5.2: Divider-Chooser
The first method we will look at is a method for continuously divisible items. This method will be familiar to many parents - it is the “You cut, I choose” method. In this method, one party is designated the divider and the other the chooser , perhaps with a coin toss. The method works as follows:
- The divider cuts the item into two pieces that are, in his eyes, equal in value.
- The chooser selects either of the two pieces
- The divider receives the remaining piece
Notice that the divider-chooser method is specific to a two-party division. Examine why this method guarantees a fair division: since the divider doesn’t know which piece he will receive, the rational action for him to take would be to divide the whole into two pieces he values equally. There is no incentive for the divider to attempt to “cheat” since he doesn’t know which piece he will receive. Since the chooser can pick either piece, she is guaranteed that one of them is worth at least 50% of the whole in her eyes. The chooser is guaranteed a piece she values as at least 50%, and the divider is guaranteed a piece he values at 50%.
Two retirees, Fred and Martha, buy a vacation beach house in Florida together, with the agreement that they will split the year into two parts.
Solution
Fred is chosen to be the divider, and splits the year into two pieces: November – February and March – October. Even though the first piece is 4 months and the second is 8 months, Fred places equal value on both pieces since he really likes to be in Florida during the winter. Martha gets to pick whichever piece she values more. Suppose she values all months equally. In this case, she would choose the March – October time, resulting in a piece that she values as \(8/12 = 66.7\%\) of the whole. Fred is left with the November – February slot which he values as 50% of the whole.
Of course, in this example, Fred and Martha probably could have discussed their preferences and reached a mutually agreeable decision. The divider-chooser method is more necessary in cases where the parties are suspicious of each other’s motives, or are unable to communicate effectively, such as two countries drawing a border, or two children splitting a candy bar.
Dustin and Quinn were given an apple pie and a chocolate cake, and need to divide them. Dustin values the apple pie at $6 and the chocolate cake at $4. Quinn values the apple pie as $4 and the chocolate cake at $10. Describe a fair division if Quinn is dividing, and specify which “half” Dustin will choose.
- Answer
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There are a lot of possible fair divisions Quinn could make. Since she values the two desserts at $14 together, a fair share in her eyes is $7. Notice since Dustin values the desserts at $10 together, a fair share in his eyes is $5 of value. A couple possible divisions:
\(\begin{array}{|l|l|l|}
\hline \text { Piece 1 } & \text { Piece 2 } & \text { Dustin would choose } \\
\hline \frac{7}{10} \text { chocolate (\$7) } & \frac{3}{10} \text { chocolate (\$3) } & \text { Piece 2. Value in his eyes: } \\
\text { No pie (\$0) } & \text { All pie (\$4) } & \$ 6+3 / 10 \cdot \$ 4=\$ 7.20 \\
\hline \text { 1/2 chocolate (\$5) } & \frac{1}{2}\text { chocolate (\$5) } & \text { Either. Value in his eyes: } \\
\frac{1}{2} \text{ pie (\$2) } & \frac{1}{2}\text { pie (\$2) } & \frac{1}{2} \cdot \$ 6+\frac{1}{2} \cdot \$ 4=\$ 5 \\
\hline
\end{array}\)
Things quickly become more complicated when we have more than two parties involved. We will look at three different approaches. But first, let us look at one that doesn’t work.