1. § 3.7 The Method of Markers

2. Motivation  We saw yesterday that the Method of Sealed Bids worked well if all of the players had enough money to play the game.  In cases where this is not possible we will need another method…

3. The Method of Markers  The first thing we will do is arrange the goods to be shared in a fixed ‘string’ called an array.  Each player will independently bid for segments of consecutive items in the array by ‘cutting’ the string.  ‘Cuts’ will be made is by placing markers along the array--each segment between a player’s two markers will represent a fair share of the loot to him/her.

4. Example: Example: Martin (A), Bart (B), Millhouse (C) and Ralph (D) stumble upon a box containing a 15 pieces of old candy. They randomly arrange the candy into an array and decide to use the method of markers to divide their find. Step 1 (Bidding). Each player writes down independently on a piece of paper exactly where he/she wants his or her markers. In this case, there are 4 players so each player needs to place 4 - 1 = 3 markers.

5. Example: Example: Martin (A), Bart (B), Millhouse (C) and Ralph (D) stumble upon a box containing a 20 pieces of old candy. They randomly arrange the candy into an array and decide to use the method of markers to divide their find. Step 1 (Bidding). Each player writes down independently on a piece of paper exactly where he/she wants his or her markers. In this case, there are 4 players so each player needs to place 4 - 1 = 3 markers. The bids are opened and the results are shown below: A1A1 A2A2 B1B1 D1D1 A3A3 B2B2 B3B3 D2D2 D3D3 C1C1 C2C2 C3C3

6. Example: Example: Martin (A), Bart (B), Millhouse (C) and Ralph (D) stumble upon a box containing a 20 pieces of old candy. They randomly arrange the candy into an array and decide to use the method of markers to divide their find. Step 2 (Allocations). Each player will be allocated one segment of the array. We will begin by scanning from left to right. Here, the first marker from left-to-right is Ralph`s (D 1 ), so we give Ralph his first segment. Now we can remove his remaining markers. A1A1 A2A2 B1B1 D1D1 A3A3 B2B2 B3B3 D2D2 D3D3 C1C1 C2C2 C3C3

7. Example: Example: Martin (A), Bart (B), Millhouse (C) and Ralph (D) stumble upon a box containing a 20 pieces of old candy. They randomly arrange the candy into an array and decide to use the method of markers to divide their find. Step 2 (Allocations). Cont`d Now we look for the first second marker. In this case, Martin has the first second marker (A 2 ). We will give Martin his second segment going from his first marker to the second. We may then remove his remaining markers. A1A1 A2A2 B1B1 A3A3 B2B2 B3B3 C1C1 C2C2 C3C3

8. Example: Example: Martin (A), Bart (B), Millhouse (C) and Ralph (D) stumble upon a box containing a 20 pieces of old candy. They randomly arrange the candy into an array and decide to use the method of markers to divide their find. Step 2 (Allocations). Cont`d Now we will look for the first third marker. This marker belongs to Bart (B 3 ). We will give Bart his third segment going from his second marker to his third. We will also now remove his markers. B1B1 B2B2 B3B3 C1C1 C2C2 C3C3

9. Example: Example: Martin (A), Bart (B), Millhouse (C) and Ralph (D) stumble upon a box containing a 20 pieces of old candy. They randomly arrange the candy into an array and decide to use the method of markers to divide their find. Step 2 (Allocations). Cont`d Finally, we will award Millhouse his final segment - the portion after his third marker. C1C1 C2C2 C3C3

10. Example: Example: Martin (A), Bart (B), Millhouse (C) and Ralph (D) stumble upon a box containing a 20 pieces of old candy. They randomly arrange the candy into an array and decide to use the method of markers to divide their find. Step 3 (Dividing the Leftovers). Cont`d Notice that there are still a few items leftover. These will be divided by some form of lottery.

11. The Method of Markers (for N players and M objects)  Step 1. Each player independently divides the array into N fair shares by placing N - 1 markers. The markers separate each fair share from the next.

12. The Method of Markers (for N players and M objects)  Step 2. Scan the array from left-to-right until the first first marker is located. The player owning that marker gets to keep his or her first segment and his/her markers are removed. Continue moving from left to right until the first second marker is found. The player owning that marker keeps his/her second segment. Continue similarly until all players receive one segment. (Ties are broken randomly.)

13. The Method of Markers (for N players and M objects)  Step 3. The leftover items can be divided among the players by some form of lottery, or if there are many more leftovers than players the method of markers can be used again.