How do two roommates run the couch auction?
Each of you writes down, privately, a price for the whole couch. Reveal both numbers
at once. The higher number keeps the couch and pays the other person
half of the average of the two numbers. That is the entire rule. It
needs no appraiser, and it is the two-person version of the auction Cramton, Gibbons and
Klemperer proved dissolves an equal partnership efficiently. The number does not have to
be your exact private value; the paper assumes both partners bid strategically and
derives a symmetric bidding strategy under which the higher bid still comes from the
person who values the couch more. That is a statement about the equilibrium, under the
paper’s assumptions, not a guarantee about every pair of numbers two people could write.
Maya writes down · her number for the whole couch $500
Dev writes down · his number for the whole couch $300
Average of the two numbers $400
Maya keeps the couch and pays Dev half the average $200
For comparison · ‘reimburse the receipt’ would have cost Maya $450
Read the result from each side, taking the two numbers as true values for the sake of
the arithmetic. Maya valued the whole couch at $500, so her half was
worth $250 to her; she bought Dev’s half for $200 and is $50 ahead of walking away. Dev
valued the whole couch at $300, so his half was worth $150 to him; he sold it for $200
and is $50 ahead. Both are better off than they were, the couch went to the person who
wanted it more, and nobody had to negotiate. The receipt-based number, $450, would have
made Maya pay more for Dev’s half than that half was worth to her. The “just take it” number, $0, would
have handed Dev’s $150 share to Maya for nothing. Both of the intuitive answers are
wrong in opposite directions; the two-number rule lands between them for a reason. The
two $50 surpluses are what this illustration produces, not what the mechanism promises.
Each roommate writes a private value for the whole item not a share, not the receipt: what it is worth to you today
The higher number keeps it so the item goes to whoever wants it more
and pays the other person half the average of the two numbers half, because the other person owned half
Averaging the two bids is what Cramton, Gibbons and Klemperer call ‘split-the-difference bidding (the average of first- and second-price)’, credited to Samuelson; the halving is their rule that the bid revenue ‘is divided equally among all the bidders’. McAfee’s 1992 paper describes two cousins that work the same way: ‘the winning bidder pay[s] half of the appropriate bid’ — the winner’s own number in his Winner’s Bid Auction, the loser’s number in his Loser’s Bid Auction. Any of the three dissolves an equal-shares partnership when there is no outside buyer; pick one before you write anything down.
Why does it work? Not because it forces honesty; it does not, and the paper does not
claim it does. Both partners will shade their numbers, and Cramton, Gibbons and Klemperer
derive the symmetric strategy each of them ends up using. In that symmetric equilibrium
both partners shade by the same strategy, so the ranking of the numbers matches the
ranking of the values and the couch lands with the person who wants it more. What the paper does not
establish is that any two bids whatsoever will do that; it establishes that the rule
gives neither of you a reason to bid in a way that breaks it. “Just agree on a price”
has no such property. The mechanism was designed for partners who cannot see each
other’s valuations, which is the situation two roommates are in, however long they have
lived together.
One number belongs on the table before either of you writes yours: what the couch would
fetch if you sold it. Sometimes the honest answer is that neither of you wants it, and a
buyer on a resale listing values it more than both of you. McAfee builds that case into
the rule. “In the event that an outside option, the possibility of sale to a third party,
exists, both the WBA and the LBA dictate sale to the third party if neither bidder’s bid
exceeds the third party’s value.” His model treats that value as known; yours is an
actual offer, net of what selling costs you, not a listing price. So the gate is: get a
real offer, and if neither private number beats it, sell the couch and split the
proceeds in half. If one number does beat it, run the auction, and use the winner’s
number rather than the average, because that is the version McAfee finds still allocates
correctly with an outside buyer in the picture: “The WBA is ex post efficient,” while
“the LBA is ex post efficient only if no outside option exists.” That is a result inside
his model; in a living room it is the sensible default, not a proof. Either way, the
offer, not the $900 receipt, is the number the argument should start from. Publication
561 already told you why.
There is a simpler rule you will hear recommended: one person names a price, and the other
chooses whether to buy the couch at that price or sell their half at it. Moldovanu calls
it “commonly known as the ‘Texas shoot-out’” and credits its appeal honestly: it “is easy
to use, since its rules do not depend on the parties’ values.” McAfee’s 1992 paper in the
Journal of Economic Theory analysed it, under the name cake-cutting, and found
the catch. “With symmetric information,” when both people can see what the couch is
worth to each other, it “produces an efficient allocation, and agents prefer to be the
proposer.” When information is private, “the mechanism is ex post inefficient.” The
person who names the price has an edge and the couch can land with the wrong person. If
you use it anyway, flip a coin for who proposes. If you would rather not have an edge to
flip for, write two numbers.
Sources: Dissolving a Partnership Efficiently, Econometrica, 1987 (the k+1-price auction, whose bid revenue is “divided equally among all the bidders”, and “split-the-difference bidding (the average of first- and second-price)”); Amicable divorce: Dissolving a partnership with simple mechanisms, Journal of Economic Theory 56(2), 1992 (the Winner’s Bid and Loser’s Bid auctions, and the cake-cutting result); How to Dissolve a Partnership, JITE, 2002 (the Texas shoot-out)