Community Auction

Some friends and I are forming a group to create a residential community, possibly by buying an apartment complex. Some fascinating economic questions naturally come up in the process, and I'm interested in any thoughts on the following:

Given some property, with some number of heterogenous units, how do we allocate the units do individuals, and how do we apportion the total purchase cost among the various units?

Note that individuals have differing preferences and budgets, and that the costs here are very large compared to their net worth, hence budget restrictions are tight.

I naturally thought of some sort of auction, basically a standard continuous auction with a twist. Standard in that the highest bidder for each unit gets the unit. But the twist comes in when we remember that the total price of all auctions is fixed. So really, bidders are bidding percentages of the purchase price, not dollars.

It would be bad to sell one unit at a time - families need to make more holistic decisions, knowing the full range of what they are getting. Hence you should have a continual auction for all units at once, until you reach an equilibrium, so no one gets shafted with more or less units than they want/need/can afford.

I figured you would start out with all N units having no bidder and a price equal to 1/N of the equity. Whenever someone bids for a unit, their marginal increase in its price is distributed among all other units in proportion to their equity. In plainer terms: when you bid more for unit unit, all others get cheaper. This is a natural consequence of the cap in total price.

This scheme seems fair to me. It also seems reasonable to implement. But would it reach equilibrium? Or are cycles possible? How can they be dealt with? I'm concerned that if the total valuation and budget of the individuals is substantially higher than the purchase price, equilibrium is impossible. Do y'all know other methods for solving this same problem?

The obvious alternative is a standard combinatorial auction using VCG, which is computationally tractable and incentive compatible, but may require a lot of work from individuals to set up their bids. Also, the total would probably not be equal to the purchase price - how do you conduct a combinatorial auction with this restriction?

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Why cap the total bids at

Why cap the total bids at the overall purchase price? Couldn't you either 1) Do the bidding before purchasing the complex, then use the sum of all bids to determine the offer price on the complex, or 2) Allow the entity which purchases the complex to profit from selling the individual units?

Perhaps you could form some sort of corporation to purchase the complex, of which all interested parties have a share, and then distribute any profits via a dividend?

Here is a suggestion: Say

Here is a suggestion:

Say there are N people and N apartments.

Let each person submit a vector of bids: [b1, ... , bN], where b_i is how much the person would be willing to pay for apartment i.

Given everyone's vector of bids, choose the allocation (one apartment per person) that maximizes revenue. If this is less than the total cost, abandon the project. If it's more than the total cost, use the surplus to endow the "community center" (or whatever, you'll need some sort of overview to pay for general improvements and whatnot).

I doubt that this scheme is incentive compatible. Do you care? If so, why? It seems "fair" in a lot of ways, and it's easy to understand.

As long as result of your

As long as result of your auction scheme is a total that exceeds the overall purchase price, you can then give a flat percentage rebate and all the buyers will be happy to have paid less than their highest bid. And it's easy to explain - tell people "bid no more than you would be willing to pay, and you may get a small discount at the end of the process."

So have people bid in dollars, use Joel's vector of sealed bids, pick an allocation that maximizes revenue, then if you've got 20% overage and you only need half that to fund the common resources, drop all bids by 10%.

Doug - The point is that

Doug - The point is that we're bidding on actual, heterogenous units based on our preferences. So we need to know what the units are. So we need to have an actual complex.

The problem with distributing the excess as gains is that it changes people's bids in an uncertain way. If you know you are getting some back, then you will bid higher, if everyone does this, bids go higher and higher without limit. Because you aren't actually paying what you bid. It seems like an unstable system. In the real world, you know that people won't make infinite bids, but they are probably smart enough to bid a little high because of this effect, so you end up having to guess what the inflation factor is, which needlessly complicates the process.

Joel - one complication is that we aren't assigning a one-to-one mapping. Some people want multiple apts, and some people optionally want multiple apts depending on price. But your scheme still has some merit. Why not charge the second-price, though, to make it more incentive compatible?

Patri, In an apartment

Patri,

In an apartment complex, there are probably a minimal number of observable objective characteristics that make the apartments non-homogeneous, such as the number of bedrooms, for example.

If you can by some approximate means assign a delta price or percentage for each such characteristic, then each apartment could be assigned a price which approximately reflected the non-homogeneous characteristics which it possessed and the total of all prices could match the total price for the entire complex.

With every apartment now having both a price and observable differences from the others, individuals could preference rank apartments knowing both what they were bidding for and what it would approximately cost. The apartments could then be assigned so that as many individuals as possible received their top choice, then their second choice, etc. I think that this would be something like preference voting.

Further adjustments could then be made by exchange or some other method.

Regards, Don

I agree with Don's comment

I agree with Don's comment that you should assign values to each of the apartments, summing to the total purchase price plus some load for an operating budget, say for transaction costs and first year's operating expense.

Then get each of the bidders to choose one or more apartments given his preference, with the understanding that this is a binding capital commitment as long as all of these apartments are available to him. Keep the bids sealed and set the ground rule that bidders must not reveal their bids, although multiple people can certainly combine to submit one bid. Hey, if you cannot trust these guys to honor that rule I'm not sure you want to be in business with them.

Once all the bids are in, give the highest bidder his preference, then go to the next highest bidder. Break ties for those bidders with the same capital commitment by randomly drawing a sequence.

If any of a bidder's apartments are no longer available at his turn, give him the opportunity to submit a new bid based upon what apartments are still available, and move him into the appropriate selection slot based upon the capital commitment of the new bid. This amounts to a pre-emptive purchasing right for a bidder who was willing to commit more capital than anyone remaining.

If you reach the end of the selections and you are still undersubscribed, repeat the process with the remaining apartments. There is an incentive to make additional bids or find additional bidders to get the deal done.

As long as you eventually become fully subscribed, this scheme will work. It gives preferential selection rights based upon capital contributed and pre-emptive purchase rights based upon capital committed. It avoids "gaming" by overbidding since the bidder must be prepared to honor the bid. You also end up with clear "shares" of ownership of the corporation based upon the capital contributions.

Low bidders may not get an apartment, but that creates opportunities through a secondary market. The price of an apartment on the secondary market would be whatever the market would bear, but the ownership share that transfers with the sale would be based upon the original capital commitment associated with that apartment.

I must admit that I have not spent much time figuring out all the possibilities for gaming this, but it does appear to be a reasonable way to approach the problem.

Congratulations, you're

Congratulations, you're solving the public goods problem. Lots of research and writing been done here... I've recently gone through a lot of it, so I'll save you some time. My surprising conclusions: 1) it's been solved, although I don't think most people realize it, and 2) the solution is ridiculously simple. All you need is to let people commit money to the project in increasing amounts until the project is funded, on the condition that commitments are released if the project is not funded by some deadline. Turns out that's enough to solve the free rider problem *and* guarantee the provision of the good when economically beneficial (and not when not).

Your problem is more like a club good than a public good, but I think the solution still applies. Note that even though the people in the club will each get something of their own (a unit), the club good in question is the complete funding of the entire project. Unless the whole complex is purchased, nobody gets any benefit.

Here's how you can implement it. Let anyone bid any time on any unit or set of units. Let them submit multiple bids if they want; each bid is a commitment to pay $X in return for unit Y or set of units W+Z (or whatever). Bids are disjoint: bidders are commited to any one of their disjoint bids, but not more than one. Bidders cannot retract or lower their bids. They can make new bids for new sets of units any time. They can increase their existing bids for a set of units any time. All bids are public.

Set a time limit for the auction. While the auction runs, continuously evaluate bids from highest to lowest. Any units claimed by a high bid will rule out any lower bids for a set of units that overlap with a higher bid. As soon as the total amount committed by the currently winning bids reaches the purchase price of the complex, the auction is over. If time expires before that happens, all bidders are released from their commitments and you can try again later, perhaps after rounding up more potential bidders. If the purchase price is reached with some units unbid for, everyone who has submitted a winning bid gets an equal ownership stake in the unbid units.

I don't have the reference handy, but some clever econ grad students put out a convincing paper showing that this works. The increasing commitment over time was the key.

I think this works out very similarly to Jeff Meh's scheme, but without sealed bids and without requiring discrete rounds, and without having to assign any value to any of the units beforehand. Also, I read Jeff's scheme as requiring a commitment of money without knowing for certain that your preference will still be available when your slot is reached - the problem there is that you may be willing to commit a large amount for one unit, but only a much smaller amount for the ones that are left when it gets down to you.

All you need is to let

All you need is to let people commit money to the project in increasing amounts until the project is funded, on the condition that commitments are released if the project is not funded by some deadline.

It seems like this view has some problems if the money is not collected ahead of time (which I assume would invalidate its usefulness). The law does not and should not be made to enforce "mere promises", like if I promise to pay you $5 and then don't. There has to be something more to a transaction than that to make it a question about justice. So if I agree to pay you $5 for your public good, and all the people except me agree on a certain date, then what stops me from simply saying I've changed my mind and won't pay you anything? Wouldn't this kill the scheme once everyone realizes it?

Stefan: I didn't make this

Stefan: I didn't make this clear, but I was assuming the parties would be contractually obligated to their commitments. Such contracts are sometimes called assurance contracts or contingent contracts - the parties contractually agree to pay their commitments if and only if the purchase price is met.

You're right, a promise of a gift is not an enforceable contract (or so a law student once told me). A mutual assurance contract is not a promise of a gift, though. The way it works is that the donating parties agree to provide payment of their commitments to some other party, and that other party agrees to provide the public good in question in exchange for the payments, with all parties' obligations being contigent upon the required total being raised before the deadline. Exchange of consideration, poof, you have an enforceable contract.

The way it works is that the

The way it works is that the donating parties agree to provide payment of their commitments to some other party, and that other party agrees to provide the public good in question in exchange for the payments, with all parties’ obligations being contigent upon the required total being raised before the deadline.

Yeah you'll notice I mentioned this possibility at the outset. I think this is assuming away the problem of public goods in a sense; if 10 people need to raise $100 and each person has paid say $4 then the process could still deadlock, with each person waiting to see if the other people are going to make any further commitment. It sounds like it might have some potential, but I'm not sure that alone will solve the public goods problems people face.

Yeah you’ll notice I

Yeah you’ll notice I mentioned this possibility at the outset.

Right, sorry about that.

if 10 people need to raise $100 and each person has paid say $4 then the process could still deadlock, with each person waiting to see if the other people are going to make any further commitment.

This was dealt with in the paper that I can't point to right now. They used game theory to show that there's no deadlock. The intuition for the explanation is that people have a time preference; they are willing to give up some of what they might gain by outwaiting the other parties in order to obtain the benefit from the public good now rather than later.

In my opinion, the key insight is that the public good problem can be restated as a bargaining problem - then you get to use lots of interesting results from bargaining theory. I only found one or two papers that mention this concept, so I think this may not be well understood by the game theorists and public choice economists.