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Adjusted Winner (AW) is an algorithm developed by Steven J. Brams and Alan D. Taylor to divide n divisible goods between two parties as fairly as possible.
The examples and other information found on this site are from the two books, Fair Division: From CakeCutting to Dispute Resolution (1996) and The WinWin Solution: Guaranteeing Fair Shares to Everybody (1999), both by Brams and Taylor.
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AW starts with the designation of goods or issues in a dispute. The parties then indicate how much they value obtaining the different goods, or "getting their way" on the different issues, by distributing 100 points across them. This information, which may or may not be made public, becomes the basis for fairly dividing the goods and issues later. Once the points have been assigned by both parties (in secret), a mediator (or a computer) can use AW to allocate the goods to each party, and to determine which good (there will be at most one) that may need to be divided.
Let's illustrate the procedure with an example. Suppose Bob and Carol are getting a divorce and must divide up some of their assets. We assume that they distribute 100 points among the five items as follows:
Item 
Carol 
Bob 
Retirement
Account 
50 
40 
Home 
20 
30 
Summer
Cottage 
15 
10 
Investments 
10 
10 
Other 
5 
10 
Total 
100 
100 
AW works by assigning, initially, the item to the person who puts more points on it (that person's points are underlined above). Thus, Bob gets the home, because he placed 30 points on it compared to Carol's 20. Likewise, Bob also gets the items in the "other" category, whereas Carol gets the retirement account and the summer cottage. Leaving aside the tied item (investments), Carol has a total of 65 (50 + 15) of her points, and Bob a total of 40 (30 + 10) of his points. This completes the "winner" phase of adjusted winner.
Because Bob trails Carol in points (40 compared to 65) in this phase, initially we award the investments on which they tie to Bob, which brings him up to 50 points (30 + 10 + 10). Now we will start the "adjusted" phase of AW. The goal of this phase is to achieve an equitable allocation by transferring items, or fractions thereof, from Carol to Bob until their points are equal.
What
is important here is the order in which items are transferred. This order
is determined by looking at certain fractions, corresponding to the items
that Carol, the initial winner, has and may have to give up. In particular,
for each item Carol won initially, we look at the fraction giving the ratio
of Carol's points to Bob's for that item:
(Number of points Carol assigned to the item)/(Number of points Bob assigned to the item)
In our example, Carol won two items, the retirement account and the summer cottage. For the retirement account, the fraction is 50/40 = 1.25, and for the summer cottage the fraction is 15/10 = 1.50.
We start by transferring items from Carol to Bob, beginning with the item with the smallest fraction. This is the retirement account, with a fraction equal to 1.25. We continue transferring goods until the point totals are equal.
Notice that if we transferred the entire retirement account from Carol to Bob, Bob would wind up with 90 (50 + 40) of his points, whereas Carol would plunge to 15 (65  50) of her points. We conclude, therefore, that the parties will have to share or split the item. So our task is to find exactly what fraction of this item each party will get so that their point totals come out to be equal.
We can use algebra to find the solution. Let p be the fraction (or percentage) of the retirement account that we need to transfer from Carol to Bob in order to equalize totals; in other words, p is the fraction of the retirement account that Bob will get, and (1p) is the fraction that Carol will get. After the transfer, Bob's point total will be 50 + 40p, and Carol's point total will be 15 + 50(1p). Since we want the point totals to be equal, we want to choose p so that it satisfies
50 + 40p = 15 + 50(1p)
Solving for p we get
90p = 15
p = 15/90 = 1/6
Thus, Bob should get 1/6 of the retirement account and Carol should get the remaining 5/6.
Recall that initially Bob is receiving: (1) the home (30 points), (2) the "other" items (10 points), and (3) the investments (10 points). Together with 1/6 of the retirement account, Bob's point total is now
30 + 10 + 10 + 40(1/6) = 50 + 40(1/6) 50 + 6.67 = 56.67
Recall that initially Carol is receiving: (1) the summer cottage (15 points). Together with 5/6 of the retirement account, Carol's point total is now
15 + 50(5/6) 15 + 41.67 = 56.67
Thus, each person receives exactly the same number of points, as he or she values their allocations.
Suppose that Bob and Carol want to fairly divide k goods, where k 2. We will illustrate the general description below with additional examples.
AW allocates goods as follows:
The AW procedure satisfies the following properties:
Suppose Bob and Carol are dividing three goods: A, B, and C.
Item

Bob's
reported values

Carol's
reported value

A

6

5

B

67

34

C

27

61

Total

100

100

X = 6+67 = 73 and Y = 61 so, initially Bob is assigned A and B, giving him 73 points, and Carol is assigned item C, giving her 61 points.
Check the properties
There are many examples of the application of AW in the Brams  Taylor books, Fair Division: From CakeCutting to Dispute Resolution and The WinWin Solution: Guaranteeing Gair Shares to Everybody. They are meant to demonstrate the reallife applicability of the AW procedure. We will illustrate two examples:
1. Panama Canal Treaty NegotiationsIn June 1974, the United States and Panama agreed on a treaty after two rounds of negotiations. The following point allocations are taken from The Art and Science of Negotiation (1982) by Howard Raiffa:
Issue 
United
States

Panama

1 US defense rights 
22

9

2 Use rights 
22

15

3 Land and water 
15

15

4 Expansion rights 
14

3

5 Duration 
11

15

6 Expansion routes 
6

5

7 Compensation 
4

11

8 Jurisdiction 
2

7

9 US military rights 
2

7

10 Defense role of Panama 
2

13

Total 
100

100

The United States wins on issues 1, 2, 4, and 6 (its points are underlined above), giving it 64 points (22 + 22 + 14 + 6), whereas Panama wins on issues 5, 7, 8, 9, and 10, giving it 53 points (15 + 11 + 7 + 7 + 13). The players tie on issue 3, which we initially give to the US for an initial allocation of 79 points.
Issue 3 has the smallest point ratio (15/15 = 1.0) and becomes the first issue used in the equitability adjustment. Let p denote the fraction of that issue that the US will retain, and (1p) the proportion that Panama will retain; then p must satisfy the following equation
64 + 15p = 53 + 15(1  p)
Solving for p, we find that p= 4/30 = 2/15. Thus, the United States should get 13.3 percent of its position (2 points), and Panama 86.7 percent of its position (13 points) on issue 3 to equalize their point totals. This results in the United States and Panama receiving each receiving 66 of their points.
2. A Hypothetical Divorce SettlementDivorces may be not only bitter and acrimonious but also very costly to the two parties as lawyers' fees accumulate. The following hypothetical divorce will illustrate how AW could be used to save heartache and money.
It is presented in Negotiating to Settlement in Divorce (1987, pp. 166  169), a manual for lawyers. Suppose Bob and Carol have decided to get a divorce. There are three major issues to be resolved: custody, alimony, and the house. Bob and Carol assign points as specified in the table below. Put the mouse over the numbers to see the justification for the point assignment. The justifications are quotations taken from Negotiating to Settlement in Divorce.
Item

Bob

Carol

Custody
(sole)

23

65

Alimony

60

25

House

15

10

Total

100

100

Under AW, Carol wins sole custody of John (their son), whereas Bob gets his way on alimony, and, initially, gets the house. However, Bob ends up with 75 points (60 + 15) and Carol with only 65 points. So Bob must give back some points on the house, which has the smallest ratio of Bob's points to Carol's points (15/10 = 1.5).
Let p denote the proportion that Bob will retain, and (1p) the proportion that Carol will retain. Then p must satisfy the following equation:
Solving for p gives p = 15/25 = 3/5. So Bob is entitled to 60 percent, and Carol to 40 percent, of the house. In terms of points, Bob will get 69 points [60 + 15(3/5) = 60 + 9 = 69], and so will Carol [65 + 10(2/5) = 65 + 4 = 69].
While the AW procedure has each party assign points independently,how can one know that the announced point assignments reflect each party's true valuations? There are certain situations, such as a divorce proceeding, in which each person will have more than an inkling of the preferences of the other person. Indeed, the intimate knowledge that a divorcing couple will have of each other's cares and concerns will frequently enable each to make rather accurate estimates of the points that the other spouse is likely to assign to the items in a divorce.
Thus we are led to ask whether the parties under AW can capitalize on their knowledge of each other's preferences. It turns out that if this knowledge is possessed by only one side  a relatively unlikely scenario  then the knowledgeable side can, in fact, exploit its informational advantage. However, if knowledge is roughly symmetric, then attempts by both sides to be strategic can lead to disaster, even without each spouse's being spiteful.
To illustrate this, let's start with a simple example. Suppose there are two paintings, a Matisse and a Picasso, and Carol thinks that the Matisse is worth three times as much as the Picasso, whereas Bob thinks the Picasso is worth three times as much as the Matisse. Thus, if Carol and Bob are sincere, then there point assignments will be as follows:
Matisse

Picasso

Total


Carol's true valuations 
75

25

100

Bob's true valuations 
25

75

100

Because of the symmetry in the preceding example, Carol will receive the Matisse and Bob will receive the Picasso, and there will be no need for an equability adjustment: both parties will receive 75 points.
Now suppose that Carol knows Bob's preferences, and that Bob does not know Carol's preferences. In the absence of any additional information, Bob will announce his true valuation (75 points for the Picasso and 25 points for the Matisse). Can Carol benefit from her knowledge?
The answer is yes. Carol should pretend that she likes the Matisse only slightly more than Bob likes the Matisse. This way, Carol will get all of the Matisse as she did before, but it will appear that she is getting only a little more than onefourth of her total value, whereas Bob is getting threefourths of his value (since he put 75 points on the Picasso). Consequently, a big equability adjustment will be required to transfer much of the value of the Picasso from Bob to Carol.
To be more precise, let's work from the numbers in this example to see the extent to which Carol can manipulate AW to her advantage. Knowing that Bob will place 25 points on the Matisse, Carol should place 26 points on this item and her remaining 74 points on the Picasso. Hence, the announced point totals, assuming Bob is sincere and Carol is not, will be as follows:
Matisse

Picasso

Total


Carol's announced valuations 
26

74

100

Bob's true valuations 
25

75

100

Initially Carol will get the Matisse, receiving 26 of her announced points, and Bob will get the Picasso, receiving 75 of his announced (and sincere) points. But now, since it appears that Bob is getting almost three times as many points as Carol does (75 to 26), there must be a large transfer from Bob to Carol.
The exact amount will be determined by solving the following equation for p.
26 + 74p = 75  75p
Solving for p, we find
149p = 49
p = 49/149 0.33
This gives Bob, in particular,
75  75(0.33) 75  25 = 50
of his points. In fact, it will appear that Carol, also, is getting the same low number of points
26 + 74(0.33) 26 + 24 = 50
However, let's consider Carol's true valuations. She is getting 75 points from winning all of the Matisse; in addition, she is getting 33% of the Picasso that she values at 25 points, which might mean that Bob would have to pay Carol onethird of the assessed value of the Picasso to keep it entirely for himself. Altogether, then, Carol is getting
75 + 25(0.33) 75 + 8.33 = 83.33
of her points. Of course, Bob could exploit Carol in the same manner if it were he, rather than Carol, who had onesided information and capitalized on his knowledge of her preferences.
This section will let you test your ability to deceive Bob. Suppose Bob and Carol are arguing over two items (A and B). Let Carol's valuation of these items be 70 points for A and 30 points for B, and Bob's valuation of these items be 50 points for A and 50 points for B.
Now suppose that Carol knows that Bob's valuation is 5050, and Bob does not know Carol's valuation. What should Carol announce in order to maximize her total point allocation (valuations are restricted to integers)?
Enter the values that you think Carol should announce in order to deceive Bob. Hit the "What is the total point allocation?" button to see the outcome of your allocation. Scroll down to see the answer.
Carol should announce 51 points for item A and 49 points for item B . Since it appears that Carol has only has a slight advantage over Bob, only a trivial fraction (1/101) of A will be transfered to Bob. Thereby Carol will end up with a generous 70  0.7 = 69.3 points (according to here true valuations).
As we have shown, a possible drawback of AW is that one player (with complete information) can exploit another player (without such information). In fact, the optimal response for any player is completely determined by the following theorem, which is proved in Fair Division.
Theorem: Assume there are two goods, G_{1} and G_{2}, and all true and announced valuations are restricted to integers. Suppose Bob's announced valuation of G_{1} is x, where x 50, and suppose Carol's true valuation of G_{1} is b. Then Carol's optimal announced valuation of G_{1} is:
x
+ 1

if

b > x

x

if

b
= x

x
= 1

if

b < x

Where can I get more information?
You can view the links page for more information.
What if the goods are not divisible?
Under AW, the only good or issue that must be divided is that on which an equitability adjustment is made. This will not be known in advance but only after the application of AW, so all goods and issues must be considered potentially divisible. In applying AW, probably the biggest problem is identifying a set of separable issues on which points are additive.
If the items being divided are not tangible property but more intangible issues, then the parties should decide before AW is applied what each would obtain if it came out the winner on an issue. Only on the one issue on which an equitability adjustment must be made will a finer breakdown actually be necessary.
This is a situation in which a mediator could play a valuable role. He or she could tell the parties the split on this issue but not which party is the relative winner. Each party, not knowing whether it got the larger or the smaller percentage, would then be motivated to reach a fairminded agreement. This could mean that one party could win entirely on that issue, or receive all of a good, but in turn it would have to pay the other party an agreedupon amount.
When exactly should AW be used?
We suggest that AW first be tried out in negotiations that involve easily specified issues or welldefined goods. Examples might include a dispute within a company over the division of job responsibilities, or the division of marital property in a divorce settlement, as illustrated in the examples page. If the procedure works well in these settings, it might be used in more complex negotiations.
We focused on divorce settlements because of the sheer magnitude of the problem  half of all marriages end in divorce in the United States. AW, in our view, provides a straightforward settlement device that takes due account of the interests of both parties. Since the settlement is not the product of protracted negotiations or court battles, it is likely to lead to a more satisfying and durable outcome as well as foster more civil future relations between the parties, which is especially important if children are involved.
But divorce settlements are not the only domain in which the application of AW seems desirable. In the political arena, negotiations over arms control or border disputes often involve a plethora of issues that AW could help to resolve. In the economic sphere, negotiations between labor and management over a new contract, or between two companies over a merger, are usually sufficiently complex that a pointallocation procedure could, we believe, prove very useful in finding a settlement that mirrors each side's most salient concerns.
All in all, what should I know before I attempt to use AW?
Given the simplicity of AW, would we need to hire a lawyer in a divorce?
The fact that AW may circumvent litigation that drags on in court and drains husbands and wives of their resources may be a social good, but it will not delight lawyers if it robs them of legal fees. We believe, however, that lawyers can play a valuable role in AW's use by
Of course, it will remain for lawyers and courts to determine what constitutes marital property, to which AW can then be applied.
How will I come up with the point assignments?
While honesty usually pays, it will not always be a simple matter to come up with point assignments that mirror one's valuations of the different issues. One way to facilitate this task is to have the parties begin by ranking the issues, from most to least important, in terms of their desire to get their way on each.
After the issues have been ranked, the parties face the problem of turning a ranking into point assignments that reflect their intensities of preferences for the different issues. In The Art and Science of Negotiation (1982), decision analyst Howard Raiffa discusses this problem in considerable detail, essentially concluding that a party must carefully weigh how much it would be willing to give up on one issue to obtain more on another.
To come up with point assignments, one option for a party would be to begin by rating the importance of winning on its highestranked issue, compared with its nexthighestranked issue, by specifying a ratio. Continuing down the list, comparing the secondhighestranked issue with the thirdhighestranked issue, and so on, parties would indicate, in relative terms, an "importance ratio" between adjacent issues.
For example, if there are three issues, and the importance ratios are 2:1 on the first issue relative to the second, and 3:2 on the second issue relative to the third, these will translate into a 6:3:2 proportion over the three issues. Rounding to the nearest integer, the point assignments would be 55, 27, and 18, respectively, on the three issues. A more systematic method for eliciting weightings, pioneered by mathematician Thomas L. Saaty and his associates and called "analytic hierarchy processing," could also be used.
Another option for a party is to begin by assigning points intuitively to items. These assignments could be "tested" by asking whether various 50point packages represent half the total value. To the extent that they do not, the initial point assignments for items would need to be modified. This process would continue until a party is satisfied that no further adjustments in its allocations of points to each item are necessary.
Will AW work with more than two players?
When there are more than two parties, there is no procedure that will simultaneously satisfy envyfreeness, efficiency, and equitability (see below for an example). However, it turns out that it is always possible to find an allocation that satisfies two of the three properties: A procedure that gives both efficiency and envyfreeness has been obtained by Dutch mathematicians J.H. Reijnierse and J.A.M. Potters; procedures (called "linear programs") that give both efficiency and equitability have been obtained by American mathematician Stephen J. Willson; and an equal division of each item to the parties gives both equitability and envyfreeness.
The following example, given by J.H. Reijnierse and J.A.M. Potters, demonstrates the impossibility of satisfying all three properties (efficiency, envyfreeness, and equitability). Suppose there are three disputants named Ann, Bob, and Carol, and assume that they allocate the following numbers of points to items X, Y, and Z:
Items

Ann

Bob

Carol

X

40

30

30

Y

50

40

30

Z

10

30

40

The only efficient and equitable allocation turns out to be to give X to Ann, Y to Bob and Z to Carol. Obviously, this 404040 allocation is equitable; it can also be shown to be efficient.
But it is not envyfree, because Ann will envy Bob for getting Y, which Ann considers to be worth 50 points. If we gave Y to Ann and X to Bob while still giving Z to Carol, this allocation would be efficient, but it would be neither equitable (because each player would get a different number of his or her points) nor envyfree (because Bob would envy Ann).
Of course, this threeperson hypothetical example does not preclude the possibility that all three properties can be satisfied in a particular situation; it says only that it is not always possible to guarantee their satisfaction when there are more than two parties. The fact that one cannot guarantee the satisfaction of efficiency, envyfreeness, and equitability, however, means that a hard choice might have to be made among them in situations with more than two parties.
I heard that the AW algorithm is patented, is this true?
Yes, the algorithm was patented in 1999 (patent number 5,983,205).