.
2
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it is repeated, most of the times the solution converges to equilibrium.
Beauty Contest Game is a simple guessing game that facilitates the evaluation of the individuals’ level of reasoning. The basic BCG is as follows: a
certain number of subjects are invited to play a game and simultaneously all
of them have to choose a number from an interval (generally [0, 100]). The
winner is the player whose number is closest to p-times the mean of all the
chosen numbers, being p > 0. The winner receives a fixed prize, the losers
get nothing. Under these rules the unique Nash Equilibrium is zero for all
players2 .
Figure 1, taken from Ho et al. (1998, pg. 951), shows the convergence
to the zero theoretical solution from a dominance iterative point of view3 .
Any number chosen between 66.6 and 100 is dominated by 66.6 (100 ∗ 2/3, if
p = 2/3), so they say that the interval [66.6, 100] corresponds to an irrational
behavior (R(0) for us). Rational individuals will always choose a number in
the [0, 66.6] interval. Applying the same reasoning, R(1) players will choose
a number below 66.6 (but above 44.4). Since 44.4 will dominate again any
number between 44.4 and 66.6, we say that any number below 44.4 (and
above 44.4 · 2/3 = 29.6) corresponds to a R(2) individual. Following this
iterative reasoning level process ad infinitum, we get the theoretical Nash
equilibrium (0, with R(∞)). Then, this game is dominance solvable4 .
2
This is not as obvious as it looks, Bosch-Domènech et al. (2000) tries an one-shot BGC
with game theoretics. The answers were very different to zero but better than the usual
ones!
3
See Rapoport and Amaldoss (2000) for an experimental study of iterative elimination
of (strongly) dominated strategies.
4
In the real world, there is even a higher rationality level. An individual who knows
the zero-solution by any of these methods, can guess that most people would not achieve
the zero solution, so the “intelligent” individuals will link their answer to their estimation
of the average rationality level. We can call this rationality level ∞-plus. Grosskopf and
Nagel (2001) argue that most individuals think that other people are not fully rational
and this is the reason why the equilibrium is not reached immediately.
3
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R(4) R(3)
···
0
13.2 19.7
29.6
R(2)
R(1)
44.4
R(0)
66.6
100
Figure 1: Iterated reasoning of individuals by eliminating dominated strategies
The BCG original idea was first mentioned by Keynes (1936) when he
wanted to express that a clever investor has to “anticipate the basis of conventional valuation a few months hence, rather than . . . over a long term of
years” (page 155), so she could act in the stock market before the rest of
investors do.
The formal game model was introduced by Moulin (1986). As we noted
above, the unique equilibrium of the game is 0 for p < 1 and it is obtained
by iterated elimination of the weakly dominated strategies.
After this basic framework, some experimental researchers started an investigation area on BCG or “p-beauty” (see Ho et al., 1998). The first experimental study is found in Nagel (1994, 1995)5 . Other works have been carried
out by Bosch and Nagel (1997a,b); Bosch-Domènech et al. (2000); Duffy and
Nagel (1997) and Ho et al. (1998). See also Nagel (1998) for a survey of the
literature.
There are different BCG experiments. Sometimes subjects are students,
other times professors or newspaper readers (which presumably have different
education levels). Some studies are one-shot, others are repeated, communication versus non-communication, laboratory versus field experiments.
Generally, BCG has been run with individual subjects. However Kocher
and Sutter (2001) compares the individual versus group behavior in this
5
The main purpose of Nagel (1994, 1995) was to contrast an iterated best-reply domi-
nance model (IBRB)
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type of game. Concretely they find that groups do not apply deeper levels
of reasoning but that they learn faster instead.
Repetition permits the individuals to learn dynamically from other people’ expected reasoning behavior. Our experiment is very similar to Ho
et al. (1998). In their design the information is based on the previous period
choices. Therefore, the learning process in Ho et al. (1998) is based on an
evolutive game. Instead, Weber (2003) argue that learning in the BCG could
happen even without feedback. Even when subjects do not receive any information between periods, authors found that there is convergence towards
the NE of the game.
Over the last years, the research on the BCG has returned to the original
idea of Keynes. Hirota and Sunder (2002) explore experimentally whether
price bubbles in security markets are generated by a beauty-contest mechanism: investors have to create beliefs about the others’ beliefs (second-order
beliefs) and these beliefs depend on the third-order beliefs which in turn depend on the fourth-order beliefs, and so on, if the dividends are paid beyond
their personal investment horizons. They conclude that when the realization dividend is distant and well beyond the investors investment horizon,
investors find difficult to induct the fundamental value of securities from the
future to the present (backward induct). This difficulty gives rise to price
bubbles because, in this case, investors adjust their expectations on the basis
of observed prices (forward induction).
Up to now, the BCG has been used to study the depth of reasoning level
of individuals. However in this paper we propose a new application of this
framework in this paper: classroom experiments.
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The experiment
The experiment involves a repeated version of the Beauty Contest Game in
several groups of students. The performance of this classroom experiment
is useful for the teaching of Nash Equilibrium; specifically to the iterated
elimination of dominated strategies. Given that Nash equilibrium is a general
topic in both intermediate microeconomics and industrial organization (and,
obviously, in Game Theory too), we expect that this pedagogical tool helps
educators.
Three aspects are crucial to run this experiment:
1. The size of the group should be large enough to reduce the effect of
any individual guess on the average (of guessing numbers); on the other
hand, very large group causes increasing monitoring cost to the instructor6 .
Nagel (1995) uses 12 and 17 subjects in each group; Ho et al. (1998)
reduces the group size to 7 subjects and 3 subjects (like Kocher and
Sutter, 2001). Following these papers we use small (5-6 subjects) groups
and large groups (10-11 subjects)7 . Each subject will play a repeated
BCG against the rest of the member of his group. There is not any
relation among groups; that is, each group has its own game.
2. In order to motivate students you should design a mechanism of rewards. In this classroom experiment we gave the winner of each round
0.25 extra-credit points as an additional mark in the midterm exam
of the final grade. In the case of several winners at the same round
(within the same group) the prize was split among them.
6
Furthermore, Ho et al. (1998) illustrates that reduced groups (3 subjects in its re-
search) need longer time to converge.
7
It is difficult to reproduce identical size groups in class. Usually, the instructor does
not have the exact number of students required.
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Our experiment was performed in Intermediate Microeconomic Theory during the 2003 course (spring semester) in three different classes:
Business (morning and afternoon groups, B-1 and B-2) and Business
+ Law (one single group, B+L). All groups shared identical subject
handout, lecturer and marks system (ranging from 0 to 10). Note that
this is an important subject in both degrees.
3. Last, another important issue is the number of rounds that experimental subjects have to play the BCG. As our argument is for students
learn Nash equilibrium through the iterated elimination of dominated
strategies, we need some rounds to observe this learning effect. Kocher
and Sutter (2001) proves that 4 rounds are enough to approximate the
theoretical prediction. Finally, our reduced group size guarantees that
four rounds are enough to observe learning.
The performance of this experiment lasts about an hour and the postexperimental session is during following half an hour8 . Therefore, the whole
class takes over 1.5 hours.
Procedures
1. The instructor must define the size of each group. It is irrelevant if one
group differs slightly in its size with respect to the others; Nevertheless,
it would be desirable to get similar sizes. Our results will show that 10
is an adequate number.
2. Given any defined size, the instructor selects randomly one monitor per
group from the students pool. They will be awarded 0.25 points.
The monitor will help the instructor recording and monitoring the experiment in his group. Each monitor will be given a calculator and
8
Note that it is worth having both sessions consecutively.
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a “monitoring sheet” (see the appendix) on which he will track the
guessed number given by each experimental subject (of his group).
Note that this monitoring sheet is crucial to explaining results after
the experimental session. Since the average and winning numbers are
recorded on this sheet, the instructor can summarize the results immediately. These easily illustrate basic concepts of Game Theory as
iterated elimination of dominated strategies.
3. The third task is to create groups, that is, to randomly allocate each
experimental subject within the group so as to avoid that friends sit
together.
4. In order to facilitate monitor’s tasks, the instructor should locate each
group in one row (or column).
5. The lecturer must discuss instructions and procedures with the monitors. Afterwards, monitors give the instruction sheet and the individual
sheet (see appendix) to the experimental subjects.
6. The instructor must explain instructions orally. It is important to avoid
numerical examples. In this kind of games the generation of focal points
is immediate. Any doubt is answered publicly. Experimental subjects
are informed that any communication is absolutely forbidden9 .
7. When everybody is fully aware of the rules, the experiment can begin.
Round 1: Experimental subjects must fill their guess number in their
answer sheet. Then, the monitor collects all guessing numbers and
calculates mean and 2/3 ∗ mean. By comparing this value to those
reported by the experimental subjects, he determines the winner. The
9
Subjects are required to keep the maximum level of confidentiality for their own sake
(note that if any subject knows his rivals guess he can use best reply rule).
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whole group is informed about the mean, 2/3 ∗ mean and the winner
guess number (not the winning subject).
Recall that, in order to analyze these results with the students during the post-experimental session, the monitors should record all this
information in the decision sheet.
8. Round 2: Experimental subjects are then informed that they will play
another round. The procedure is identical to round 1. The subjects
are informed that this second round is independent of round one.
9. Round 3 and 4: Identical procedures.
Post Experimental Discussion
Immediately after the experimental session, we begin the post-experimental
discussion. This second part introduces the theoretical background which
involves three topics:
a. First of all, we present students graphically the iterative elimination
of dominated strategies underlying the BCG by means of graphics (1).
By explaining this process, students can understand that if rationality
is common knowledge, nobody will choose a number within the interval
[66.6 − 100] because this subset of numbers is dominated by the [0 −
66.6] set. Following this reasoning ad infinitum —as it was explained
previously— students fully capture this idea.
b. Once common knowledge rationality is clearly defined, it is time to
explain best response. Students observe that their own performance
depends on their beliefs about other players’ actions. Furthermore,
they realize that the other players’ behavior also relies on beliefs about
their own actions and so on.
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As an example, we present some preliminary results of their own responses. We can use some monitoring sheets (see appendix) to illustrate
the average behavior of these groups. Since the average chosen number
is decreasing in the course of the successive rounds, students realize
that GT approximately predicts their performance.
c. Now we have all the required ingredients to cook the Nash Equilibrium
concept. Since students have experienced the recursive “dynamics” of
the NE, we explain them what the abstract concept implies and how
this can be reconciled with the real world. In the theoretical model, we
suppose that everything is instantaneously adjusted; the BCG must be
explained as a slow motion picture: the repetition lets individuals reach
the theoretical solution step by step. Perfectly rational agents do not
need this, but nobody is fully rational. This is the second remarkable
point: repetition could be a good substitute for rationality.
Finally, as an anecdote, we told about the Nash movie, A Beautiful Mind.
The film tries to explain the NE to the regular audience with an allegorical
sequence. We see John with friends in the pub when some girls appear; if
all the fellows tried to get off with the most beautiful girl, all the females
would fly away and none of the males would get her. The BCG can be seen
as a similar symbolic game: individuals have to compete to outguess the rest
of the participants, but without cooperation, costly errors take place. The
knowledge of game theory concepts will reduce these costs10 .
10
It is important to remind students about the importance of the intellectual achieve-
ment they have reached: John Nash won the Nobel Prize thanks to this simple discovery.
Myerson (1999) gives an interesting historical perspective of this portentous innovation.
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Results
The experimental BCG was performed in three different sessions: two samples of undergraduate first-course students in Business (B-1 and B-2) and
a sample of undergraduate first-course students in a mixed Business+Law
degree (B+L). Table 1 shows the distribution of subjects within groups participating in the experimental sessions.
Number of Subjects per group & session
Session B-1
Session B-2
Session B+L
Group 1
10
6
10
Group 2
11
6
8
Group 3
9
6
9
Group 4
9
5
8
Group 5
10
6
10
Group 6
10
6
—
Session Pop.
59
35
45
Table 1: Distribution of groups size
Our whole sample consists of 139 subjects. Each session consists of 6
(5 in B+L) groups of students. All groups performed four rounds of the
BCG. As our main objective is the analysis of a classroom procedure and
not the analysis of experimental results, we will only consider the data at an
aggregate level. Table 2 shows the average guessed number (and standard
deviation) in terms of sessions and rounds.
As one might expect the average of chosen numbers —in each session—
is decreasing throughout the successive rounds11 . This is true for the three
cases. Although we should observe that the path to the convergence (to zero)
11
Although the initial average observed in our experiment is similar to that of Kocher
and Sutter (2001), the final one is very different! Their averages are approximately around
7 while ours are clearly higher (from 9.7 to 19.1).
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Session B-1
Session B-2
Session B+L
Round Average St. Dev. Average St. Dev. Average St. Dev.
1
38.885
19.348
20.782
17.611
37.067
18.891
2
32.087
14.235
17.060
12.362
29.008
11.554
3
22.660
16.750
13.042
7.318
22.712
11.577
4
14.419
13.360
9.748
7.503
19.142
13.944
Table 2: Average number and standard deviation per round
is not identical. Figure 2 illustrates each experimental session trend (data
from table 2).
At first sight, sessions B-1 and B+L behave similarly but B-2 trend shows
a flatter slope. As our main interest is to study the learning effect we should
analyze the average variation round after round. We define speed of convergence as: (µi − µi−1 )/µi−1 being µi the average guessed number in round i
(i = 1, . . . , 4). This speed is shown in absolute value in table 3.
B-1
B-2
B+L
(µ2 − µ1 )/µ1
0, 17 0, 18
0, 22
(µ3 − µ2 )/µ2
0, 29 0, 24
0, 22
(µ4 − µ3 )/µ3
0, 36 0, 25
0, 16
Table 3: Speed per Rounds and Sessions
Interestingly, all sessions share a similar initial speed of convergence close
to 20%. From round 2 to 3 (and from 3 to 4) this speed varies dramatically
in each group. So, we observe learning effects in all rounds and sessions.
After this descriptive approximation, we analyze statistically the results
obtained. Remember that our main objective is to contrast if playing a
BCG repeatedly a certain number of times induces students to modify their
behavior towards the Nash equilibrium. Therefore, we expect the average
number in a round to be different to the following one and, moreover, the
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50
40
30
20
10
0
round 1
round 2
round 3
B-1
B-2
round 4
B+L
Figure 2: Evolution of the session averages
difference between one round and the following one to be greater than zero,
that is, to be decreasing. In order to contrast that, we formulate as null
hypothesis:
H0 : µij − µkj = 0, i < k and i, k = 1, ..., 4
and as alternative hypothesis:
H1 : µij − µkj > 0, i < k and i, k = 1, ..., 4
In the appendix we develop the statistical analysis in depth. Summarizing, we check if each population follows the Normal distribution; given
that some of them do not follow it, we use two alternative statistical tests
(parametric and non parametric ones) to test our null hypothesis. Our main
results show that:
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• In session B-1 we do observe learning effect in the whole experiment.
• In session B-2 we do observe learning effect in the last two rounds.
• In session B+L we also observe learning during all the whole rounds.
Consequently, these results indicate that this tool is powerful to induce
students to reach the Nash Equilibrium.
Final Remarks
In this paper we have formulated an interesting classroom experiment: a
repeated version of the Beauty Contest Game. Our experience is that this
kind of games are a good way to introduce the intricateness of the equilibrium
concept. Not only does it permits students to see the mechanism in action
but also to appreciate the difference between a theoretical abstraction and
the real world dynamics towards equilibrium.
The BCG is one of those tricky games worth playing: although it is
plain simple in its formulation and the solution is always obvious ex-post, by
amusing oneself in the game one realizes the difficulties of outsmarting other
people.
The classroom experiment needs some careful preparation to make conveniently mixed groups. However, once everything is ready to start, if the
monitors are well trained, rounds should run quite smoothly and increasingly
fast. Although for our purposes four rounds are enough, it would be easy to
reach ten rounds if necessary. Continuous repetitions could be very rewarding for students, as this will let most of them to reach for themselves the
theoretical solution before the proper explanation is given.
Once the BCG is done and the post-experimental session is concluded
we can run other experiments such as the Traveler’s dilemma (that we ac-
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tually did). Students realize how game theory training helps them to solve
interesting (and sometimes lucrative) puzzles.
Appendix
Experimental Instruments
All these instructions were originally written in Spanish.
Instructions
You are going to participate in a microeconomic experiment. You have been
randomly assigned to a (6-10) person group. In this game you will have to
make decisions repeatedly in four rounds. Your partners will be the same
during the four stages. Moreover, your group has been assigned to a monitor
that will control the procedure.
The rules of the game in each period are the following: you should choose
a (integer or decimal) number in the interval [0, 100]. Zero and one hundred
are allowed choices. Once the monitor has collected the choices of your group,
the winner will be determined. The winning number is the closest to 2/3 of
the average of all group numbers:
2
x= ·
3
n
i=1
n
xi
.
The winning prize will be 0.25 extra points in each round for your final Micro
II exam. If two or more people are equally close to x, the prize will be split
equally among them. A person so lucky as to guess the right number the
four times will get a whole point!
First you must write down the group and code you were given at the top
line of the attached strip. Then at each round, you must select your number.
When all of you have finished, the monitor will pick up the responses. Afterwards, you will receive back your strip with the additional information: the
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average of your group, 2/3 of this and the winning number. This procedure
will be repeated four times.
If you have any question, please raise your hand and the instructor will
come to you. You are not allowed to speak during the experiment.
Monitoring Sheet
Monitor:
Group:
Full Name
Session:
Round 1
Subject 1
Subject 2
Subject 3
Subject 4
Subject 5
Subject 6
Subject 7
Subject 8
Subject 9
Subject 10
Average
2/3 of average
16
Round 2
Round 3
Round 4
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Individual Sheet
Code:
Group:
First Round
Group Average:
Your Choice:
2/3 of the average
Winning Number:
Second Round
Group Average:
Your Choice:
2/3 of the average
Winning Number:
Third Round
Group Average:
Your Choice:
2/3 of the average
Winning Number:
Fourth Round
Group Average:
Your Choice:
2/3 of the average
Winning Number:
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Statistical Analysis
Let Xij be the number chosen by a student at round i in session j, where
1 = 1, ..., 4 and j = 1, 2, 3. Let Fij (X) be the distribution function associated
to each variable Xij .
First, we contrast if each Fij (X) follows a normal distribution with average µij and standard deviation σij . For that, we use the Kolmogorov-Smirnof
goodness-of-fit test. The following table shows the results:
B-1
B-2
B+L
Round
Z-statistic
p-value
Z-statistic p-value Z-statistic p-value
1
1.116
0.165
0.937
0.344
1.135
0.152
2
1.063
0.208
0.713
0.690
0.681
0.742
3
1.579
0.014(*)
0.384
0.998
0.792
0.556
4
1.285
0.074
0.709
0.696
0.867
0.440
(*) significant at 5%
Table 4: Kolmogorov-Smirnof goodness-of-fit results
As we can see in table 4, individual choices adjust to a normal distribution
except for the first session (B-1) at the third round of the game.
Using a parametric test for the equality of means between paired rounds,
the results are the following:
B-1
t-stat.
B-2
p-value t-stat.
B+L
p-value t-stat.
p-value
Round 1 vs. 2
2.315
0.012
1.252
0.109
2.951
0.002
Round 2 vs. 3
-
-
2.107
0.021
3.148
0.001
Round 3 vs. 4
-
-
2.859
0.003
1.758
0.042
Round 2 vs. 4
7.586
0
-
-
-
-
Table 5: Parametric test for the mean equality
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Observe that for the first session (B-1) at the third round it is not possible
to apply this test because it does not follow a normal distribution.
Table 5 shows that the difference between successive average choices is
statistically positive in all cells except for the first two rounds in the second session (B-2). In this last case, the null hypothesis of mean equality is
accepted.
So as to involve the third round of the first session in the statistical
analysis, we apply the Wilcoxon non-parametric test using the same former
hypothesis. Results are summarized in the following table:
B-1
Z-stat.
B-2
p-value Z-stat.
B+L
p-value Z-stat.
p-value
Round 1 vs. 2
-2.378
0.008
-1.077
0.1414
-2.467
0.006
Round 2 vs. 3
-4.612
0
-2.001
0.022
-2.871
0.002
Round 3 vs. 4
-5.225
0
-3.459
0
-3.45
0
Table 6: Wilcoxon non-parametric test for the mean equality
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