Application of Pascal’s Triangular Plus Monte Carlo Analysis to Appraise the Wisdom of Crowds

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/application-of-pascals-triangular-plus.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/student-life/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, Cut-offs, 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
0	1186
3.55E-06	1192
1%	1195
10%	1197
40%	1201
90%	1210
100.00%	1210

10% error
Cut offs	Weight
0	1078
1.45E-53	1180
1%	1188
10%	1196
40%	1208
90%	1257
100.00%	1318
20% error
Cut offs		Weight
0%		958
1.2E-105		1172
1%		1184
10%		1195
40%		1212
90%		1284
1		1438

30% error
Cut offs	Weight
0	838
1.5E-157	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 Average 1213.951188 STDEV 1.096048697 CV 0.000902877 CV (max) 0.10% CV (Ave.) 0.000954337 STDEV (ave.) 1.158530439 STDEV (max) 1.216094929 Weight (Ave.) 1213.96 Real weight 1198 Error -1.33% 100 people Average 1213.854 STDEV 3.335317 CV 0.002748 CV (max) 0.28% Cv (average) 0.26%
50people
Average		1214.281
STDEV		4.601357
CV		0.003789
CV (max)		0.39%
Cv (average)		0.38%

10people Average 1213.519 STDEV 10.28537 CV 0.008476 CV (max) 0.87% Cv (average) 0.84%

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.”

Application of Pascal’s Triangular Plus Monte Carlo Analysis to Calculate the Risk of Expected Utility

Introduction

Following to the article of “Application of Pascal’s Triangular in Corporate Financial Strategy” on link: http://emfps.blogspot.com/2012/04/application-of-pascals-triangular-in.html and the article of “Monte Carlo Analysis on Case of Nike, Inc.: Cost of Capital” on link: http://emfps.blogspot.com/2012/01/monte-carlo-analysis-on-case-of-nike.html,  the purpose of this article is to use a new simulation model which has been made by the combination of the Pascal’s Triangular and Monte Carlo method for calculating the risk of Expected Utility. The most important finding is to track and control the changes of the risk and Expected Utility when we change our trials. Another finding is to improve our primary estimation about the probability for Utility function extracted from some macroeconomic indicators such as inflation rate, unemployment, CPI, PPI, GDP, interest rate and so on. As the matter of fact, I have proved that by increasing of the trials, we will have the constant Expected Utility but the risk of Expected Utility will decrease. Finally, I am willing to tell you that one of the best ways to solve the complex problems (Multi Dimension problems) by using of Monte Carlo Simulation Model (due to the limitation of CPU of our PC) is to breakdown the problem and to utilize the combination of many methods such as Markov Chain, Fuzzy Logic and so on accompanied by Monte Carlo Simulation Model. I think that this simulation model can be used by someone who have been involved in business of  Saving Energy, Stock Markets, Forex trading, Shopping malls such as Carrefour, Giant Hypermarket, IKEA, AEON, Mydin, Parkson, SOGO, Tesco and so on but the most important usage is to estimate the risk of deficit financing  in macroeconomic where it was implied in article of “Case Analysis of GAINESBORO MACHINE TOOLS CORPORATION (CON): A New Financial Simulation Model” on link: http://emfps.blogspot.com/2012/04/case-analysis-of-gainesboro-machine.html

Expected Utility Theory

In the reference with Wikipedia on below link: http://kleene.ss.uci.edu/lpswiki/index.php/Expected_Utility_Theory

The definition of Expected Utility Theory, Utility and Expected Utility Calculations are as follows:

Expected utility theory is a tool aimed to help make decisions amongst various possible choices. It is a way to balance risk versus reward using a formal, mathematical function.

When faced with a number of different choices, expected utility theory recommends that you calculate the expected utility of each choice and then choose the one with highest expected utility.

Utility

Utility is simply a measure of a person's preferences amongst different things. From these preferences (if they are rational!) we can deduce a utility function which represents preferences by order relations between numbers.

This only works if a person's preferences are, in a certain sense, rational. If someone prefers the Angels to the Dodgers, then they shouldn't also prefer the Dodgers to the Angels. And, if they prefer the Angels to the Dodgers, and the Dodgers to the Giants, then they shouldn't prefer the Giants to the Angels. We also require that, for any two things, a person prefers one to the other, or is indifferent between the two.

If a person's preferences are rational in the above sense, then we can define a utility function as follows:

u(⋅) is a function that assigns numbers to things (represented by variables x,y,z,…)

u(x)>u(y) if and only if this person prefers x to y

u(x)=u(y) if and only if this person is indifferent between x and y

The numbers assigned by u(⋅) should also match how much one thing is preferred to another. If someone assigns u(Angelswin)=100 and u(Dodgerswin)=1, then they'd prefer to see an Angels win 100 times more than a Dodgers win.

Example of Utility Functions

Suppose we want to create a utility function for a fan of the "Planet of the Apes" movies. There were five movies in the series: "Planet of the Apes", "Beneath the Planet of the Apes", "Escape From the Planet of the Apes", "Conquest of the Planet of the Apes" and "Battle for the Planet of the Apes".

Our fan likes "Escape" the best out of the five, prefers "Escape" to "Beneath," is indifferent between "Beneath" and "Conquest," prefers "Beneath" to "Planet," and prefers "Planet" to "Battle." Here is a utility function that could represent our fan's preferences:

u(Escape)=10,u(Beneath)=u(Conquest)=8,u(Planet)=5,u(Battle)=1

One way to think of utility is in terms of how much you would pay for each of these things, or how much these things are worth to you.

Expected Utility Calculations

How appealing a certain choice is depends not only on the payoffs of that choice, but how likely those payoffs are. The multi-million dollar payoff of a lottery is certainly appealing, but it is so unlikely that buying a lottery ticket is virtually a waste of money. Expected utility calculations are meant to balance risk versus reward.

We think of an act (like buying a lottery ticket) as having a number of possible outcomes (in this case, winning or losing). Given a person's utility function (see above) and their degrees of belief in each of the possible outcomes, we can figure out the expected utility of any act. This is done as follows:

Let the act in question be labelled A. Let o1,o2,…,on be the various possible outcomes of A (there needs to be at least one outcome, but there could be many).

To each outcome oi is an associated probability Pr(oi) which measures how likely that outcome is, and a utility u(oi) which measures that outcome's spot in this person's preference relation.

The expected utility of A is:

E(A)=u(o1)⋅Pr(o1)+u(o2)⋅Pr(o2)+…u(on)⋅Pr(on)

Now, when faced with a choice between multiple acts A1,A2,…An, expected utility theory says that a person should choose the act with the highest expected utility. That is, calculate E(A1),E(A2),…E(An) and then choose the act with the highest associated utility.

Therefore, if we obtain the probability and Utility function for each outcome, we are able to calculate Expected Utility for each action.

Methodology

The methodology to make this simulation model has been explained step by step as follows:

1) To define the acts

2) To determine the numbers of the total trials or the numbers of the possible outcomes

3) To make the appropriate trials of Pascal’s Triangular in reference with the numbers of the total trials (the total possible outcomes) by using of Excel spreadsheet (referred to step 2)

4) To calculate the total amount of outcomes for each possible outcome

5) To calculate the total sum of outcomes for the total trials

6) To divide step 4 to step 5 for each possible outcome to obtain accumulative probability

7) To find out the consequence of all possible outcomes (which is the same the Binomial Probability Distribution) by using of accumulative probability (step 6)

8) To find a range for Utility function by using of marketing research

9) To estimate a primary probability for each Utility function (primary range) by using of macroeconomic indicators such as PPI, CPI, GDP, interest rate, unemployment rate, inflation, exchange currency and so on

10) To calculate Cut-offs or accumulative primary probability (which is utilized to conduct a Monte Carlo analysis)

 11) Using of the Rand formula = Rand () on our excel spreadsheet for all Utility functions

12) Using of the VLOOKUP formula = VLOOKUP (Rand cell, Cut-off, 2) for all Utility functions

13) To multiply each probability of the possible outcome to each Utility function

14) To sum all results obtained from step 13

15) 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 200 or 400 or 1000 and independent variable is the result of step 14

16) To calculate the standard deviation (STDEV), coefficient of variation (CV) and the average of Utility function.

Finding and discussion

In this case, I started by an example as follows:

1) I defined the acts as The Purchasing and Selling

2) Referring to the article of “Application of Pascal’s Triangular in Corporate Financial Strategy” on link: http://emfps.blogspot.com/2012/04/application-of-pascals-triangular-in.html, I considered the total trials or the total possible outcomes in four levels as follows:

Ø  999 trials (possible outcomes) where the total number of the people (participants) were 997 plus Purchasing and Selling

Ø  500 trials (possible outcomes) where the total number of the people (participants) were 498 plus Purchasing and Selling

Ø  100 trials (possible outcomes) where the total number of the people (participants) were 98 plus Purchasing and Selling

Ø  10 trials (possible outcomes) where the total number of the people (participants) were 8  plus Purchasing and Selling

Here is the consequence of the outcomes for each level in the format of binomial probability distribution curve as follows:

-The level of 999 trials:

- The level of 500 trials:

-The level of 100 trials:

- The level of 10 trials

3) I considered the primary range for Utility function and probability for Purchasing and Selling as follows:

Probability of Purchasing			U (Oi)P
	0.1		30
	0.15		35
	0.35		40
	0.4		45
		Cut offs	U (Oi)P
		0	30
		0.1	35
		0.25	40
		0.6	45

Probability of Selling			U (Oi)S
0.1		45
0.15		40
0.35		35
0.4		30
	Cut offs	U (Oi)S
	0	45
	0.1	40
	0.25	35
	0.6	30

As you can see, the probability for Purchasing is exactly vice versa with Selling.

The Results

I calculated the standard deviation (STDEV), coefficient of variation (CV) and the average of Utility function for each level of trial as follows:

-The level of 999 trials:

Purchasing
STDEV (Ave.)						0.657515
CV (max)						0.017365
CV (average)						0.016342
E (Pur.) Total Ave.						40.23528
Selling
STDEV (Ave.)				0.649111
CV (max)				0.019577
CV (average)				0.018688
E (Pur.) Total Ave.				34.73502

- The level of 500 trials:

Purchasing

STDEV (Ave.)		0.784547
CV (max)		0.02199
CV (average)		0.019497
E (Pur.) Total Ave.		40.23969

Selling

STDEV (Ave.)		0.784866
CV (max)		0.024444
CV (average)		0.022575
E (Sellig) Total Ave.		34.76781

-The level of 100 trials:

Purchasing

STDEV (Ave.)			1.175161
CV (max)			0.03183
CV (average)			0.029202
E (Pur.) Total Ave.			40.24228

Selling

STDEV (Ave.)		1.143864
CV (max)		0.034977
CV (average)		0.032905
E (Sellig) Total Ave.		34.76331

- The level of 10 trials:

Purchasing

STDEV (Ave.)		2.066192
CV (max)		0.053684
CV (average)		0.051285
E (Pur.) Total Ave.		40.28859

Selling

STDEV (Ave.)		2.043727
CV (max)		0.063607
CV (average)		0.058879
E (Sellig) Total Ave.		34.7068

I compared the changes of risk for each level of possible outcomes where this finding has been included in below diagram:

As you can see, the risk will decrease if we increase the level of possible outcomes while the Expected Utility will be the constant for each level of possible outcomes.

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.”

 

How many gallons of gasoline would it take to charge an iPhone?

How many gallons of gasoline would it take to charge an iPhone?

Pressure drop field

You can review below link which is about System Advisor Model (SAM):

https://sam.nrel.gov/content/pressure-drop-field

Here is also very fascinating report for who are interested in working on Energy Saving field:

http://www.unene.ca/un803-pwp/UN0803_System_Thermal_Hydraulics_Written_Report.pdf