The Method of Markers The method of markers is a discrete fair-division procedure...

The Method of Markers

The method of markers is a discrete fair-division procedure that is used when there are many more items than there are players and the similarly valued items can be lined up. Players mark sections of the lineup that they feel are equal in value. The method is explained here for three people, but it can easily be extended to include more people.

Suppose that Ally, Babs, and Cal will split up a collection of 17 snacks. The snacks are lined up, as shown next, and the players secretly consider the value of sections of the lineup. (Pictures taken from http://office .microsoft.com/clipart/default.aspx.)

STEP 1: Each player will place two markers so that they divide the lineup into three parts the player considers to be of equal value. Ally will place markers A₁ and A₂. Babs will place markers B₁ and B₂. Cal will place markers C₁ and C₂. (If there had been four people, then they would each have placed three markers.) The placement of markers is shown next.

STEP 2: Sections of items will be allocated to the players in the following way. Locate the leftmost of all the first markers (A₁, B₁, and C₁). We see the leftmost marker belongs to Babs. Babs will keep all of the items to the left of her first marker: a banana, cookies, a bunch of grapes, an orange, and a carrot. Now that Babs has a collection of snacks that she feels is fair, her markers are removed, as shown next, and she may go and enjoy her food.

STEP 3: Next, locate the leftmost of all of the second markers (A₂ and C₂). We see the marker belongs to Ally. Ally will keep all of the items between her first and second markers: a bunch of grapes, a pear, a slice of watermelon, a banana, cookies, and a carrot. Now that Ally has a collection of snacks that she feels is fair, her markers are removed, as shown next, and she may go and enjoy her food.

STEP 4: Finally, Cal is the last remaining player. He will get the snacks to the right of his second marker: a pear, a slice of watermelon, an orange, and a slice of cake. Notice that two items are left over, namely, a slice of cake and an orange.

STEP 5: The leftover items will need to be divided. In general, if there are more leftover items than there are players, then the players can apply the method of markers again, otherwise, they will have to use another method to divide the remaining items.

Two couples spend the weekend camping in the woods. As the foursome prepares to leave their campground for a hike, they decide to take a collection of snacks to keep their energy up while they are on the trail. They spread out snacks on a table. Suppose the four people place markers as indicated. Carry out the method of markers and describe a fair allocation of the snacks. List the leftover items and explain how to fairly divide them.