01:640:103 Chap.8 Fair Distribution: Sealed Bids and Markers
Fair Distribution: Sealed Bids and Markers
Indivisible goods
A house, a car or a painting cannot be cut up without destroying its value, so each item must go whole to one player. Fairness still means that each player ends with at least one Nth of the total value by their own measure.
The chapter covers two methods: sealed bids, which uses cash to even out the shares, and markers, which uses the order of a row of small items.
Sealed bids in five stages
Each player privately bids a value for every item and divides their own total by how many players there are, whatever the count of items, to find their minimum fair share. Each item goes to its highest bidder.
In the first settlement a player whose items exceed their fair share pays the excess into a common account and a player below it draws the difference, so everyone holds exactly a minimum fair share. Whatever is left in the account is the surplus, split equally among all the players in the final settlement.
Cash paid in must equal cash paid out.
Markers for many small items
In the Method of Markers the items are laid out in a row and each of the N players places N minus 1 markers, cutting the row into N stretches each worth exactly a minimum fair share to them.
Scanning from the left, the owner of the first first-marker takes their first stretch, then among those remaining the first second-marker wins a second stretch, and so on.
Items outside every allocated stretch are leftovers, shared by lottery, by sealed bids or any other way.
Why everyone ends with at least a fair share
After the first settlement of sealed bids, every player holds exactly a minimum fair share: items plus or minus cash worth one Nth of their own total.
The surplus is never negative, because the highest bidder values each item at least as much as the others do, so the final settlement only adds to everyone's position. In the Method of Markers, each allocated stretch lies between two of the owner's own markers, so it is worth exactly a minimum fair share to them.
The one requirement is enough items to place the markers; the method suits many small items, not a few large ones.
What this chapter covers
- 01
Discrete goods and fair shares
- 02
Bidding, allocation and the first settlement
- 03
Surplus and the final settlement
- 04
Placing markers in a row
- 05
Allocating stretches and leftovers
Worked example · free
Two partners, three items
- 2Totals are $10,800 and $10,400, so fair shares are $5,400 for Ray and $5,200 for Sam.
- 2Ray takes the van ($9,000) and pays $3,600; Sam takes the printer and desk ($2,400) and draws $2,800.
- 2Surplus = $3,600 − $2,800 = $800, so each partner gets $400.
- 1Final: Ray pays $3,200 and Sam receives $3,200; paid in equals paid out.
Key terms
- Sealed bid
- A sealed bid is a player's private valuation of an item, hidden from the other players.
- First settlement
- The first settlement is the cash exchange that leaves every player holding exactly a minimum fair share.
- Surplus
- The surplus is the cash left in the common account after the first-settlement payouts.
- Method of Markers
- The Method of Markers divides a row of small items using markers each player places at fair-share points.
- Leftover item
- A leftover item is an item in the row that falls inside no allocated stretch under the Method of Markers.
Fair Distribution: Sealed Bids and Markers FAQ
Do I divide by the number of items or the number of players?
By the number of players. A minimum fair share is one Nth of a player's own total valuation, where N counts players, regardless of how many items are being distributed.
What happens to leftover items in the Method of Markers?
The method itself does not say. Players may draw lots, run sealed bids or use any other rule, because every player already holds a fair share before the leftovers are handed out.
Can a player receive cash but no items under sealed bids?
Yes. A player who is never the highest bidder receives their whole minimum fair share in cash in the first settlement, plus one Nth of the surplus in the final settlement.
Exam move
Draw the settlement as a table with columns for items, value received, first settlement and final settlement, and close every problem by matching cash in with cash out.
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