Item	Bob	Carol
Custody (sole)	23	65
Alimony	60	25
House	15	10
Total	100	100

60 + 15p = 65 + 10(1 - p)

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].

Manipulation

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:

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 one-fourth of her total value, whereas Bob is getting three-fourths 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:

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 one-third 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 one-sided information and capitalized on his knowledge of her preferences.

How to Optimally Deceive Bob

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 50-50, 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 for item A

Carol should announce for item B

Carol's total points (based on announced valuation): Carol's total points (based on true valuation): Bob's total points:

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, G₁ and G₂, and all true and announced valuations are restricted to integers. Suppose Bob's announced valuation of G₁ is x, where x 50, and suppose Carol's true valuation of G₁ is b. Then Carol's optimal announced valuation of G₁ is:

x + 1	if	b > x
x	if	b = x
x = 1	if	b < x

Frequently Asked Questions

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 fair-minded 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 agreed-upon amount.

When exactly should AW be used?

We suggest that AW first be tried out in negotiations that involve easily specified issues or well-defined 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 point-allocation 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 highest-ranked issue, compared with its next-highest-ranked issue, by specifying a ratio. Continuing down the list, comparing the second-highest-ranked issue with the third-highest-ranked 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 50-point 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 envy-freeness, 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 envy-freeness 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 envy-freeness.

The following example, given by J.H. Reijnierse and J.A.M. Potters, demonstrates the impossibility of satisfying all three properties (efficiency, envy-freeness, 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 40-40-40 allocation is equitable; it can also be shown to be efficient.

But it is not envy-free, 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 envy-free (because Bob would envy Ann).

Of course, this three-person 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, envy-freeness, 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).

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