Adjusted Winner
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 
To see a justification for these points click here
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.
Adjusted Winner (General Description)
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

Check the properties