Following to the article of “Application
of Pascal’s Triangular Plus Monte Carlo Analysis to Calculate the Risk of
Expected Utility” on link: http://emfps.blogspot.com/2012/05/applicationofpascalstriangularplus.html, the purpose of this article is to appraise the wisdom of crowds
by using of a simulation model which is the combination of Pascal’s Triangular
method and Monte Carlo simulation model. The most important finding is to be
proved the research study conducted in 1906 by the British scientist Sir
Francis Galton (1822 – 1911) in the case of the Wisdom of Crowds in which we
can say there is a very good adaptability between the mathematical
models and Human’s sense. I think that we can also use from this simulation
model for Market Predictions.
According
the book of “The Wisdom of Crowds, Why the Many Are Smarter Than the Few and
How Collective Wisdom Shapes Business, Economics, Societies, and Nations”
by James Surowiecki (2004), he stated the story of research study conducted by Sir Francis Galton in 1906 in related to the wisdom Crowds. I have
uploaded the details of this story for your convenience as follows:
British Scientist Sir Francis Galton (1822 – 1911)
In the meanwhile, you can also read
this story on below link:
http://www.hsph.harvard.edu/studentlife/orientation/files/the_wisdom_of_crowds_reading.pdf
Methodology
Let me explain you the methodology
and the procedure of making this simulation model step by step as follows:
1) I used the data included in the
story of the research study done by Sir Francis Galton in 1906 as my example to
make this simulation model where the real weight of ox was equal to 1198 pounds
and average weight of ox announced by 787 people (participants) was equal to
1197.
2) I assumed the percentage of
errors equal to 1% (plus/minus), 10% (plus/minus), 20% (plus/minus), and 30%
(plus/minus) where we have:
1% error 10% error 20% error 30% error
Plus 1209.98 1317.8 1437.60 1557.40
Minus 1186.02 1078.2 958.40 838.60
3) The difference between Plus and
Minus will be equal to the total trials (the possible outcomes) as follows:
1% error 10% error 20% error 30% error
Trials (n) 24 240 480 719
As the matter of fact, each number
of trials between 0 and 24, 0 and 240, 0 and 480, 0 and 719 has been assigned
to each weight of ox between 1210 and 1186, 1318 and 1078, 1438 and 958, 1557
and 838. There are two situations in which we assign zero to the least weight;
in this case, we will have all variances from right side of real weight (all
percentages are negative). Another situation is that we assign zero to the most
weight, in this case, we will have all variances from left side of real weight
(all percentages are positive). It does not make any different.
4) According to the total trials
(the possible outcomes), I used from Pascal’s Triangular to find nominal
coefficient (C (n, r)) respectively.
5) To calculate the total amount of
outcomes for each possible outcome
6) To calculate the total sum of
outcomes for the total trials
7) To divide step 5 to step 6 for
each possible outcome to obtain cumulative probability
8) To find out the consequence of
all possible outcomes (which is the same the Binomial Probability Distribution)
by using of accumulative probability (step 7)
9) Using of the Binomial Probability
Distribution to make the basic feeds for Monte Carlo Simulation Model. In this
case, we should make all consequences as the assigned probability or cumulative
probability (cut –offs) for using into our Monte Carlo Simulation Model. If we
want to make very exact Cut – Offs, we can utilize from excel formulas of
“= NORMINV (Probability, mean,
STDEV) and = NORMDIST (x, mean, STDEV, true)”
But, at the first we should
normalize our Binomial Probability Distribution by using of below formula
instead of “mean and STDEV” as follows:
mean = n / 2 , n =
trials
STDEV = ((n) ^0.5) / 2
10) After making Cut – Offs by using
of step (9), Using of the Rand formula = Rand () on our excel spreadsheet for
all participants (787 people)
11) Using of the VLOOKUP formula =
VLOOKUP (Rand cell, Cutoffs, 2) for all participants (787 people)
12) To calculate the average of all
participants (787 people), 100 people, 50 people, and 10 people
13) The increase iterative
calculations by using of a two –way Table just like Sensitivity analysis in
which row is the numbers of 1 to 10; column is the numbers of 1 to 400 or 1000
and independent variable is the result of step 12
13)
To calculate the average of ox’s weight, Standard deviation (STDEV),
Coefficient of Variation (CV) or Risk percentage and Variance of the
real weight (%) for 1% error (plus/minus), 10% error (plus/minus), 20%
error (plus/minus), 30% error (plus/minus) and also for 787 participants, 100
participants, 50 participants, and 10 participants.
Finding
and discussion
According to the step 1 to 9, I calculated all Cut – Offs as
follows:
1% error


Cut offs

Weight

1186


3.55E06

1192

1%

1195

10%

1197

40%

1201

90%

1210

100.00%

1210

10%
error


Cut offs

Weight


1078


1.45E53

1180


1%

1188


10%

1196


40%

1208


90%

1257


100.00%

1318


20%
error


Cut offs

Weight


958


1.2E105

1172


1%

1184


10%

1195


40%

1212


90%

1284


1

1438


30%
error


Cut offs

Weight

0

838

1.5E157

1166

1%

1180

10%

1194

40%

1215

90%

1304

100%

1557

Regarding
to the step 10 to 13, the final results are below cited:
30% Error
787 people
100 people


50people


Average

1214.281


STDEV

4.601357


CV

0.003789


CV (max)

0.39%


Cv (average)

0.38%

10people


20% Error
787 people
Average

1209.92


STDEV

0.792926


CV

0.000655


CV (max)

0.07%


CV (Ave.)

0.0007


STDEV (ave.)

0.846342


STDEV (max)

0.891621


Weight (Ave.)

1209.892


Real weight

1198


Error

0.99%

100 people
Average

1211.146


STDEV

2.659586


CV

0.002196


CV (max)

0.22%


Cv (average)

0.21%

50 people
Average

1211.209


STDEV

3.708314


CV

0.003062


CV (max)

0.32%


Cv (average)

0.31%

10 people
Average

1210.929


STDEV

7.793489


CV

0.006436


CV (max)

0.70%


Cv (average)

0.67%

10% Error
787 people
Average

1207.204


STDEV

0.655779


CV

0.000543


CV (max)

0.06%


CV (Ave.)

0.000531


STDEV (ave.)

0.640865


STDEV (max)

0.6879


Weight (Ave.)

1207.215


Real weight

1198


Error

0.77%

100 people
Average

1207.214


STDEV

1.74593


CV

0.001446


CV (max)

0.16%


Cv (average)

0.15%

50 people
Average

1207.27


STDEV

2.682693


CV

0.002222


CV (max)

0.23%


Cv (average)

0.21%

10 people
Average

1206.959


STDEV

5.516085


CV

0.00457


CV (max)

0.51%


Cv (average)

0.47%

1% Error
787 people
Average

1200.064927


STDEV

0.144784429


CV

0.000120647


CV (max)

0.01%


CV (Ave.)

0.01%


STDEV (ave.)

0.141034486


STDEV (max)

0.145644979


Weight (Ave.)

1200.072824


Real weight

1198


Error

0.17%

100 people
Average

1200.078


STDEV

0.399969


CV

0.000333


CV (max)

0.04%


Cv (average)

0.033%

50 people
Average

1200.086


STDEV

0.552151


CV

0.00046


CV (max)

0.05%


Cv (average)

0.047%

10 people
Average

1200.144


STDEV

1.314832


CV

0.001096


CV (max)

0.11%


Cv (average)

0.107%

I
compared the changes of Error (plus/minus) to Risk (CV) on below graph:
As
you can see, there is a significant increase on Risk from 1% error to 10% error
then it will go up slowly.
In
the case of the changes on error to variance of real weight and average weight,
I can say that the results are the same above mentioned. Please see below
graphs:
The most
important finding is about the changes of Risk to increase of participants
(people) on below graph:
This
graph shows that the Risk will sharply decrease, if the participants (people)
increase more than 50 people. This result is always the constant for the
changes on error’s percentages. I can say that the research study conducted by Sir
Francis Galton is approximately approved, if the range of error does not
increase more than 1%.
Note:
“All spreadsheets and calculation notes are available. The people, who are
interested in having my spreadsheets of this method as a template for further
practice, do not hesitate to ask me by sending an email to: soleimani_gh@hotmail.com or call
me on my cellphone: +989109250225. Please be informed these spreadsheets are
not free of charge.”