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Mathematics LibreTexts

8.2: Continuous Methods 1 - Divider/Chooser and Lone Divider Methods

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    The Divider/Chooser method and the Lone Divider method are two fairly simple methods for dividing a continuous set S. They can be used to split up a cake or to split up a piece of land. The Lone Divider method works for three or more players but works best with only three or four players. The Divider/Chooser method is a special case of the Lone Divider method for only two players.

    Divider/Chooser Method

    If you have siblings you probably used the Divider/Chooser method for fair division as a kid. Remember when Mom told one child to break the candy bar in half and then the other child got to choose which half to take: That was the Divider/Chooser method. It is a very simple method for dividing a single continuous item between two players. Simply put, one player cuts and the other player chooses. Call the object to be divided S. The divider is the player who cuts the object S. The divider is forced to cut the object S in a way that he/she would be satisfied with either piece as a fair share. The chooser then picks the piece that he/she considers a fair share. Once the chooser picks a piece the divider gets the remaining piece. The divider always gets exactly half the value of S. The chooser sometimes ends up with more than half of the value of S. This sounds contradictory but remember that each player has his/her own value system.

    Example \(\PageIndex{1}\): Divider/Chooser Method with a Pizza

    Bill and Ted want to divide a pizza that is half cheese and half pepperoni. Bill likes cheese pizza but not pepperoni and Ted likes all pizza equally.

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    Figure \(\PageIndex{1}\): Half Cheese and Half Pepperoni Pizza
    1. If Bill cuts and Ted chooses, describe the fair division.

    Bill likes cheese but not pepperoni so he sees all the value of the pizza in the cheese part. He cuts the pizza in a way that half of the cheese part ends up in each piece. The most obvious way to do this is to cut it in half vertically. You might think that Bill would choose half the pizza as the cheese side and half as the pepperoni side in the hopes he would end up with the entire cheese side. However, that would not be a division that results in two equal halves in his eyes. In other words, he could end up with the entire pepperoni side which he does not like.

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    Figure \(\PageIndex{2}\): Bill’s Division of the Pizza

    Since Ted likes all pizza equally and both parts are the same it does not matter which piece Ted chooses. Let’s say he chooses the piece on the right.

    Ted is happy because he got half of the pizza, a fair share in his value system.

    Bill is happy because he got half of the cheese part of the pizza, half of the value (or a fair share) in his value system.

    1. If Ted cuts and Bill chooses, describe three different fair divisions.

    Remember that one of our assumptions is that Ted does not know that Bill only likes the cheese part of the pizza. Since Ted likes all pizza equally, he should cut the pizza in half in terms of the volume (for our two-dimensional pizza, cut it in half in terms of the area).

    1. Ted could cut the pizza in half vertically just like Bill did in part (a). It would not matter which piece Bill chose since both pieces are the same.
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    Figure \(\PageIndex{3}\): Ted’s Division of the Pizza, #1
    1. Ted could cut the pizza in half horizontally so that one piece was all cheese and the other piece was all pepperoni.
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    Figure \(\PageIndex{4}\): Ted’s Division of the Pizza, #2

    Bill would choose the cheese half and Ted would get the pepperoni half. Bill is happy because he gets 100% of the value of the pizza in his value system. Ted is happy because he gets 50% of the value of the pizza in his value system.

    1. Ted could cut the pizza at an angle so that each piece is part pepperoni and part cheese.
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    Figure \(\PageIndex{5}\): Ted’s Division of the Pizza, #3

    Since Bill only likes the cheese part, he should choose the piece on the left with the 75% of the cheese part of the pizza. Bill is happy because he gets 75% of the value of the pizza in his value system. Ted is happy because he gets 50% of the value of the pizza in his value system.

    Example \(\PageIndex{2}\): Divider/Chooser Method with a Sub Sandwich (Example \(\PageIndex{4}\) Continued)

    In Example \(\PageIndex{4}\), George and Ted want to split a 12-inch sandwich worth $9. Half the sandwich is vegetarian and half the sandwich is meatball. George does not eat meat at all. Ted likes the meatball part twice as much as vegetarian part. We already figured out how each player should cut the sandwich.

    1. If George cuts which piece should Ted choose?
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    Figure \(\PageIndex{6}\): George’s Division of the Sandwich

    Ted sees the meatball part with a value of $6 and the vegetarian part with a value of $3. Half of the vegetarian part would be worth $1.50 to him. The larger part of the sandwich would have a value of $6.00 + $1.50 = $7.50 and the smaller part of the sandwich would have a value of $1.50. He should choose the larger part of the sandwich.

    1. If Ted cuts which piece should George choose?
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    Figure \(\PageIndex{7}\): Ted’s Division of the Sandwich

    George does not eat meat so the smaller all meatball piece is worth $0 to him. The larger piece contains all the vegetarian part of the sandwich so it contains all the value to him. George should choose the larger piece which is worth $9 to him.

    Note that in Example \(\PageIndex{2}\), part (a), Ted’s piece was worth $7.50 to him and in part (b) George’s piece was worth $9 to him. In both situations, the chooser ends up with more than a fair share. The divider always gets exactly a fair share. Given the choice, it is always better to be the chooser than the divider.

    Lone Divider Method:

    The Divider/Chooser method only works for two players. For more than two players we can use a method called the Lone Divider method. The basic idea is that a divider cuts the object into pieces. The rest of the players, called choosers, bid on the pieces they feel are fair shares. Each chooser is given a piece he/she considers a fair share with the remaining piece going to the divider. As we saw in the Divider/Chooser method, the divider always gets exactly a fair share but the choosers may get more than a fair share.

    Example \(\PageIndex{3}\): Lone Divider Method, Basic Example

    Three cousins, Russ, Sam, and Tom want to divide a heart-shaped cake. They draw straws to choose a divider and Russ is chosen. Russ must divide the cake into three pieces. Each piece must be a fair share in his value system. Assume Russ divides the cake as shown in the following figure.

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    Figure \(\PageIndex{8}\): Russ’s Division of the Cake

    Sam and Tom now bid on each piece of the cake. They privately and independently determine a value for each piece of the cake according to their value system.

    Sam sees the value of the cake as: Piece A – 40%, piece B – 30%, and piece C – 30%.

    Tom sees the value of the cake as: Piece A – 35%, piece B – 35%, and piece C – 30%.

    Since there are three players, a fair share would be 1/3 or 33.3%.

    Each player writes down which pieces they would consider a fair share of the cake. These are called the bids.

    Sam would bid {A} and Tom would bid {A, B}.

    Neither Sam nor Tom consider piece C to be a fair share so piece C goes to Russ, the divider.

    Sam only considers piece A to be a fair share so give Sam piece A.

    Tom would be satisfied with either piece A or B. Since piece A was given to Sam, Tom gets piece B.

    Notice that Sam believes his piece is worth 40% of the value and Tom believes his piece is worth 35% of the value so both of them got more than a fair share. The divider Russ got a piece worth exactly 33.3% or a fair share in his opinion. The divider always receives exactly a fair share using this method.

    Summary of the Lone Divider Method:

    1. The n players use a random method to choose a divider. The other n-1 players are all choosers.
    2. The divider divides the object S into n pieces of equal value in his/her value system.
    3. Each of the choosers assigns a value to each piece of the object and submits his/her bid. The bid is a list of the pieces the player would consider a fair share.
    4. The pieces are allocated using the bids. Sometimes, in the case of a tie, two pieces must be combined and divided again to satisfy all players.

    Example \(\PageIndex{4}\): Lone Divider Method with a Cake, No Standoff

    A cake is to be divided between four players, Ian, Jack, Kent, and Larry. The players draw straws and Ian is chosen to be the divider. Ian divides the cake into four pieces, S1, S2, S3, and S4. Each of these pieces would be a fair share to Ian. The other three players assign values to each piece as summarized in Table \(\PageIndex{9}\).

    Table \(\PageIndex{9}\): Players’ Valuation of Shares

    S1

    S2

    S3

    S4

    Ian

    25%

    25%

    25%

    25%

    Jack

    40%

    30%

    20%

    10%

    Kent

    15%

    35%

    35%

    15%

    Larry

    40%

    20%

    20%

    20%

    Since there are four players a fair share is 25% of the cake. The three choosers submit their bids as follows:

    Jack: {S1, S2}, Kent: {S2, S3}, and Larry: {S1}

    The distribution is fairly straightforward. Larry gets S1 since it is the only piece he considers a fair share. With S1 taken Jack will get S2, his only remaining possible fair share. With S2 taken Kent will get S3, his only remaining possible fair share. That leaves S4 for the divider Ian.

    Example \(\PageIndex{5}\): Lone Divider Method with a Piece of Land, Simple Standoff

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    Figure \(\PageIndex{10}\): A Map of the Land

    Amy, Bob and Carly want to divide a piece of land using the lone-divider method. They draw straws and Bob is chosen as the divider. Bob draws lines on the map to divide the land into three pieces of equal value according to his value system.

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    Figure \(\PageIndex{11}\): Bob’s Division of the Land

    Amy and Carly bid on the pieces of land that they would consider fair shares. Both of them like the beach and the fields but not the trees so their bids are Amy: {B, C} and Carly: {B, C}.

    Since neither Amy nor Carly want piece A with the trees, that piece will go to the divider Bob.

    Both Amy and Carly would be happy with either of the remaining pieces. A simple way to allocate the pieces is to toss a coin to see who gets piece B with the beach. The other player would get piece C with the fields.

    Example \(\PageIndex{6}\): Lone Divider Method with a Piece of Land, More Complicated Standoff

    Let’s look at the land in Example \(\PageIndex{5}\) again. This time let’s assume the bids are Amy: {B} and Carly: {B}.

    Since both Amy and Carly want the same piece of land we have a standoff. Neither of the women want pieces A and C so give one of them to the divider Bob. Toss a coin to choose which piece he gets. Let’s assume the toss results in Bob getting piece A.

    To resolve the standoff we combine pieces B and C to make one large piece.

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    Figure \(\PageIndex{12}\): Combining Pieces B and C

    We now have one piece of land to be divided equally between two players. Amy and Carly can use the Divider/Chooser method to the finish the division. Toss a coin to determine the divider. Assume Amy is chosen to divide and divides the land as shown in Figure \(\PageIndex{13}\).

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    Figure \(\PageIndex{13}\): Amy’s Division of the Piece

    Let’s assume that Carly picks piece E, leaving piece D for Amy, to complete the fair division.

    You can see from the previous examples that sometimes the lone divider method is very straight forward and other times it can be more complicated. Imagine how complicated the method could become with 10 players. Regardless of the number of players or how complicated the division is, one fact remains. The choosers always get at least a fair share while the divider only gets an exact fair share. It is better to be a chooser than the divider.

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