13.2 13.1 The Adjusted Winner Procedure
To illustrate the adjusted winner procedure, we will consider an application to the multibillion-dollar world of business mergers. It turns out that one of the most elusive ingredients in the success of a merger is what deal-makers call social issues—how power, position, sacrifice, and status are allocated between the merging companies and their executives.
As a case in point, let’s revisit the 1998 proposed merger between two giant pharmaceutical companies, Glaxo Wellcome and SmithKline Beecham. While most of the details underlying this aborted deal are still unknown to outsiders, the role of social issues is clearly underscored by reports that the companies saw nearly $19 billion of stock market value vanish in the clash of two corporate egos.
Exactly what kinds of issues might bring on a “clash of two corporate egos”? While not privy to the details of the Glaxo Wellcome-SmithKline Beecham merger attempt, we can speculate as to their nature. For purposes of illustration, let’s assume that the following five social issues were paramount:
- The name that the combined company would use
- The location of the headquarters of the combined company
- The question of who would serve as chair of the combined company
- The question of who would serve as CEO of the combined company
- The question of where the necessary layoffs would come from
Each of these five social issues is known to have been a major factor in other recent proposed mergers. For example, when Chrysler merged with Daimler-Benz in 1998, the issue of the choice of a name for the combined company was described as a “standoff” before both sides finally agreed to DaimlerChrysler.
So let’s assume that these were the five social issues confronting Glaxo Wellcome and SmithKline Beecham, and let’s see how the adjusted winner procedure would have suggested a resolution. The starting point—and something that is quite difficult when dealing with issues (as in a negotiation) as opposed to objects (as in a divorce)—is to have each side quantify the importance it attaches to getting its own way on each of the issues.
With the adjusted winner procedure, quantification is done by having each side—independently and simultaneously—spread 100 points over the issues in a way that reflects the relative worth of each issue to that party. In our present example, let’s assume that the companies allocated their 100 points as shown in Table 13.1. The adjusted winner procedure is now used to decide which side gets its way on which issues, but the procedure requires that a compromise of sorts may have to be reached on one of the issues.
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| Point Allocations | ||
|---|---|---|
| Issue | Glaxo Wellcome | SmithKline Beecham |
| Name | 5 | 10 |
| Headquarters | 25 | 10 |
| Chair | 35 | 20 |
| CEO | 15 | 35 |
| Layoffs | 20 | 25 |
| Total | 100 | 100 |
Here’s how the procedure works. Suppose that we have two parties and a list of either issues to be resolved in one party’s favor or the other’s (as in our merger example) or objects to be awarded either to one party or to the other (as in a divorce or a two-person inheritance). To have a single word covering both issues and objects, we will often speak of “items.” The adjusted winner procedure follows these basic steps:
Basic Steps in the Adjusted Winner Procedure PROCEDURE
- Step 0. As described earlier, each party distributes 100 points over the items in a way that reflects each item’s relative worth to that party.
- Step 1. Each item on which the assigned points differ is initially given to the party that assigned it more points. Add up the total number of points each party feels that he or she has received. The party with the fewest points is now given all the items on which both parties placed the same number of points. Once again, add up the total number of points each party feels that he or she has received. The party with the most points is called the initial winner; the other party is called the initial loser.
- Step 2. For each item given to the initial winner, calculate the point ratio.
- Step 3. Start moving items from the initial winner to the initial loser in ascending order of point ratio. Stop when you get to an item whose move will cause the initial winner to have fewer points than the initial loser. This item will need to be split or shared and is thus called the shared item.
- Step 4. Let represent the fractional part of the shared item that will be moved from the initial winner to the initial loser. Write a formula that equates each party’s total points after the sharing of this item.
- Step 5. Solve the equation and state the final division of items between the two parties.
Let’s demonstrate the adjusted winner procedure by continuing with our analysis of the proposed merger between Glaxo Wellcome and SmithKline Beecham. Why the order of transfer given in Step 3 is so important will be explained later.
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- Step 0. Assume that Glaxo Wellcome and SmithKline Beecham have given us the point assignments shown in Table 13.1.
- Step 1. Because Glaxo Wellcome has placed more points on headquarters (25) and chair (35), it is initially “given” these issues, while SmithKline Beecham is initially given name (10), CEO (35), and layoffs (25). Notice that SmithKline Beecham now has of its points, whereas Glaxo Wellcome has only of its points.
Step 2. We now calculate the point ratio of the three issues held by the initial winner, SmithKline Beecham:
- Step 3. We now start transferring issues from SmithKline Beecham to Glaxo Wellcome until the point totals of the two sides are equal. Because the layoff issue has the lowest point ratio, we start to transfer that item first. But we now see that transferring the entire layoff issue (worth 25 to SmithKline Beecham and 20 to Glaxo Wellcome) gives Glaxo Wellcome more points than SmithKline Beecham has . Thus, the entire layoff issue cannot be transferred. Glaxo Wellcome and SmithKline Beecham will need to compromise on the issue of layoffs (since it is the shared item). But compromise may not mean meeting each other halfway. Our goal is to equalize points between the two companies, and a little algebra will tell us exactly the extent to which Glaxo Wellcome and SmithKline Beecham should get their way on the issue of layoffs.
Step 4. Let be the fractional part of the layoff issue that will be transferred from SmithKline Beecham to Glaxo Wellcome. Because SmithKline Beecham had 25 points on the layoff issue, it loses points; because Glaxo Wellcome had 20 points on the layoff issue, it gains points.
Hence, the original point totals of 70 for SmithKline Beecham and 60 for Glaxo Wellcome become, after the transfer, and , respectively. Thus, if we want a fraction that will make SmithKline Beecham’s total points equal Glaxo Wellcome’s total points, then we need to solve the following equation:
Algebra Review Appendix
Linear Equations in One Variable
Step 5. We use algebra to solve this equation.
Reducing the fraction, we see that . Inserting 2/9 back into the equation, we see that
or approximately 64 points for each side. Thus, equality of points is achieved when SmithKline Beecham gives up two-ninths of what it wanted on the issue of layoffs and Glaxo Wellcome gets two-ninths of its way.
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Self Check 1
Suppose SmithKline Beecham had placed 25 points on CEO and 35 points on Layoffs, with all other point assignments as in Table 13.1. SmithKline Beecham is still the initial winner and still holds the same three items: Layoffs, Name, and CEO. What are the new point ratios? What is the new order in which issues are transferred?
- Layoffs now has a point ratio of ; Name has a point ratio of ; and CEO has a point ratio of . The order of transfer is now CEO, and then Layoffs, and finally Name.
Having seen how the adjusted winner procedure works, we must now ask the following question: Exactly what is it about the allocation produced by this scheme that would make someone want to use it? To answer this question, we need three definitions.
Equitable DEFINITION
A fair-division procedure is said to be equitable if each player believes he or she received the same fractional part of the total value.
Envy-free DEFINITION
A fair-division procedure is said to be envy-free if each player has a strategy that can guarantee him or her a share of whatever is being divided that is, in the eyes of that player, at least as large (or at least as desirable) as that received by any other player, no matter what the other players do.
Pareto-Optimal DEFINITION
A fair-division procedure is said to be Pareto-optimal if it produces an allocation of the property such that no other allocation achieved by any means whatsoever can make any one player better off without making some other player worse off.
The answer to our earlier question is given by the following theorem (whose proof can be found in Fair Division by Brams and Taylor, listed in the Suggested Readings on page 570):
Properties of the Adjusted Winner Allocation THEOREM
For two parties, the adjusted winner procedure produces an allocation based on each player’s assignment of 100 points over the items to be divided that has the following properties:
- The allocation is equitable.
- The allocation is envy-free.
- The allocation is Pareto-optimal.
Economists consider Pareto optimality (named after Vilfredo Pareto) to be an extremely important property, and the order of transfer in Step 3 on page 541 of the adjusted winner procedure is so important because it guarantees that the outcome is Pareto-optimal. The fact that the adjusted winner procedure produces an allocation that is efficient in this sense leads us to hope that it can and will play a future role in real-world dispute resolution.
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