Understanding Utility Function And Standard Gamble For Assessing Utility Values
- December 21, 2023/ Uncategorized
Utility Function and Its Assessment
Utility function is a twice differentiable function of wealth which satisfies the principle of non-satiation and risk aversion. We can assess the utility function by mainly two methods, that is interview method and probability encoding. In interview method, a person is asked a series of questions from which his/her utility function is assessed while in probability encoding probabilities are attached to uncertain possible outcomes depending on an individual’s preference.
A standard gamble is a technique for managing outcomes by attaching probability values to an individual’s preferences. The standard gamble is used in determining utility values by asking a series of leading questions the answer to which will help determine the utility values of the individual as they indicate an individual’s attitude towards a particular risk.
- Decision matrix
|
Good |
Fair |
Bad |
|
|
Share market |
11400 |
10800 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
- Optimist criterion
|
Good |
Fair |
Bad |
Maximum outcome |
|
|
Share market |
11400 |
10800 |
10000 |
11400 |
|
Government bonds |
10900 |
10900 |
10900 |
10900 |
The optimist will take the option of the maximum outcome which is to trade shares under good market which its outcome is $11400
- Pessimist criterion
|
Good |
Fair |
Bad |
Minimum outcome |
|
|
Share market |
11400 |
10800 |
10000 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
10900 |
The pessimist will take the option of maximum in the possible minimum outcome which is to trade in government bond in all seasons which results in $10900
- Regret criterion
|
Good |
Fair |
Bad |
Minimum outcome |
Maximum outcome |
|
|
Share market |
11400 |
10800 |
10000 |
10000 |
11400 |
|
Government bonds |
10900 |
10900 |
10900 |
10900 |
10900 |
Regret criterion can choose to go for both the minimax and maximin take the option of trading in government bond which gives $10900.
- Expected monetary value
|
Good |
Fair |
Bad |
|
|
Share market |
11400 |
10800 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
|
Probability |
0.4 |
0.2 |
0.4 |
The optimal action in the expected monetary value will be to trade in government bond which its outcome is $10900.
- Expected value of perfect information
|
Good |
Fair |
Bad |
|
|
Share market |
11400 |
10800 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
|
Probability |
0.4 |
0.2 |
0.4 |
The highest expected monetary value is $10900
Expectation for maximizing profit
11100
Knowing the direction the market will go is worth $200
- The probability of unfavorable market is 0.5
Since the expected monetary value is $20000 he can go ahead and start the producing new type of razor.
- The friend’s probability
|
Favorable |
Unfavorable |
|
|
Correct prediction |
0.7 |
0.2 |
|
Incorrect prediction |
0.3 |
0.8 |
The prior probabilities are 0.5 for favorable market and 0.5 for unfavorable market
The joint and marginal probability will be;
|
Favorable |
Unfavorable |
||
|
Correct prediction |
0.35 |
0.1 |
0.45 |
|
Incorrect prediction |
0.15 |
0.4 |
0.55 |
|
0.5 |
0.5 |
Posterior probability
|
Favorable |
Unfavorable |
|
|
Correct prediction |
0.78 |
0.73 |
|
Incorrect prediction |
0.22 |
0.27 |
- Posterior probability
Posterior probability when the friend predict unfavorable market is 0.22
- In engaging friend; under correct prediction
Expected value of sample information (EVSI) = emv with sample information – emv without sample information.
emv with sample information =
= 19920
Emv without sample information
He should not engage his friend because the value of the information is less than the cost of the information.
|
MODEL |
|
|
|
|
|
|
|
|
|
|
|
Selling |
|
Profit |
Fixed |
|
|
Month |
RN1 |
Demand |
Price |
RN2 |
Margin |
Costs |
Profit |
|
1 |
0.23297 |
107 |
$180 |
0.22763 |
20% |
2000 |
6408 |
|
2 |
0.794052 |
147 |
174 |
0.233362 |
30% |
2000 |
7956 |
|
3 |
0.395857 |
139 |
163 |
0.40283 |
70% |
2000 |
11536 |
|
4 |
0.320458 |
196 |
174 |
0.240464 |
13% |
2000 |
3164 |
|
5 |
0.245738 |
134 |
165 |
0.360996 |
90% |
2000 |
24192 |
|
6 |
0.422416 |
153 |
178 |
0.890802 |
40% |
2000 |
12408 |
|
7 |
0.038463 |
148 |
176 |
0.452313 |
90% |
2000 |
19890 |
|
8 |
0.730945 |
168 |
178 |
0.737935 |
80% |
2000 |
23780 |
|
9 |
0.503711 |
192 |
171 |
0.878873 |
40% |
2000 |
8514 |
|
10 |
0.726539 |
149 |
171 |
0.221567 |
60% |
2000 |
18795 |
|
11 |
0.179755 |
141 |
166 |
0.241994 |
90% |
2000 |
15998 |
|
12 |
0.412426 |
105 |
163 |
0.142019 |
30% |
2000 |
7164 |
|
DATA |
|||||||
|
Prob |
Cum prob |
Demand |
Selling |
Price |
$180 |
$220 |
|
|
0.05 |
0 |
100 |
Monthly |
Fixed cost |
$2,000 |
||
|
0.1 |
0.05 |
120 |
Profit |
Margin |
22% |
32% |
|
|
0.2 |
0.15 |
140 |
|||||
|
0.3 |
0.35 |
160 |
|||||
|
0.25 |
0.65 |
180 |
|||||
|
0.1 |
0.9 |
200 |
|
MODEL |
|||||||
|
Selling |
Profit |
Fixed |
|||||
|
Month |
RN1 |
Demand |
Price |
RN2 |
Margin |
Costs |
Profit |
|
1 |
0.23297 |
192 |
$187 |
0.22763 |
20% |
2000 |
4700 |
|
2 |
0.158694 |
103 |
218 |
0.479955 |
30% |
2000 |
8259 |
|
3 |
0.179437 |
126 |
186 |
0.672223 |
70% |
2000 |
17550 |
|
4 |
0.285063 |
187 |
191 |
0.671108 |
13% |
2000 |
2978 |
|
5 |
0.482977 |
187 |
197 |
0.800577 |
90% |
2000 |
37080 |
|
6 |
0.636752 |
132 |
199 |
0.133095 |
40% |
2000 |
12408 |
|
7 |
0.395724 |
168 |
203 |
0.651491 |
90% |
2000 |
19890 |
|
8 |
0.19685 |
116 |
214 |
0.334298 |
80% |
2000 |
23780 |
|
9 |
0.226749 |
113 |
188 |
0.495291 |
40% |
2000 |
8514 |
|
10 |
0.592614 |
150 |
185 |
0.459049 |
60% |
2000 |
18795 |
|
11 |
0.223638 |
123 |
199 |
0.774456 |
90% |
2000 |
15998 |
|
12 |
0.15323 |
142 |
210 |
0.031765 |
30% |
2000 |
7164 |
Report of the changes observed
It can be seen than the changes cause the increase of profit from $159805 to $177116. This will bring about the benefit to the company as it obvious the profit margin also increased. The drastic effect it can cause is that with time the increase in price of the tires can lower the demand leading to low sales which will eventually lower the profit. This can force the profit margin to reduce. Some players can come in the market to take the opportunity to sell tires at relatively lower price which can bring stiff competition lowering the profits, demand even further. Therefore, increasing the varieties of tires customers can choose from can make them understand the increase in prices setting in a continuous trend which can be realized by customers in later days. This will ensure continuous demand keeping our increase in profit margin. (Working Group on Radiative Corrections and Monte Carlo Generators for Low Energies, 2010)
- Regression between Overhead cost against machine hours
The cost equation I from Overhead cost and Machine hours is
From the above table we see the p-value is less than 0.05 which shows that in this case the variable Machine Hours is statistically significant and can be used for future projections
The regression analysis between Overhead cost and batches
|
Regression Statistics |
||||||||
|
Multiple R |
0.999144163 |
|||||||
|
R Square |
0.998289059 |
|||||||
|
Adjusted R Square |
0.998098954 |
|||||||
|
Standard Error |
6299.920723 |
|||||||
|
Observations |
11 |
|||||||
|
ANOVA |
||||||||
|
df |
SS |
MS |
F |
Significance F |
||||
|
Regression |
1 |
2.08417E+11 |
2.08417E+11 |
5251.262 |
9.17576E-14 |
|||
|
Residual |
9 |
357201010.1 |
39689001.12 |
|||||
|
Total |
10 |
2.08775E+11 |
||||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
119.1919192 |
2317.767593 |
0.051425311 |
0.96011 |
-5123.962644 |
5362.34648 |
-5123.96 |
5362.346482 |
|
X Variable 1 |
267.345679 |
3.689277512 |
72.46559201 |
9.18E-14 |
258.9999535 |
275.691405 |
259 |
275.6914046 |
The cost equation I from Overhead cost and Batches is
From the above table we see the p-value is less than 0.05 which shows that in this case the variable Batches is statistically significant and can be used for future projections
The regression analysis between Overhead cost both Machine hours and Batches
|
SUMMARY OUTPUT |
||||||||
|
Regression Statistics |
||||||||
|
Multiple R |
0.999169 |
|||||||
|
R Square |
0.998338 |
|||||||
|
Adjusted R Square |
0.997923 |
|||||||
|
Standard Error |
6585.241 |
|||||||
|
Observations |
11 |
|||||||
|
ANOVA |
||||||||
|
df |
SS |
MS |
F |
Significance F |
||||
|
Regression |
2 |
2.08428E+11 |
1.04214E+11 |
2403.15558 |
7.62471E-12 |
|||
|
Residual |
8 |
346923227.2 |
43365403.4 |
|||||
|
Total |
10 |
2.08775E+11 |
||||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
66.4435 |
2425.159896 |
0.027397575 |
0.97881375 |
-5525.985247 |
5658.872 |
-5525.99 |
5658.872249 |
|
X Variable 1 |
1.018835 |
2.092790071 |
0.486830774 |
0.63943573 |
-3.807147947 |
5.844817 |
-3.80715 |
5.844817169 |
|
X Variable 2 |
253.6505 |
28.39446871 |
8.93309405 |
1.9576E-05 |
188.1726972 |
319.1282 |
188.1727 |
319.1282217 |
The cost equation from Overhead cost and both Machine hours and Batches is
From the above table the p-value of Batches is less than 0.05 which makes it statistically significant to the equation and can be used for future projections while the p-value of Machine Hours is more than 0.05 making it statistically insignificant hence can be dropped for future projection.
- I will use the simple regression of Overhead cost and Batches. Batches proves to be more statistically significance than Machine Hours because when they are regressed with Overhead cost the p-value of Batches is very smaller than 0.05.
References
Colleen M. Norris, W. A. (2006). Ordinal regression model and the linear regression model were superior to the logistic regression models. 9.
Working Group on Radiative Corrections and Monte Carlo Generators for Low Energies, S. A. (2010). Quest for precision in hadronic cross sections at low energy:. Monte Carlo tools vs. experimental data, 102.
Understanding Utility Function And Standard Gamble For Assessing Utility Values
- December 21, 2023/ Uncategorized
Utility Function and Its Assessment
Utility function is a twice differentiable function of wealth which satisfies the principle of non-satiation and risk aversion. We can assess the utility function by mainly two methods, that is interview method and probability encoding. In interview method, a person is asked a series of questions from which his/her utility function is assessed while in probability encoding probabilities are attached to uncertain possible outcomes depending on an individual’s preference.
A standard gamble is a technique for managing outcomes by attaching probability values to an individual’s preferences. The standard gamble is used in determining utility values by asking a series of leading questions the answer to which will help determine the utility values of the individual as they indicate an individual’s attitude towards a particular risk.
- Decision matrix
|
Good |
Fair |
Bad |
|
|
Share market |
11400 |
10800 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
- Optimist criterion
|
Good |
Fair |
Bad |
Maximum outcome |
|
|
Share market |
11400 |
10800 |
10000 |
11400 |
|
Government bonds |
10900 |
10900 |
10900 |
10900 |
The optimist will take the option of the maximum outcome which is to trade shares under good market which its outcome is $11400
- Pessimist criterion
|
Good |
Fair |
Bad |
Minimum outcome |
|
|
Share market |
11400 |
10800 |
10000 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
10900 |
The pessimist will take the option of maximum in the possible minimum outcome which is to trade in government bond in all seasons which results in $10900
- Regret criterion
|
Good |
Fair |
Bad |
Minimum outcome |
Maximum outcome |
|
|
Share market |
11400 |
10800 |
10000 |
10000 |
11400 |
|
Government bonds |
10900 |
10900 |
10900 |
10900 |
10900 |
Regret criterion can choose to go for both the minimax and maximin take the option of trading in government bond which gives $10900.
- Expected monetary value
|
Good |
Fair |
Bad |
|
|
Share market |
11400 |
10800 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
|
Probability |
0.4 |
0.2 |
0.4 |
The optimal action in the expected monetary value will be to trade in government bond which its outcome is $10900.
- Expected value of perfect information
|
Good |
Fair |
Bad |
|
|
Share market |
11400 |
10800 |
10000 |
|
Government bonds |
10900 |
10900 |
10900 |
|
Probability |
0.4 |
0.2 |
0.4 |
The highest expected monetary value is $10900
Expectation for maximizing profit
11100
Knowing the direction the market will go is worth $200
- The probability of unfavorable market is 0.5
Since the expected monetary value is $20000 he can go ahead and start the producing new type of razor.
- The friend’s probability
|
Favorable |
Unfavorable |
|
|
Correct prediction |
0.7 |
0.2 |
|
Incorrect prediction |
0.3 |
0.8 |
The prior probabilities are 0.5 for favorable market and 0.5 for unfavorable market
The joint and marginal probability will be;
|
Favorable |
Unfavorable |
||
|
Correct prediction |
0.35 |
0.1 |
0.45 |
|
Incorrect prediction |
0.15 |
0.4 |
0.55 |
|
0.5 |
0.5 |
Posterior probability
|
Favorable |
Unfavorable |
|
|
Correct prediction |
0.78 |
0.73 |
|
Incorrect prediction |
0.22 |
0.27 |
- Posterior probability
Posterior probability when the friend predict unfavorable market is 0.22
- In engaging friend; under correct prediction
Expected value of sample information (EVSI) = emv with sample information – emv without sample information.
emv with sample information =
= 19920
Emv without sample information
He should not engage his friend because the value of the information is less than the cost of the information.
|
MODEL |
|
|
|
|
|
|
|
|
|
|
|
Selling |
|
Profit |
Fixed |
|
|
Month |
RN1 |
Demand |
Price |
RN2 |
Margin |
Costs |
Profit |
|
1 |
0.23297 |
107 |
$180 |
0.22763 |
20% |
2000 |
6408 |
|
2 |
0.794052 |
147 |
174 |
0.233362 |
30% |
2000 |
7956 |
|
3 |
0.395857 |
139 |
163 |
0.40283 |
70% |
2000 |
11536 |
|
4 |
0.320458 |
196 |
174 |
0.240464 |
13% |
2000 |
3164 |
|
5 |
0.245738 |
134 |
165 |
0.360996 |
90% |
2000 |
24192 |
|
6 |
0.422416 |
153 |
178 |
0.890802 |
40% |
2000 |
12408 |
|
7 |
0.038463 |
148 |
176 |
0.452313 |
90% |
2000 |
19890 |
|
8 |
0.730945 |
168 |
178 |
0.737935 |
80% |
2000 |
23780 |
|
9 |
0.503711 |
192 |
171 |
0.878873 |
40% |
2000 |
8514 |
|
10 |
0.726539 |
149 |
171 |
0.221567 |
60% |
2000 |
18795 |
|
11 |
0.179755 |
141 |
166 |
0.241994 |
90% |
2000 |
15998 |
|
12 |
0.412426 |
105 |
163 |
0.142019 |
30% |
2000 |
7164 |
|
DATA |
|||||||
|
Prob |
Cum prob |
Demand |
Selling |
Price |
$180 |
$220 |
|
|
0.05 |
0 |
100 |
Monthly |
Fixed cost |
$2,000 |
||
|
0.1 |
0.05 |
120 |
Profit |
Margin |
22% |
32% |
|
|
0.2 |
0.15 |
140 |
|||||
|
0.3 |
0.35 |
160 |
|||||
|
0.25 |
0.65 |
180 |
|||||
|
0.1 |
0.9 |
200 |
|
MODEL |
|||||||
|
Selling |
Profit |
Fixed |
|||||
|
Month |
RN1 |
Demand |
Price |
RN2 |
Margin |
Costs |
Profit |
|
1 |
0.23297 |
192 |
$187 |
0.22763 |
20% |
2000 |
4700 |
|
2 |
0.158694 |
103 |
218 |
0.479955 |
30% |
2000 |
8259 |
|
3 |
0.179437 |
126 |
186 |
0.672223 |
70% |
2000 |
17550 |
|
4 |
0.285063 |
187 |
191 |
0.671108 |
13% |
2000 |
2978 |
|
5 |
0.482977 |
187 |
197 |
0.800577 |
90% |
2000 |
37080 |
|
6 |
0.636752 |
132 |
199 |
0.133095 |
40% |
2000 |
12408 |
|
7 |
0.395724 |
168 |
203 |
0.651491 |
90% |
2000 |
19890 |
|
8 |
0.19685 |
116 |
214 |
0.334298 |
80% |
2000 |
23780 |
|
9 |
0.226749 |
113 |
188 |
0.495291 |
40% |
2000 |
8514 |
|
10 |
0.592614 |
150 |
185 |
0.459049 |
60% |
2000 |
18795 |
|
11 |
0.223638 |
123 |
199 |
0.774456 |
90% |
2000 |
15998 |
|
12 |
0.15323 |
142 |
210 |
0.031765 |
30% |
2000 |
7164 |
Report of the changes observed
It can be seen than the changes cause the increase of profit from $159805 to $177116. This will bring about the benefit to the company as it obvious the profit margin also increased. The drastic effect it can cause is that with time the increase in price of the tires can lower the demand leading to low sales which will eventually lower the profit. This can force the profit margin to reduce. Some players can come in the market to take the opportunity to sell tires at relatively lower price which can bring stiff competition lowering the profits, demand even further. Therefore, increasing the varieties of tires customers can choose from can make them understand the increase in prices setting in a continuous trend which can be realized by customers in later days. This will ensure continuous demand keeping our increase in profit margin. (Working Group on Radiative Corrections and Monte Carlo Generators for Low Energies, 2010)
- Regression between Overhead cost against machine hours
The cost equation I from Overhead cost and Machine hours is
From the above table we see the p-value is less than 0.05 which shows that in this case the variable Machine Hours is statistically significant and can be used for future projections
The regression analysis between Overhead cost and batches
|
Regression Statistics |
||||||||
|
Multiple R |
0.999144163 |
|||||||
|
R Square |
0.998289059 |
|||||||
|
Adjusted R Square |
0.998098954 |
|||||||
|
Standard Error |
6299.920723 |
|||||||
|
Observations |
11 |
|||||||
|
ANOVA |
||||||||
|
df |
SS |
MS |
F |
Significance F |
||||
|
Regression |
1 |
2.08417E+11 |
2.08417E+11 |
5251.262 |
9.17576E-14 |
|||
|
Residual |
9 |
357201010.1 |
39689001.12 |
|||||
|
Total |
10 |
2.08775E+11 |
||||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
119.1919192 |
2317.767593 |
0.051425311 |
0.96011 |
-5123.962644 |
5362.34648 |
-5123.96 |
5362.346482 |
|
X Variable 1 |
267.345679 |
3.689277512 |
72.46559201 |
9.18E-14 |
258.9999535 |
275.691405 |
259 |
275.6914046 |
The cost equation I from Overhead cost and Batches is
From the above table we see the p-value is less than 0.05 which shows that in this case the variable Batches is statistically significant and can be used for future projections
The regression analysis between Overhead cost both Machine hours and Batches
|
SUMMARY OUTPUT |
||||||||
|
Regression Statistics |
||||||||
|
Multiple R |
0.999169 |
|||||||
|
R Square |
0.998338 |
|||||||
|
Adjusted R Square |
0.997923 |
|||||||
|
Standard Error |
6585.241 |
|||||||
|
Observations |
11 |
|||||||
|
ANOVA |
||||||||
|
df |
SS |
MS |
F |
Significance F |
||||
|
Regression |
2 |
2.08428E+11 |
1.04214E+11 |
2403.15558 |
7.62471E-12 |
|||
|
Residual |
8 |
346923227.2 |
43365403.4 |
|||||
|
Total |
10 |
2.08775E+11 |
||||||
|
Coefficients |
Standard Error |
t Stat |
P-value |
Lower 95% |
Upper 95% |
Lower 95.0% |
Upper 95.0% |
|
|
Intercept |
66.4435 |
2425.159896 |
0.027397575 |
0.97881375 |
-5525.985247 |
5658.872 |
-5525.99 |
5658.872249 |
|
X Variable 1 |
1.018835 |
2.092790071 |
0.486830774 |
0.63943573 |
-3.807147947 |
5.844817 |
-3.80715 |
5.844817169 |
|
X Variable 2 |
253.6505 |
28.39446871 |
8.93309405 |
1.9576E-05 |
188.1726972 |
319.1282 |
188.1727 |
319.1282217 |
The cost equation from Overhead cost and both Machine hours and Batches is
From the above table the p-value of Batches is less than 0.05 which makes it statistically significant to the equation and can be used for future projections while the p-value of Machine Hours is more than 0.05 making it statistically insignificant hence can be dropped for future projection.
- I will use the simple regression of Overhead cost and Batches. Batches proves to be more statistically significance than Machine Hours because when they are regressed with Overhead cost the p-value of Batches is very smaller than 0.05.
References
Colleen M. Norris, W. A. (2006). Ordinal regression model and the linear regression model were superior to the logistic regression models. 9.
Working Group on Radiative Corrections and Monte Carlo Generators for Low Energies, S. A. (2010). Quest for precision in hadronic cross sections at low energy:. Monte Carlo tools vs. experimental data, 102.
