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### Chapter 14

Methods of Fair Division

Adjusted Winner Procedure

- We are given a discrete set of items (objects and/or issues).
- There are two interested parties with equal claim to the set of items.
- The adjusted winner procedure provides a method to divide up the items among the interested parties in a way that both will consider fair.
- It might happen that one item will need to be divided among the two parties.

Adjusted Winner Procedure – Example #1

- Example #1

Calvin and Hobbes find a buried treasure. The items they find are as follows:

Cannon, Anchor, Treasure Chest, Doubloon, Figurehead, Sword, Cannon Ball, Wooden Leg, Flag, Crow’s Nest

Calvin and Hobbes each have equal claim to the treasure and decide to use the adjusted winner procedure to divide up these objects among themselves.

Adjusted Winner Procedure – Example #1

- The procedure begins with each party distributing a total of 100 points among the given items.
- Higher point assignments mean an item is valued more by that party.

Suppose Calvin and Hobbes each distribute their 100 points as given in the table.

Adjusted Winner Procedure – Example #1

- We go through the list of items and initially divide the items among the two parties according to who had placed a higher point value on each item. The adjusted winner procedure could stop here if each party receives an equal number of points.

Adjusted Winner Procedure – Example #1

- Generally the point totals received by each party will not equal - as in this case. The totals listed at the bottom of the table indicate the total points received by each party according to that party’s initial point distribution.

Note that point values are based on

that party’s valuation of a given item.

If there are any items that both parties

valued equally, we transfer

those items to the party with the smaller

point total. If a fraction of an item had

to be transferred at this stage to make the

totals equal, we would use the algebraic

process shown later in this example.

Because the point totals are not equal,

we will now begin to transfer items, or

a part of one item, from one party to the

other to equalize the point totals.

Adjusted Winner Procedure – Example #1

Point ratio = ( A’s valuation / B’s valuation )

where A is the party with the greater

point total, in this case Hobbes.

Notice that because we will only transfer items from Hobbes to Calvin, we calculate point ratios for items belonging to Hobbes only.

Adjusted Winner Procedure – Example #1

- We are interested only in the point ratios for items belonging to Hobbes.
- Following the adjusted winner procedure, we will transfer items from Hobbes to Calvin in order of increasing point ratio (from smallest to largest point ratio).
- Because it has the lowest point ratio, we transfer the Doubloon first.

Adjusted Winner Procedure – Example #1

- Notice the Doubloon was worth 14 to Hobbes but only 11 to Calvin. We adjust the point totals accordingly.
- We continue to transfer items from Hobbes to Calvin until the point totals are equal.
- By increasing point ratio, the next item to transfer will be the chest.

Adjusted Winner Procedure – Example #1

- Note that if we transfer the entire chest from Hobbes to Calvin then Calvin (with 60) will have more points than Hobbes (with 50).
- Therefore, we will not transfer the entire chest to Calvin.
- We will use an algebraic procedure to find the fraction x of the chest that will be transferred from Hobbes to Calvin.

Adjusted Winner Procedure – Example #1

- Let x be the fraction of the chest that Hobbes will keep and therefore 1 – x will represent the fraction that Calvin will receive.
- For example, if x = 2/3 then 1 – x = 1/3.
- In the transfer of the chest, Hobbes will keep 20x points from the chest and Calvin will receive 15(1-x) points.

Adjusted Winner Procedure – Example #1

- Note that Calvin currently has 55 points and will receive 15(1-x) points.
- Note that Hobbes has 50 points plus will keep some portion of the chest, which is represented by 20x.
- To equate the point totals, we write the equation:
- 55 + 15(1-x) = 50 + 20x

Adjusted Winner Procedure – Example #1

Now, solving this linear equation, we get …

55 + 15(1-x) = 50 + 20x

55 + 15 – 15x = 50 + 20x

70 – 15x = 50 + 20x

20 = 35x

x = 20/35

x = 4/7

Therefore, Hobbes will keep 4/7 of the chest and Calvin will receive 3/7 of the chest.

Adjusted Winner Procedure – Example #1

Notice the significance of the solution to this equation …

Given the equation: 55 + 15(1-x) = 50 + 20x

If x = 4/7 then each side becomes …

55 + 15 (3/7) = 50 + 20 (4/7)

61.43 = 61.43

Each party will receive an equal number of points.

This is where the adjusted winner procedure ends, with a partial transfer of the chest, all of the items have been divided in a way that each party will consider equitable.

Adjusted Winner Procedure – Example #1

- The result of the adjusted winner procedure is the distribution of items as shown in the table.
- Calvin gets the cannon, doubloon, sword, cannon ball, wooden leg, flag, crow’s nest and 3/7ths of the chest.
- Hobbes gets the anchor, figurehead and 4/7ths of the chest.

Adjusted Winner Procedure – Example #2

Example #2: A Corporate Merger

Suppose two companies will negotiate a merger and must agree on the following issues:

Company name, location of headquarters, choosing a

chairperson, choosing a chief executive officer, deciding

which company will face layoffs from any redundancies.

These companies could use the adjusted winner procedure to facilitate the negotiations.

Adjusted Winner Procedure – Example #2

Example #2: A Corporate Merger

Suppose these companies distribute their 100 points as shown below based on their priorities in the negotiations.

Adjusted Winner Procedure – Example #2

We initially give each item to the party who gave that item the higher point value.

Adjusted Winner Procedure – Example #2

Without yet distributing those items that were valued equally, we see that XYZ currently has the lower point total.

Note that, in this example, there are still two items (CEO and Layoffs) to be distributed. We could interrupt the adjusted winner procedure and end the negotiations if it was possible to distribute those items in such a way as to make the point totals equal. However, in this case that is not possible.

We continue with the adjusted winner procedure and, initially, give both of the remaining items to XYZ, Inc.

Adjusted Winner Procedure – Example #2

Because XYZ initially had the lower point total, they receive both items which were equally valued.

Now we decide which items will be transferred from XYZ to ABC.

We calculate point ratios for items currently belonging to XYZ by calculating XYZ’s valuation divided by ABC’s valuation of each item.

HQ = 25/10 = 2.5

Chair = 10/5 = 2

CEO = 30/30 = 1

Layoffs = 15/15 = 1

Adjusted Winner Procedure – Example #2

We have point ratios of …

HQ = 25/10 = 2.5

Chair = 10/5 = 2

CEO = 30/30 = 1

Layoffs = 15/15 = 1

Both items CEO and Layoffs have equal point ratio, and we must decide which item will be transferred first. Following the adjusted winner procedure, we can not transfer the CEO item first. This is because then ABC would have more points than XYZ and we’d then need to transfer points back to XYZ, which is not consistent with the adjusted winner procedure. Following the adjusted winner procedure we will only transfer points in one direction until equality is achieved.

Adjusted Winner Procedure – Example #2

We have point ratios of …

HQ = 25/10 = 2.5

Chair = 10/5 = 2

CEO = 30/30 = 1

Layoffs = 15/15 = 1

Following the adjusted winner procedure, we will transfer the item Layoffs first and then determine a partial transfer of the item CEO.

In a realistic situation, either the negotiations fail and the merger is cancelled (and at least the adjusted winner procedure quickly identified an incompatibility between the companies) or some negotiations are needed beyond the adjusted winner procedure.

Adjusted Winner Procedure – Example #2

We have point ratios of …

HQ = 25/10 = 2.5

Chair = 10/5 = 2

CEO = 30/30 = 1

Layoffs = 15/15 = 1

It may be possible to reach an agreement which in some way that provides for a compromise on the CEO issue.

Following the adjusted winner procedure, we can determine a fraction representing what we could call “control” of the issue of the CEO for each company.

Adjusted Winner Procedure – Example #2

We have point ratios of …

HQ = 25/10 = 2.5

Chair = 10/5 = 2

CEO = 30/30 = 1

Layoffs = 15/15 = 1

We will say that XYZ, Inc., will retain some fraction of the CEO issue, which we can call x. Then ABC, Co., receives the other fraction of the issue, which would be 1 – x. To equate point totals we write the following equation:

55 + 30(1-x) = 35 + 30x

Adjusted Winner Procedure – Example #2

Now, solving for x, we get …

55 + 30(1-x) = 35 + 30x

55 + 30 – 30x = 35 + 30x

85 – 30x = 35 + 30x

50 = 60x

x = 50/60 = 5/6

The conclusion here is that XYZ will retain 5/6 of the CEO issue.

Adjusted Winner Procedure – Example #2

The conclusion of the adjusted winner procedure, in this example, is shown in the table.

ABC, Co., is given ownership of the Name and Layoffs issues and a small part ownership in the issue of the CEO.

XYZ, Inc., has ownership over the issue of the Headquarters and the Chairperson and the larger part of the ownership over the issue of the CEO.

This conclusion canprovidea guideline as to finalizing negotiations over the CEO.

Adjusted Winner Procedure – Example #2

The conclusion of the adjusted winner procedure, in this example, is shown in the table.

Beyond the adjusted winner procedure, to finalize the negotiations, suppose, for example, that additional negotiations went as follows ...

Originally, ABC valued the chair at 5 points. Perhaps they would accept ownership of the chairperson issue in place of a 1/6 ownership in the CEO issue. In exchange for relinquishing the chair issue which XYZ valued at 10 points, XYZ could receive full ownership of the issue of the CEO and perhaps could negotiate some other consideration to make up for the loss of 5 points.

Adjusted Winner Procedure – Example #2

A possible agreement?

XYZ gets their way on their two biggest issues but will need to concede more on layoffs. ABC gets to decide the name, gets the chairperson, and has more control over layoffs

Just for fun, we might imagine the table shown above as a possible resolution to the negotiations.

Practically speaking, it may be easier to divide the issue of layoffs between the two companies. We could transfer back some points on the issue of layoffs to XYZ to make up for the deficit of 5 points in the agreement on Chair and CEO.

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