House rental, showers, and auctions
I'm currently in the prologue of the house-rental process. Our sights are set on a four-bedroom house with an interesting property: the room resources are of differing value — not in terms of square-footage, but other, less easily scalable factors.
One bedroom has an attached bathroom and a slightly larger closet (the master bedroom).
One bedroom is on its own floor with a half-bathroom. [*]
The other two share a full bathroom, whose shower is also a resource shared with the bedroom mentioned above.
So how do we dole out the resources of unequal value in a manner that everybody considers fair?
(Talk about first-world problems...)
Three systems for deciding the room allocations have been proposed, and one has been rejected:
Randomly assign choosing priority: this was dismissed because the person with the last priority inevitably feels "stuck with" a room.
Entirely random: we take four playing cards from a deck, associate each card with a room, shuffle the cards until everybody is satisfied, then deal each person a card. That's the room that they get. Everybody is equally "stuck with" a room, and not due to the actions of any other person.
Bidding system: the auction would move from perceived-most-valuable-room to perceived-least-valuable-room. Bidding starts with the master bedroom at the evenly-divided rental rate. Once a winner/price is determined for that room, the process is repeated for the room with the half-bathroom, with a newly-divided rental rate. Once a point is reached where nobody is willing to place a bid on a room, the rest are assigned at random. Each person, through their own decision to bid or not to bid, ends up with the room of their choosing.
I'm personally a fan of the simplicity and total lack of competition in the random system, but if the bidding system is more likely to lead to the optimal outcome, it must be considered! (Of course, care must be taken to address possible social ramifications.)
During casual discussion we noted that the bidding system would be strategically interesting. If a person were not really interested in getting a high-value room but didn't care all that much if they did, it is clearly in their favor to drive bids on the early rooms as high as possible — this would entail a decrease in the payment rate for a later room.
I perused the Wikipedia entry on auction theory and realize that we were assuming an open/ascending bid auction, but a first-price/sealed-bid auction might be less socially stressful. [†] My main fear is that open bidding would get carried away (it's exciting, after all) and participants would come to regret their bids.
In any case, it's an interesting problem I thought I would share. Suggestions are welcome; otherwise, I'll write a follow-up entry as to how it turns out.
This means it has no shower. Stupid terminology, but I suppose it has a nicer ring than "toilet room".
As an aside, the game theory described on that page also seems to discuss provable equilibrium of a single auction, whereas our auctions have dependent equilibrium values.