01:640:103 Chap.7 Fair Division of Continuous Goods
Fair Division of Continuous Goods
Fair means fair to each player
Fair division problems involve players with equal claims to goods they value differently. With N players, a portion is a fair share for a player if that player values it at no less than one Nth of the whole, measured by that player's own values; a minimum fair share is worth exactly one Nth.
Because values are personal, every player can end with more than a fair share of what they care about, as when two people split a pizza that is half vegetarian.
The course assumes players act only on their own preferences, without strategy or negotiation.
Methods for two and for many
In Divider-Chooser a coin flip picks a Divider, who cuts the goods into two shares of equal value by the Divider's own measure, and the Chooser takes either one. The Divider always ends with exactly half and the Chooser with at least half.
The Lone Divider method extends this to any number of players: one Divider cuts N equal shares, everyone else writes a bid list of shares they consider fair, and shares are assigned from the lists.
Standoffs and the Lone Chooser
When two players can accept only the same share, the Lone Divider method sets that share aside, satisfies everyone else, merges what is left into a supershare, and lets the two run Divider-Chooser on it; three or more players in a standoff run Lone Divider again.
The Lone Chooser method has two players split the goods by Divider-Chooser, each subdivide their half into three, and the third player take one piece from each, never two from the same person.
What each method guarantees
Each continuous method comes with a guarantee that holds whatever the other players value. In Divider-Chooser the Divider gets exactly half by their own measure and the Chooser at least half.
In Lone Divider every player who is not in a standoff receives a share from their own bid list, and players in a standoff are guaranteed at least half of the supershare they divide, which is always worth more than their fair share of the whole. In Lone Chooser each Subdivider keeps two of three equal pieces of a block worth at least half, and the Chooser picks the best piece from each.
What this chapter covers
- 01
Fair shares and minimum fair shares
- 02
Continuous division versus discrete distribution
- 03
The Divider-Chooser method
- 04
Lone Divider, bid lists and standoffs
- 05
The Lone Chooser method
Worked example · free
Is this division fair?
- 1Nadia's total is $40,000 + $70,000 + $50,000 = $160,000.
- 2With 3 players her minimum fair share is $160,000 ÷ 3 ≈ $53,333.33.
- 1X is worth only $40,000 to her, below that threshold.
Key terms
- Fair share
- A fair share is a portion worth at least one Nth of the goods to the player receiving it.
- Divider-Chooser
- Divider-Chooser is the two-player method in which one player cuts equal halves and the other picks first.
- Bid list
- A bid list is the set of shares a player considers fair shares in the Lone Divider method.
- Standoff
- A standoff occurs when players cannot all receive a share from their own bid lists.
- Supershare
- A supershare is the merged remainder of the goods that the players in a standoff divide again.
Fair Division of Continuous Goods FAQ
Why does the Divider cut the goods into exactly equal halves?
The Divider knows nothing about the Chooser's values. Making the two halves equally valuable to themselves is the only cut that protects them whichever half the Chooser takes.
What happens to shares in a standoff?
The disputed share is set aside, every other player receives a share from their bid list, and the remaining shares are merged into one supershare, which the players in the standoff divide again.
Can every player end up with more than a fair share?
Yes. Fairness is measured by each player's own values, so when players value parts of the goods differently, several of them can each receive more than one Nth of what they care about.
Why must the Chooser in Lone Chooser take one piece from each Subdivider?
Each Subdivider's guarantee depends on keeping two of their three equal pieces. If the Chooser took two pieces from the same person, that person would keep only one third of their block and could fall below a fair share.
Exam move
For every division problem, first write each player's total and minimum fair share in a small table, then check each assigned share against that player's own number.
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