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Chapter 14: Fair Division
Part 3 – Taking Turns and Divide and Choose
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Taking Turns We only consider division by using the taking turns method when the division is made among two parties. Somehow an initial decision must be made as to which party will go first. Each party takes turns selecting the available items, until all items are distributed. Some questions related to this procedure are the following: How do we decide who chooses first? Choosing first can be a significant advantage. Could we adjust the procedure to compensate the party that chooses second? Based on knowledge of the other player’s preferences are there any strategies a player could follow that would be in their best interest.
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Some questions related to this procedure are the following:
Taking Turns Some questions related to this procedure are the following: How do we decide who chooses first? Flip a coin ? Bid ? Choosing first can be a significant advantage. Could we adjust the procedure to compensate the party that chooses second? Many possibilities – answer could depend on the situation. Based on knowledge of the other player’s preferences are there any strategies a player could follow that would be in their best interest. If a player knew the other’s preferences, it may be to their advantage to choose strategically. One strategy is the “bottom-up strategy”. This strategy might involve some risk.
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Taking Turns – Strategy
Suppose two players will divide a list of items (goods or issues) using the taking turns method. Suppose neither player knows the preferences of the other before the procedure begins. For example, suppose Bob and Carol will divide the following items: A house, pension, investments and vehicles. Suppose their preferences are as follows: Not knowing the preferences of the other, the only reasonable strategy for each player is to choose their most preferred item from those remaining. Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles
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Taking Turns – Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Suppose each player does not know the preferences of the other. Therefore, at each turn, each player will choose their most preferred item from those that remain. In this case, if Bob went first, the result would be as follows: Bob: Pension Carol: House Bob: Investments Carol: Vehicles Bob gets the pension and investments and Carol gets the house and vehicles.
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Taking Turns – Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Now, if Carol went first, and each player followed the same strategy, the result would be as follows: Carol: House Bob: Pension Carol: Investments Bob: Vehicles Carol gets the house and investments and Bob gets the pension and vehicles.
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Now, we consider a strategy that both players could use if they knew the preferences of the other player. Remember, if Bob went first, without knowing Carol’s preference, the result was: Bob gets pension and investments and Carol gets the house and vehicles. By knowing the other’s preferences, could either player do better than the result above?
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles For example, if Bob knew Carol put the pension as her third choice and she really wasn’t going to pick it right away, he might pick the house first. Then the result could be as follows: Bob: House Carol: Investments Bob: Pension Carol: Vehicles In a sense, Bob played insincerely (because he did not pick according to his real preferences) but by doing this he actually did better according to those preferences: he ended up with his first two choices instead of his first and third as he would have by playing sincerely.
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Taking Turns – the Bottom Up Strategy
There is a strategy both players could use that would potentially provide a better result for each player in this type of fair division. The strategy is called the “bottom-up strategy”. It was discovered in 1969 by two mathematicians named Kohler and Chandrasekarean. To use the strategy a player must know the other’s preferences. If the players provided each other with their preferences before taking turns diving them up, both could use the strategy simultaneously. The strategy is as follows for each player: Complete a table representing choices at each turn by alternating filling in a player’s last choice with the other’s least preference starting with whichever player chooses last and working backwards to whichever goes first.
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles For example, given the preferences in the table, assuming he’ll go first, Bob could plan his strategy as follows: (using the bottom up strategy) Bob: ______ ______ Carol: _______ _______ Being first, Bob would have an advantage. He knows that the choices will fill in the blanks above. And he knows Carol would be last. He will leave his last preference for her on her last choice.
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Bob fills in his lowest preference for Carol’s last choice. Then he fills in his last choice with Carol’s lowest preference. Bob: ______ __pension__ Carol: _______ __vehicles_ Basically, this is because he expects she will not pick that item first, since it is lower on her list of preferences. Bob continues his planning by filling in choices from right to left…
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Next, Bob will fill in Carol’s next choice (moving to the left) with his next lowest preference. Bob: ______ __pension__ Carol: __investments___ __vehicles_ Finally, Bob fill’s in his next lowest choice with Carol’s next lowest preference, which is the house. At each step, as we move from the right to left filling in the diagram, selecting preferences from the bottom of each list of preferences and moving up. For each player, we select the next available preference from the bottom of the list moving up, but from the other player list.
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Finally, Bob fills in the next choice (moving backwards – left to right) with the next lowest available item (which is actually the first item) from Carol’s preferences. Bob: __house__ __pension__ Carol: __investments___ __vehicles_ By following this “bottom-up strategy” Bob is able to do better than by simply choosing based on his preferences. Of course, he still had a significant advantage by going first, which has nothing to do with his use of this specific strategy.
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles We can now ask, suppose Carol could choose first, and she planned her choices based on the bottom-up strategy. What would be the result? Carol ________ __________ Bob _________ __________ We start with whoever has the last choice. In this case Bob chooses last since there are 4 items and Carol goes first. Following the bottom-up strategy, Carol will put her last preference as Bob’s last choice.
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Carol ________ _investments__ Bob _________ __vehicles___ Carol continues by putting Bob’s next to last preference as her last choice. That is, moving from the bottom-up of what is available, Carol chooses for her last choice the lowest available preference of Bob’s. She puts investments as her last choice.
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Carol: ________ _investments__ Bob: __pension_ __vehicles___ Moving right to left, we fill in Carol’s choice with Bob’s next lowest preference that is still available. That is, Carol will fill in the house for her first choice…
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Taking Turns – the Bottom Up Strategy
Rank Bob Carol 1st Pension House 2nd Investments 3rd 4th Vehicles Carol _house__ _investments__ Bob __pension_ __vehicles___ Therefore, Carol takes the house and investments, her first two choices, and Bob is left with the pension and vehicles, his first and last choice. Clearly, there is an advantage to using the bottom up strategy but also there is an advantage in being the first to choose.
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Divide and Choose is an ancient method of fair-division.
The method is simple for two players. There are variations of divide and choose for more than two players (we will study three of these variations later) For two players: One player divides the object (or objects) and the other player chooses his or her share. Strategies: Player 1 must divide the object in a way that he or she thinks both are fair. Player 2 chooses the part containing the most value in his or her estimation.
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Divide and Choose Both players use a strategy to insure that they receive a “fair” share. Player 1 divides the object to insure he or she would be satisfied with either piece – that is, would consider either piece to be a fair share. Player 2 chooses a piece that he or she considers a fair share. An interesting example of Divide and Choose is the Convention of the Law of the Sea. The Convention is an international agreement arranged by the United Nations for managing international interests in ocean resources. For example, it protects the mining interests of developing countries. If an industrialized country wants to develop or mine a portion of the seabed, that country must propose a division of that portion into two tracts. After that, an organization within the U.N. known as the International Seabed Authority, which represents the interests of developing countries, chooses one tract to be reserved for later use by developing countries.
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Divide and Choose There are some considerations that could arise naturally with divide and choose which go beyond a brief and general analysis. For example, is it better to be the divider or the chooser? The answer may depend on the situation. Depending on the situation, it is conceivable that the divider could make a division which he or she considered fair and yet the chooser felt that neither piece was fair.
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