3 The Mathematics of Sharing

2. 3 The Mathematics of Sharing 3.1 Fair-Division Games 3.2 Two Players: The Divider-Chooser Method 3.3 The Lone-Divider Method 3.4 The Lone-Chooser Method 3.5 The Last-Diminsher Method 3.6 The Method of Sealed Bids 3.7 The Method of Markers

3. The Method of Markers The method of markers is a discrete fair-division method proposed in 1975 byWilliam F. Lucas, a mathematician at the Claremont Graduate School. Themethod has the great virtue that it does not require the players to put up any oftheir own money. On the other hand, unlike the method of sealed bids, thismethod cannot be used effectively unless (1) there are many more items to bedivided than there are players in the game and (2) the items are reasonably closein value.

4. The Method of Markers In this method we start with the items lined up in a random but fixed sequence called an array. Each of the players then gets to make an independent bidon the items in the array. A player’s bid consists of dividing the array into segments of consecutive items (as many segments as there are players) so that eachof the segments represents a fair share of the entire set of items.

5. The Method of Markers For convenience, we might think of the array as a string. Each player thencuts” the string into N segments, each of which he or she considers an acceptable share. (Notice that to cut a string into N sections, we need N – 1 cuts.) Inpractice, one way to make the “cuts” is to lay markers in the places where the cutsare made. Thus, each player can make his or her bids by placing markersso that they divide the array into N segments.

6. The Method of Markers To ensure privacy, no player shouldsee the markers of another player before laying down his or her own. The final step is to give to each player one of the segments in his or her bid. The easiest way to explain how this can be done is with an example.

7. Example 3.11 Dividing the Halloween Leftovers Alice, Bianca, Carla, and Dana want to divide the Halloween leftovers shown inFig. 3-16 among themselves. There are 20 pieces, but having each randomlychoose 5 pieces is not likely to work well–the pieces are too varied for that.Their teacher, Mrs. Jones, offers to divide the candy for them, but the childrenreply that they just learned about a cool fair-division game they want to try, andthey can do it themselves, thank you.

8. Example 3.11 Dividing the Halloween Leftovers Arrange the 20 pieces randomly in an array.

9. Example 3.11 Dividing the Halloween Leftovers Step 1 (Bidding) Each child writes down independently on a piece of paperexactly where she wants to place her three markers. (Three markers divide thearray into four sections.) The bids are opened, and the results are shown on the next slide. The A-labels indicate the position of Alice’s markers (A1 denotesher first marker, A2 her second marker, and A3 her third and last marker).

10. Example 3.11 Dividing the Halloween Leftovers Step 1 (Bidding)

11. Example 3.11 Dividing the Halloween Leftovers Step 1 (Bidding) Alice’s bid means that she is willing to accept one of the following as a fairshare of the candy: (1) pieces 1 through 5 (first segment), (2) pieces 6 through11 (second segment), (3) pieces 12 through 16 (third segment), or (4) pieces 17through 20 (last segment). Bianca’s bid is shown by the B-markers and indicates how she would break up the array into four segments that are fairshares; Carla’s bid (C-markers) and Dana’s bid(D-markers).

12. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) This is the tricky part, where we are going to give to eachchild one of the segments in her bid. Scan the array fromleft to right until the first first marker comes up. Here the first first marker isBianca’sB1.

13. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) This means that Bianca will be the first player to get her fairshare consisting of the first segment in her bid (pieces 1 through 4).

14. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) Bianca is done now, and her markers can be removed since they are nolonger needed.Continue scanning from left to right looking for the firstsecond marker. Here the first second marker is Carla’s C2, so Carla will bethe second player taken care of.

15. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) Carla gets the second segment in her bid(pieces 7 through 9). Carla’s remaining markers can now be removed.

16. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) Continue scanning from left to right looking for the first third marker.Herethere is a tie between Alice’sA3 and Dana’sD3.

17. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) As usual, a coin toss is usedto break the tie and Alice will be the third player to go–she will get the thirdsegment in her bid (pieces 12 through 16).

18. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) Dana is the last player and gets the last segment in her bid (pieces 17 through 20,).

19. Example 3.11 Dividing the Halloween Leftovers Step 2 (Allocations) At thispoint each player has gotten a fair share of the 20 pieces of candy. The amazing part is that there is leftover candy!

20. Example 3.11 Dividing the Halloween Leftovers Step 3 (Dividing the Surplus) The easiest way to divide the surplusis to randomly draw lots and let the players take turns choosingone piece at a time until there are no more pieces left. Here theleftover pieces are 5, 6, 10, and 11 The players now drawlots; Carla gets to choose first and takes piece 11. Dana choosesnext and takes piece 5. Bianca and Alice receive pieces 6 and 10,respectively.

21. The Method of Markers Generalized The ideas behind Example 3.11 can be easily generalized to anynumber of players. We now give the general description of the method of markerswith N players and M discrete items. Preliminaries The items are arranged randomly into an array. For convenience, label the items 1 through M, going from left to right.

22. The Method of Markers Generalized Step 1 (Bidding) Each player independently divides the array into N segments (segments 1, 2,. . . ,N) by placing N – 1 markers along the array.These segments are assumed to represent the fair shares of the array in theopinion of that player.

23. The Method of Markers Generalized Step 2 (Allocations) Scan the array from left to right until the first first markeris located. The player owning that marker (let’s call him P1) goes first and getsthe first segment in his bid. (In case of a tie, break the tie randomly.) P1’smarkers are removed, and we continue scanning from left to right, looking forthe first second marker.

24. The Method of Markers Generalized Step 2 (Allocations) The player owning that marker (let’s call her P2) goessecond and gets the second segment in her bid. Continue this process, assigning to each player in turn one of the segments in her bid. The last player getsthe last segment in her bid. Step 3 (Dividing the Surplus) The players get to go in some random order andpick one item at a time until all the surplus items are given out.

25. The Method of Markers: Limitation Despite its simple elegance, the method of markers can be used only undersome fairly restrictive conditions: it assumes that everyplayer is able to divide the array of items into segments in such a way that eachof the segments has approximately equal value. This is usually possible whenthe items are of small and homogeneous value, but almost impossible to accomplish when there is a combination of expensive and inexpensive items (goodluck trying to divide fairly 19 candy bars plus an iPod using the method ofmarkers!).