Example Chapter 14 #3 Suppose Mike and Phil are roommates in college, and they encounter serious conflicts during their first week of school. Their resident advisor has taken Survey of Math, and decides to use the adjusted winner procedure to resolve the dispute. The issues agreed upon, and the independently assigned points are the following:

Use the adjusted winner procedure to resolve this dispute.

Solution

The first step in the adjusted winner procedure is to have Mike and Phil distribute 100 points over the items that reflects the items worth to them. The RA has already had Phil and Mike do this.

Each item is initially given to the party that assigned it more points, leaving out items which received the same points by each person, and the points are totaled for the items they received. I've used hearts (♥) in the following table to indicate who gets what item at this stage.

Items which received the same number of points are given to Phil, since he has the lower point total (this will make Phil's point total higher than Mike's).

If the point totals are equal we are done, since both Mike and Phil will have what they perceive to be equal 'worth' of the issues they agreed to divide up (note this worth could be greater than 50%). This is not the case, so there is more to do! We are going to have to find the issues that we can transfer from Phil to Mike to make their point totals equal. This will result in a fractional transfer of an item.

To decide which item to transfer, we need to take into account the relative importance of the items to the two. We can do this by taking the ratio of Phil's points to Mike's for the the items Phil has, and transferring items with the lowest point ratios first (see box below for why we do this). We always put the point value from the person we are transferring something away from (i.e., the one with the highest point total) in the numerator; this means the ratio should always be greater than or equal to 1.

Since the alcohol policy has the lowest point ratio, it is what we will transfer. If we transferred this entirely from Phil to Mike, Mike would have more points that Phil. So we only want to transfer part of the alcohol policy to Mike. Let's let x be the amount we want to transfer to Mike. So Mike will have x of the alcohol policy, and Phil will have the remaining (1-x) of the policy. Since we want their points to be equal, we have:

66+15x = 55 + 15(1-x)

Which we can solve for x = 0.13. The points for Mike and Phil are both 68 (within rounding).

Therefore, Mike gets to decide the room party policy, cleanliness, lights out time and will get 13% say in the alcohol policy. Phil gets 87% say in the alcohol policy, and gets to decide the stereo level, smoking rights, phone time, and visitor policy.

How can you get 13% of something? If it is an item, it could be sold and Mike would get 13% of the sale price. Here we have an issue, and not a item, so we can convert the item to be divided between Mike and Phil. Mike and Phil could each decide what they would like as a policy, and then the policy could be set closer to Phil's liking than Mike's.

In the previous example, Mike and Phil ended up with 68 points each. What if they had shared something other than the alcohol policy? What would their point total be then?

Let's say they shared the stereo level. If we transferred this entirely from Phil to Mike, Mike would have more points that Phil. So we only want to transfer part of the stereo level to Mike. Let's let x be the amount we want to transfer to Mike. So Mike will have x of the stereo level, and Phil will have (1-x) of the stereo level policy. Since we want their points to be equal, we have:

66+4x = 48 + 22(1-x)

Which we can solve for x = 0.15. The points for Mike and Phil are both 66.6 (within rounding).

Although the points are equal, they are not as high as if we had transferred the smoking policy. Since higher points means more satisfaction, transferring the smoking policy was a better solution.

Transferring the item that has the lowest point ratio is always the best choice, since the person with the most points will have their point score reduced the least for each point the other person gains. This will result in a higher final point total for each.

Example Mike and Phil found that they got along well after their first year of college. They rented a house and purchased some big ticket items together, splitting the cost equally on each item. Now, they each have jobs after getting their degree, and are dividing up the contents of their household. Remembering fondly the procedure the RA used to get them over their initial hostilities, they decide to use the adjusted winner procedure to distribute the goods equitably, and without envy.

The items they have are the following, with 100 points distributed by each over the items:

They assign the items initially as follows, and figure out their point totals:

Since the point totals are not equal, they know they will have to figure out an item to transfer from Mike to Phil. They do this by constructing point ratios for the items Mike has been assigned.

The item with the lowest ratio is the TV. To be equitable and envy free, they should sell the TV, and divide the money between them based on the following calculation.

Let x be the amount of the TV we want to transfer from Mike to Phil. Therefore, Phil gets 25x towards his point total and Mike gets 30(1-x). Their point totals become:

50 + 30(1-x) = 50 + 25x

Which we solve for x = 0.54. Mike and Phil both get 63.3 points (within rounding).

Therefore they should sell the TV, and give 0.54 of the money they get to Phil, who also gets the BBQ and couch. Mike gets 0.46 of the money from the TV, as well as the stereo and recliner. Both Mike and Phil were treated equitably (received the same number of points) under this procedure, and neither would be happier with what the other person got (envy-free).