13.4 Taking Turns
For many of us, an early lesson in fair division happens in elementary school with the choosing of sides for a kickball team or some such thing. Surprisingly, the same fair­division procedure­­
taking turns­­is often used in more serious situations such as property settlement in a divorce.
Taking turns is fairly self­explanatory. With two parties, one party selects an object, then the other party selects one, the the first party again, and so on. But in this context, there are some interesting questions that should be considered:
1) How do we decide who chooses first?
2) Should a player always choose the object he or she wants the most or are there strategic considerations that should be taken into account?
The answer to question 1) could be "toss a coin" or bid for the right to go first, as in an auction.
Question 2) is the one we will consider with the following example. 1
Suppose that Bob and Carol are getting a divorce, and their four main possessions, ranked from best to worst by each, are as follows:
Assuming that they do not know what each other's preferences are, we assume that they will choose sincerely, choosing their most preferred item first, etc., which means if Bob chooses first, the items will be allocated as follows:
Bob:
Pension
Investments
Carol:
House
Vehicles
So Bob gets his 1st and 3rd favorites, Carol gets her 1st and 4th. But suppose they know what each other's preferences are. Then the strategy for choosing their items may be different. In this case, they can use something called the bottom­up strategy, discovered by the mathematicians D. A. Kohler and R. Chandrasekaran in 1969.
The intuition is fairly simple. Let’s make two assumptions about rational players: 1) A rational player will never willingly choose his or her least­preferred alternative, and
2) A rational player will avoid wasting a choice on an object that he or she knows the other player doesn’t want.
The bottom­up strategy starts at the last blank (last item chosen) and puts the opposite player's last choice there. The next­to­last blank (next­to­last item chosen) would be the other player's last choice, and so on, until the blanks are all filled. This seems counter­intuitive, but it works!
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With these assumptions in mind, let’s look at the example again and think about the mental calculation that Bob will go through, if he is the first to choose. He knows the eventual sequence of choices will fill in all of the following blanks:
Bob:
Carol:
__________
1
__________
___________
3
2
___________
4
Knowing that Carol did not rank Pension in her top two, Bob can save his first choice (Pension) for last and choose his second choice (House) first. He assumes that Carol will choose one of her top favorites first. So it plays out differently if Bob uses a bottom­up strategy.
What if Carol chooses first? What should her strategy be? 3
Suppose that Bob and Carol are choosing between five fruits­­Apple, Banana, Kiwi, Orange, and Watermelon­­and Bob is choosing first. Suppose that Bob and Carol have the following rankings of the fruits:
Bob
1st Apple
2nd Banana
3rd Kiwi
4th Orange
5th Watermelon
Carol
Kiwi
Watermelon
Orange
Apple
Banana
If Bob chooses first, consider the mental calculation he will go through. He knows the eventual sequence of choices will fill in all of the following blanks:
Bob:
Carol:
__________
___________
___________
5
3
___________
___________
1
2
4
He can consider his first and second choices fairly safe since they are Carol's last two choices, so he saves Banana for last and Apple for next to last and picks Kiwi first. Meanwhile, Carol figures that Watermelon is safe, so she can save that for her last pick. Now suppose Carol picks first. What will the allocation be?
Carol:
Bob:
__________
___________
___________
___________
___________
How would the picks have been different if they had not known the other person's choices?
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Example: Jim and Ed attend a car auction and together win the bidding for a collection of four classic cars: a ’59 Chevy, a ’48 Packard, a ’61 Austin­Healy, and a ’47 Hudson. They will take turns to split the cars between them. Their preferences are shown in the table below.
Use the bottom­up strategy to allocate the cars if
a) Jim picks first.
Jim: ________ __________ 1
3
Ed: __________
____________
2
4
b) Ed picks first.
Ed: ________ __________ Jim: __________
____________
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Another example: Kathy and Anne are sisters and are splitting up the movies they have. Here are the titles: Titanic (T), Dumb & Dumber (D), Breakfast at Tiffany’s (B), Monty Python (M), and What About Bob? (W)
Their preferences are shown below.
Use the bottom­up strategy to allocate the movies if
a) Kathy picks first.
Kathy: ________ __________ Anne: __________
_____________ ____________
b) Anne picks first.
Anne: ________ Kathy: __________
__________ _____________ ____________
Homework: Read 13.4 and do HW #18 (p. 479: 18­26)
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#22) Suppose that Donald and Ivana Trump decide to settle their divorce (described in Exercise 1 on p. 476) by taking turns. Assume that they both use the bottom­up strategy, and Donald chooses first. Determine Donald's first choice and the final allocation. (Assume that Donald values the Connecticut estate slightly more than he values the Trump Plaza apartment.)
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