When the demands or desires of one party are in conflict with those of another, be it a divorce, a labor­management negotiation, an international dispute, or an inheritance, to name a few, no one wants to be treated unfairly. It turns out that mathematics can help in the search for procedures that ensure fair and equitable resolutions of such conflicts. We begin in 13.1 with The Adjusted Winner Procedure, a procedure that was developed in the mid­1990's. The first example we will consider comes from the multi­billion dollar world of business mergers. In 1998, there was a proposed merger between two giant pharmaceutical companies, Glaxo Wellcome and SmithKline Beecham. The deal ultimately failed due to a clash of two corporate egos, costing the companies billions of dollars in stock. For purposes of illustration, let's assume that the following five social issues were a large part of the problem:
1) The name that the new company would use
2) The location of the headquarters of the new company
3) The question of who would serve as chair of the combined company
4) The question of who would serve as CEO of the combined company
5) The question of where the necessary layoffs would come from
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The Adjusted Winner Procedure:
Step 1: Each player independently divides 100 points over the items in a way that reflects the item’s worth to that player.
Step 2: Determine which items (objects) each player receives. An item is given to the player that assigned it more points.
*Items tied in points go to the player with the lower initial point total.
Step 3: Find each player’s initial point total.
Step 4: Determine which player has the higher point total and call him/her the initial winner. The other player is called the initial loser.
Step 5: Move or transfer items from the initial winner to the initial loser by ascending or increasing point ratio.
*Only calculate the point ratio for the initial winner’s items.
Step 6: Move or transfer items according to the formula
This fraction goes to the Initial Loser.
Step 7: Calculate how much of the moved or transferred item stays with the initial winner. *Fractions must add to one.
Step 8: Write the Final Division including the point totals for each player.
*Point totals for both players must be the same.
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The first step in the Adjusted Winner Procedure is that each party distributes 100 points between the items or issues at stake, as shown in the table below.
The next step is to determine which items each party receives. Place a check next to the winner of each item. Tied items go to the party with the lower initial point total.
Determine which party has the higher point total at this point and call him/her the initial winner. The other player is the initial loser.
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Next we calculate the point ratio for each item that the initial winner got. The formula for calculating the point ratio is: Round the point ratio to the nearest hundredth.
Move or transfer items from the initial winner to the initial loser by ascending or increasing point ratio, according to the formula This fraction of the item at stake goes to the initial loser.
Now calculate how much of the item stays with the initial winner. Remember that the two fractions must add up to 1. 4
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The last step is to write the final division including the point totals for each player. The point totals for both parties must be the same (hence fair division). 6
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There are three properties of the adjusted winner allocation that are important.
1) The allocation is equitable (each player believes that he or she receives the same fractional part of the total value).
2) The allocation is envy­free (each player gets a share of whatever is being divided that is at least as large, or at least as desirable, as that received by any other player).
3) The allocation is Pareto­optimal (no other allocation achieved by any means can make one player better off without making some other player worse off).
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Homework: Read section 13.1 and do HW #15.
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