I am moving into an eight-bedroom flat with seven friends in a few months and need to decide fair rates for each of the rooms. Assuming some people want cheaper rent and some want nicer rooms, what is the fairest way to split the total monthly rent of £3,200 and decide who gets what?
Student, St Andrews
I have answered a very similar question before, but for three people rather than eight. My solution then – a modification of “one cuts, the other chooses” – would not work for you. Therefore I propose a simultaneous ascending auction of the largest seven rooms; whoever does not win one of the largest seven gets the smallest room at whatever rent is necessary to bring the total to £3,200.
Bidding proceeds in rounds. In the first round, any student may choose to bid on any of the seven largest rooms, in increments of £5. Ties are broken with the toss of a coin. In each subsequent round, students without rooms must submit a bid to exceed the current high bid by £5. Any incumbents thus dislodged can bid on any room in the following round.
The auction ends when seven students are incumbents in the seven rooms, and the eighth student does not wish to outbid any of them, but would rather take the small room and pay the balance of the £3,200. (If the bidding is frenetic enough, she may be paid to take the small room.)
At any time, roomless students have an incentive to bid on whichever room offers the best combination of price and quality – or to drop out if they think the smallest room looks cheap. But beware: such an auction is neither foolproof nor, if some players decide to collude, cheat-proof. Still, perfection is for mathematicians, not economists.
Also published at ft.com.