# Division of a Will

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An old man left his fortune and goods to his four children. In the will it stated that each item should be divided up fairly so that each person should have an equal share. Unfortunately, the lawyer found that each person appraised the objects' worth differently and no official appraiser was nearby. The lawyer had each of the children complete a simple little informal appraisal of each of the major items. The results are as follows:

Albert:

Piano= 1,000

Ring= \$120

Home= 2,000,000

Appliances= \$500

Wardrobe= \$50

Eric:

Piano= 1,500

Ring= \$20

Home= 2,600,000

Appliances= \$200

Wardrobe= \$10

Theresa:

Piano= 900

Ring= \$75

Home= 1,500,000

Appliances= \$350

Wardrobe= \$10

Sonya:

Piano= 1,150

Ring= \$150

Home= 4,000,000

Appliances= \$750

Wardrobe= \$0

The deceased also had a net value of \$12,000,000. How should the lawyer divide up the assets using only this given information to where each person walks away with an equal (or better than equal) percentage of the deceased's worth?

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An old man left his fortune and goods to his four children. In the will it stated that each item should be divided up fairly so that each person should have an equal share. Unfortunately, the lawyer found that each person appraised the objects' worth differently and no official appraiser was nearby. The lawyer had each of the children complete a simple little informal appraisal of each of the major items. The results are as follows:

Albert:

Piano= 1,000

Ring= \$120

Home= 2,000,000

Appliances= \$500

Wardrobe= \$50

Eric:

Piano= 1,500

Ring= \$20

Home= 2,600,000

Appliances= \$200

Wardrobe= \$10

Theresa:

Piano= 900

Ring= \$75

Home= 1,500,000

Appliances= \$350

Wardrobe= \$10

Sonya:

Piano= 1,150

Ring= \$150

Home= 4,000,000

Appliances= \$750

Wardrobe= \$0

The deceased also had a net value of \$12,000,000. How should the lawyer divide up the assets using only this given information to where each person walks away with an equal (or better than equal) percentage of the deceased's worth?

Love this problem. Here's how I would approach it.

So the assets to be divided are

Piano

Ring

Home

Appliances

Wardrobe

\$12,000,000.

Here's a method that guarantees each person be happy from his/her perspective.

* Divide the \$12,000,000 into 4. Each person gets 3,000,000.

* Hold an auction among the 4 heirs for each of the other assets. Under perfect scenario, this will result in a sale price slightly higher than the second highest valuation of the property. Give the property to the winning person and then divide the proceeds evenly by 4.

* For instance, let's say we auction the house among the 4 heirs. Sonya would bid \$2,600,001 and win the house. She then pays \$2,600,001 to the lawyer and get ownership of the house. The lawyer will then divide \$2,600,001 into 4 - one portion for each heir.

* Repeat with the other assets.

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Quick question: when you say "equal (or better than equal) percentage"...from whose perspective are we talking?

If we are saying that each PERSON believes they have AT LEAST equal percentage of the worth, then we first look at what each person believed the old man was worth:

ALBERT: \$14,001,670...which means "equal" to him would be everyone getting \$3,500,417.50...
ERIC: \$14,601,730...which means "equal" to him would be everyone getting \$3,650,432.50
THERESA: \$13,501,335...which means "equal" to her would be everyone getting \$3,375,333.75
SONYA: \$16,002,050...which means "equal" to her would be everyone getting \$4,000,512.50

If we use those, we can simply give Sonya the house and all of the belongings...and then divide the remaining \$12,000,000 in cash evenly between the rest of the siblings.

To Sonya, she got more than an equal share (\$4,002,050) and everyone else got \$4,000,000...so she's happy

To Theresa, she and everyone else but Sonya got more than an equal share (\$4,000,000) while Sonya only got \$1,501,335...so she's happy

To Eric, he and everyone else but Sonya got more than an equal share (\$4,000,000) while Sonya only got \$2,601,730...so he's happy

To Albert, he and everyone else but Sonya got more than an equal share (\$4,000,000) while Sonya only got \$2,001,670...so he's happy

Doing this gives everyone the perception that they had a more than equal share of the total worth...

However, from the lawyer's perspective, he would see Sonya getting much less than equal, simply because the value of the house most likely is closer to the average (\$2,275,000)...which means with this approach, Sonya would have much less than an equal share...

That's why I'm wondering from whose perspective we are defining "equal".

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Posted · Report post

Quick question: when you say "equal (or better than equal) percentage"...from whose perspective are we talking?

If we are saying that each PERSON believes they have AT LEAST equal percentage of the worth, then we first look at what each person believed the old man was worth:

ALBERT: \$14,001,670...which means "equal" to him would be everyone getting \$3,500,417.50...
ERIC: \$14,601,730...which means "equal" to him would be everyone getting \$3,650,432.50
THERESA: \$13,501,335...which means "equal" to her would be everyone getting \$3,375,333.75
SONYA: \$16,002,050...which means "equal" to her would be everyone getting \$4,000,512.50

If we use those, we can simply give Sonya the house and all of the belongings...and then divide the remaining \$12,000,000 in cash evenly between the rest of the siblings.

To Sonya, she got more than an equal share (\$4,002,050) and everyone else got \$4,000,000...so she's happy

To Theresa, she and everyone else but Sonya got more than an equal share (\$4,000,000) while Sonya only got \$1,501,335...so she's happy

To Eric, he and everyone else but Sonya got more than an equal share (\$4,000,000) while Sonya only got \$2,601,730...so he's happy

To Albert, he and everyone else but Sonya got more than an equal share (\$4,000,000) while Sonya only got \$2,001,670...so he's happy

Doing this gives everyone the perception that they had a more than equal share of the total worth...

However, from the lawyer's perspective, he would see Sonya getting much less than equal, simply because the value of the house most likely is closer to the average (\$2,275,000)...which means with this approach, Sonya would have much less than an equal share...

That's why I'm wondering from whose perspective we are defining "equal".

Is there a way to ensure that all five parties are satisfied (lawyer included)?

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An old man left his fortune and goods to his four children. In the will it stated that each item should be divided up fairly so that each person should have an equal share. Unfortunately, the lawyer found that each person appraised the objects' worth differently and no official appraiser was nearby. The lawyer had each of the children complete a simple little informal appraisal of each of the major items. The results are as follows:

Albert:

Piano= 1,000

Ring= \$120

Home= 2,000,000

Appliances= \$500

Wardrobe= \$50

Eric:

Piano= 1,500

Ring= \$20

Home= 2,600,000

Appliances= \$200

Wardrobe= \$10

Theresa:

Piano= 900

Ring= \$75

Home= 1,500,000

Appliances= \$350

Wardrobe= \$10

Sonya:

Piano= 1,150

Ring= \$150

Home= 4,000,000

Appliances= \$750

Wardrobe= \$0

The deceased also had a net value of \$12,000,000. How should the lawyer divide up the assets using only this given information to where each person walks away with an equal (or better than equal) percentage of the deceased's worth?

Love this problem. Here's how I would approach it.

So the assets to be divided are

Piano

Ring

Home

Appliances

Wardrobe

\$12,000,000.

Here's a method that guarantees each person be happy from his/her perspective.

* Divide the \$12,000,000 into 4. Each person gets 3,000,000.

* Hold an auction among the 4 heirs for each of the other assets. Under perfect scenario, this will result in a sale price slightly higher than the second highest valuation of the property. Give the property to the winning person and then divide the proceeds evenly by 4.

* For instance, let's say we auction the house among the 4 heirs. Sonya would bid \$2,600,001 and win the house. She then pays \$2,600,001 to the lawyer and get ownership of the house. The lawyer will then divide \$2,600,001 into 4 - one portion for each heir.

* Repeat with the other assets.

If one of the people got the house which in Sonya's case means the house is worth 2.6 million should she also get 1/4 of the value back? Sounds like she got an incredible discount.

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Posted (edited) · Report post

I should have stated in the OP that the will dictates that someone must have some if not all of the items so no selling it to a third party option. Thank you to Pickett and Bushindo for assuming correctly in this regard.

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Posted · Report post

Love this problem. Here's how I would approach it.

So the assets to be divided are

Piano

Ring

Home

Appliances

Wardrobe

\$12,000,000.

Here's a method that guarantees each person be happy from his/her perspective.

* Divide the \$12,000,000 into 4. Each person gets 3,000,000.

* Hold an auction among the 4 heirs for each of the other assets. Under perfect scenario, this will result in a sale price slightly higher than the second highest valuation of the property. Give the property to the winning person and then divide the proceeds evenly by 4.

* For instance, let's say we auction the house among the 4 heirs. Sonya would bid \$2,600,001 and win the house. She then pays \$2,600,001 to the lawyer and get ownership of the house. The lawyer will then divide \$2,600,001 into 4 - one portion for each heir.

* Repeat with the other assets.

If one of the people got the house which in Sonya's case means the house is worth 2.6 million should she also get 1/4 of the value back? Sounds like she got an incredible discount.

I don't think that would be a problem

Technically, before the sale Sonya has a 25% ownership in the property, so she should get 1/4 of the proceeds. Essentially, she is paying 75% of the sale price \$2,600,001 to the other 3 heirs for their 75% ownership.

As to whether she got a discount. Yes, she did from her perspective. From Albert and Theresa's perspectives, Sonya overpaid but I doubt they would be complaining since they get a bigger share of the pie from their point of view. Eric receives a share that is 25 cents above his valuation, so he should be happy too.

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What is the best means to dividing these items up? Let's define Best as the approach that gives the greatest benefit to the recipients. The submissions will be judged by examining the highest percentage over the expected individual's share of the individual who made the least over their expectation.

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What is the best means to dividing these items up? Let's define Best as the approach that gives the greatest benefit to the recipients. The submissions will be judged by examining the highest percentage over the expected individual's share of the individual who made the least over their expectation.

Can you elaborate on the bolded part? I'm not sure that I can parse that correctly.

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Posted · Report post

Just a comment that Busindo's solution is attractive in that the inheritors have a hand in determining their benefit.

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What is the best means to dividing these items up? Let's define Best as the approach that gives the greatest benefit to the recipients. The submissions will be judged by examining the highest percentage over the expected individual's share of the individual who made the least over their expectation.

Can you elaborate on the bolded part? I'm not sure that I can parse that correctly.

There are many ways to determine a 'best' answer in this question. Both you and Pickett found effective answer and there are in fact more answers that would work in providing everyone at least 25% of the fair share of the goods and money. I am now seeking the answer that provides everyone the most profit. I do not want the average percentage of perceived benefit from the will, I want to award the 'best' solution to the one who can give the most to the person who received the least.

For example: (I am making these percentages up by the way)

Remember each person expects to receive 1/4 of the value of the old man's wealth

strategy 1's allocation strategy gave the following outcomes

person 1=27%

person 2=36%

person 3=40%

person 4=30%

strategy 2's allocation strategy gave the following outcomes

person 1= 30%

person 2= 29%

person 3= 28%

person 4= 29%

though the average gain in strategy 1 is 33.25% which is higher than strategy 2, person 1 only made 2% more than expected so this strategy is not preferred when compared to strategy 2 since its lowest recipient got 28% of the perceived wealth or 3% more than expected. So in these hypothetical cases, solution number 2 is perceived as better since the minimum beneficiary is better than the other cases minimum beneficiary.

**and of course these percentages are perceived percentages of value they placed on the old man's wealth.

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What is the best means to dividing these items up? Let's define Best as the approach that gives the greatest benefit to the recipients. The submissions will be judged by examining the highest percentage over the expected individual's share of the individual who made the least over their expectation.

Can you elaborate on the bolded part? I'm not sure that I can parse that correctly.

There are many ways to determine a 'best' answer in this question. Both you and Pickett found effective answer and there are in fact more answers that would work in providing everyone at least 25% of the fair share of the goods and money. I am now seeking the answer that provides everyone the most profit. I do not want the average percentage of perceived benefit from the will, I want to award the 'best' solution to the one who can give the most to the person who received the least.

For example: (I am making these percentages up by the way)

Remember each person expects to receive 1/4 of the value of the old man's wealth

strategy 1's allocation strategy gave the following outcomes

person 1=27%

person 2=36%

person 3=40%

person 4=30%

strategy 2's allocation strategy gave the following outcomes

person 1= 30%

person 2= 29%

person 3= 28%

person 4= 29%

though the average gain in strategy 1 is 33.25% which is higher than strategy 2, person 1 only made 2% more than expected so this strategy is not preferred when compared to strategy 2 since its lowest recipient got 28% of the perceived wealth or 3% more than expected. So in these hypothetical cases, solution number 2 is perceived as better since the minimum beneficiary is better than the other cases minimum beneficiary.

**and of course these percentages are perceived percentages of value they placed on the old man's wealth.

If you are trying to maximize the minimum perceived percentage of the value, then you can divide the property in the following manner

Albert: Wardrobe + 3,855,971

Eric: Piano + 4,019,776

Theresa: 3,718,230

Sonya: Ring + Appliances + House + 406,020

From each person's point of view, he/she is getting 27.534% of the old man's wealth.

I think one more constraint that you will need to place on the division is that from each person's point of view, no one else is getting more money than he/she is. From the case above, Theresa is getting 27.5% of the old man's wealth from her point of view, but she thinks her brother Eric is getting almost 300,000 more than her (and he did nothing to deserve this except for appraising the house higher). Human nature being what it is, that is a prime cause for litigation.

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Brilliant! We will add your extra restraint. Find a way to best maximize your minimum score while keeping the perceived share as balanced as possible.

What is the best means to dividing these items up? Let's define Best as the approach that gives the greatest benefit to the recipients. The submissions will be judged by examining the highest percentage over the expected individual's share of the individual who made the least over their expectation.

Can you elaborate on the bolded part? I'm not sure that I can parse that correctly.

There are many ways to determine a 'best' answer in this question. Both you and Pickett found effective answer and there are in fact more answers that would work in providing everyone at least 25% of the fair share of the goods and money. I am now seeking the answer that provides everyone the most profit. I do not want the average percentage of perceived benefit from the will, I want to award the 'best' solution to the one who can give the most to the person who received the least.

For example: (I am making these percentages up by the way)

Remember each person expects to receive 1/4 of the value of the old man's wealth

strategy 1's allocation strategy gave the following outcomes

person 1=27%

person 2=36%

person 3=40%

person 4=30%

strategy 2's allocation strategy gave the following outcomes

person 1= 30%

person 2= 29%

person 3= 28%

person 4= 29%

though the average gain in strategy 1 is 33.25% which is higher than strategy 2, person 1 only made 2% more than expected so this strategy is not preferred when compared to strategy 2 since its lowest recipient got 28% of the perceived wealth or 3% more than expected. So in these hypothetical cases, solution number 2 is perceived as better since the minimum beneficiary is better than the other cases minimum beneficiary.

**and of course these percentages are perceived percentages of value they placed on the old man's wealth.

If you are trying to maximize the minimum perceived percentage of the value, then you can divide the property in the following manner

Albert: Wardrobe + 3,855,971

Eric: Piano + 4,019,776

Theresa: 3,718,230

Sonya: Ring + Appliances + House + 406,020

From each person's point of view, he/she is getting 27.534% of the old man's wealth.

I think one more constraint that you will need to place on the division is that from each person's point of view, no one else is getting more money than he/she is. From the case above, Theresa is getting 27.5% of the old man's wealth from her point of view, but she thinks her brother Eric is getting almost 300,000 more than her (and he did nothing to deserve this except for appraising the house higher). Human nature being what it is, that is a prime cause for litigation.

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Brilliant! We will add your extra restraint. Find a way to best maximize your minimum score while keeping the perceived share as balanced as possible.

If we want to maximize the minimum perceived percentage plus making sure that no one thinks someone else is receiving more money, then here's an approximate strategy

Albert: Wardrobe + 3906852.6

Eric: Piano + 3905490.1

Theresa: 3906902.6

Sonya: Ring + Appliances + House + 280754.6

Eric and Sonya each thinks that he/she is getting 26.75% of the old man's wealth. Albert and Theresa think that they are getting 27.9% and 28.94%, respectively.

However, the most important thing is that from each person's point of view, he/she has the most money of all the siblings.

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