The Lone Chooser Method
This is another three step process, and again we can extend it to more than three players. After a chooser and two dividers are chosen at random, they proceed as follows.
- Step 1. First Division--The two dividers split the pizza by the divider-chooser method.
- Step 2. Second Division--Each divider now divides his part into three parts he considers equal.
- Step 3. Selection--The chooser picks one piece from each divider, and each divider keeps whatever he has left. Again, we have fair shares for everybody.
As an example, we will let Don, Dora, and Cal divide a pizza, which they agree is worth $18 (since each was willing to put up $6). Again, they get a half-anchovie half-pepperoni pizza. They will use the Lone Chooser Method, and by flipping coins with odd person out, Cal becomes the chooser leaving Don and Dora the dividers. When Don and Dora flip again to see who is chooser in Step 1, Dora wins. As is usually the case, each player has a different value system.
- Cal can't stand anchovies.
- Don likes anchovies and pepperoni equally well.
- Dora likes anchovies three times as much as pepperoni.
Here is what the pizza looks like to the three players.
Picture could not be drawn. Visualize the picture or try drawing the picture.
Don must make the first cut, and since he has no preference between pepperoni and anchovies, all he needs to do is split the pizza into two equal-sized pieces. Suppose he cuts so that one piece is one-third pepperoni and two-thirds anchovies. In his value system, each piece has identical value. Now Dora must choose between these two pieces. Here is what they look like to Dora.
Visualize the picture or try drawing the picture.
It is pretty obvious to Dora which is the better deal--the $10.50 worth of pizza is clearly preferable to the $7.50 piece. The next step is for Don and Dora each to cut their pieces into thirds. Again, it does not make much difference to Don how to go about it except that each piece should be the same size. The most straightforward way for him to do this is to leave one entirely anchovie piece and two entirely pepperoni pieces. Each of these is the same size and each is 1/6 of the original pizza. It is somewhat more complicated for Dora; she needs to make each piece worth $3.50. One way to do that is to put the $1.50 worth of pepperoni pizza with $2 worth of anchovie pizza and then split equally what is left of the anchovie part. Two dollars worth of the anchovie part is is 2/9, and since Dora's anchovie part is 1/3 of the original pizza, the part that remains attached to the pepperoni is 2/27 of the original pizza. Since Dora's pepperoni part was originally 1/6 of the pizza, that makes the mixed piece 13/54 of the original, somewhat bigger than 1/6. That makes the two anchovie pieces (after she splits what was left) 7/54 of the original pizza.
And finally it is time for Cal to choose one piece from Don's three pieces and one from Dora's three pieces. Since Don's pieces are all the same size and two of them are pepperoni, Cal takes a pepperoni piece from Don. Since only one of Dora's pieces has any pepperoni at all, that one is obviously Cal's choice of what she has to offer. Now let us look at what everybody ended up with.
|Share||18/54 P and 4/54 A||9/54 P and 9/54 A||no P and 14/54 A|
|Value to Cal||$12.00||$6.00||$0.00|
|Value to Don||$7.33||$6.00||$4.67|
|Value to Dora||$5.00||$6.00||$7.00|
Notice that both Cal and Dora feel that they got more than a fair share ($6 worth) and that Cal, being the lone chooser, did best of all (by his estimation). Also notice the Don, who had to divide twice, gotonlya fair share, and if he looks around after the division, he will think that Cal got a better deal than he did.
We emphasize again that getting a fair share is all a fair division scheme promises to deliver; it doesnotpromise the best share. We also point out that things would have come out different (but still fair) had the initial coin tosses come out different.