### Formulas for fairness

#### Applying the math of cake cutting to conflict resolution

By IVARS PETERSON

### Envyfree Cake Division

Suppose Alice, Bob, and Carol want to divide up a cake. Alice starts by cutting it into three pieces that look equal to her. If Bob views one piece as being largest, he trims it to look equal to the piece he sees as second largest. This leaves one trimmed piece and two untrimmed pieces. Carol chooses one of the three pieces. Bob picks next and must take the trimmed piece if it's available. Alice gets the last piece.

By choosing first, Carol can't lose because she picks the piece she likes best. Bob can't lose because he can choose one of the two pieces he made sure were tied for largest. Alice ends up with one of the two untrimmed pieces, both of which are better in her eyes than the one Bob trimmed.

This procedure can then be repeated with the trimmings until the crumbs are so small that no one cares anymore.

For four players, the cake cutting has to start with an extra piece: Alice must slice the cake into five pieces. The extra piece ensures that no player is forced into taking second best. The number of extra pieces escalates for more people, the researchers discovered. For example, nine pieces are needed for five players, 17 pieces for six, and 2n-2 + 1 for n players.

It's interesting to note that after World War II, Great Britain, France, the United States, and the Soviet Union divided Germany into four zones of occupation, with Berlin, which fell within the Soviet zone, as a valuable "trimming" that was itself divided into four zones.

### Divorce points

In the adjusted winner method of fair division, the husband and wife secretly prepare lists showing how much they value each of the disputed items, ranking them by allocating a total of 100 points among the items.

For example, husband and wife might have come up with these allocations:

```Marital property                Husband                     Wife

Paris apartment                   35                         55
Paris studio                       6                          1
New York City coop                 8                          1
Farm                               8                          1
Cash and receivables               5                          6
Securities                        18                         17
Profit-sharing plan               15                         15
Life insurance policy              5                          4
----                       ----
Total                            100                        100
```

In this example, based on an actual divorce case, the husband and wife initially win the items that one person rated higher than the other. This gives the husband 45 points to the wife's 61 points. The profit-sharing plan, valued equally by both (15/15), goes to the husband, but that still puts him short of the wife's total. The next closest item is the cash and receivables, which can be divided up so that each party ends up with an equal number of points (in this case, 60.5 points-boldface indicates which party received each asset).

In this way, Brams argues, both husband and wife achieve a more satisfactory result than that actually decreed by the courts.

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