# An auction process for preserving incentives to improve land in Jubilee

I wrote previously about how to allocate land in Jubilee. One of the problems to be solved by this procedure was that people in Jubilee might have no incentive to improve land that they had no assurance they would continue to enjoy access to in future. I described strategies that owners of improvements (and their bidding rivals) might use, and concluded that owners of improvements who wanted to retain access to the land could arrange to pay for only the unimproved value of the land.

However, I’m not satisfied with the strategies I described there. First, they rely on good estimates of the value that other bidders might place on the land and improvements (separately and together). Second, I considered only one rival bidder other than the owner of the improvements; in reality, there may be several bidders, some placing a high value on the land with the improvements and a low value on the land alone, and others placing a high value on the land alone.

So, can we do better?

I think so. Consider this variant of a Vickrey–Clarke–Groves auction, where we auction the land and improvements at the same time, with the possibility that they are sold together, if they’re more valuable together than separately.

First, we have a set $I$ of participants in the auction, including the participant $0$ who owns the improvements before the auction begins. Each participant $i$ has a cost $b(i)$ which they’re indifferent to paying in exchange for the land and improvements together; if it costs more to buy them, they won’t want to pay that much, and if it costs less, they’ll consider themselves better off if they buy them. Similarly, each participant has a cost $l(i)$ which they’re indifferent to paying in exchange for the land alone (after the owner of the improvements has taken away everything worth taking). The owner of the improvements also has a salvage value $t$ that they place on the improvements, if they are going to lose access to the land; $0$ will be willing to sell the improvements in-place if they’re paid at least $t$, but not if they’re paid less.

For any $J \subseteq I$, let $B(J) = \max_{i \in J} b(i)$ and $L(J) = \max_{i \in J} l(i)$; that is, $L(J)$ is the highest value placed on the land alone by any of the participants in the subset $J$; the explanation is similar for $B(J)$.

Now, each participant $i \neq 0$ submits to the auctioneer their values of $b(i)$ and $l(i)$. (We’ll see later that it’s in their interests to be truthful.) The participant $0$ submits $b(0)$ and $t$.

If $B(I) > L(I) + t$, then the land and improvements are worth more together than separately. Let $m$ denote the participant who submitted $b(m) = B(I)$. They are assigned the land and improvements together, for which they pay $\max\lbrace B(I - \lbrace m \rbrace), L(I - \lbrace m \rbrace) + t \rbrace$ — that is, the highest value placed on the land and improvements (together or separately) by participants other than themselves. Of what they pay, $L(I - \lbrace m \rbrace)$ is paid to the auctioneer, and the remainder to participant $0$.

This ensures that participant $0$ receives at least $t$ for the improvements, and if $m = 0$, then they retain access to the land by paying only for its unimproved value.

If, on the other hand, $L(I) + t > B(I)$, then the land and improvements are worth more separately than together. Let $n$ denote the participant who submitted the highest land-only bid $l(n) = L(I)$. They are assigned the land, for which they pay $\max\lbrace B(I - \lbrace n \rbrace) - t, L(I - \lbrace n \rbrace)\rbrace$ — that is, the highest value placed on the land by other participants (individually or collectively). The entire payment goes to the auctioneer, and participant $0$ is free to salvage the improvements before participant $n$ takes over the land.

Now, as in a normal Vickrey–Clarke–Groves auction, for participants other than $0$, changing their bids can only affect whether or what they win in the auction; it cannot affect what they pay if they win the same package as they would with truthful bids. This is because the payment is determined only by the bids placed by other participants. An under-bid risks missing out on what the participant thinks is a good deal, without reducing the price if they win, and an over-bid risks that the participant will be forced to pay more than they think the package is worth to them, without increasing the chance that they’ll win at an advantageous price, so it is in their interests to bid truthfully, even if other participants bid irrationally.

There’s a little bit of a hitch with participant $0$, though, since if someone else wins the land and trees together, then the payment to $0$ — which is $\max\lbrace B(I - \lbrace m \rbrace) - L(I - \lbrace m \rbrace), t \rbrace$ — depends in part on the bids that $0$ submitted, so there may be some incentive to falsify their bids.

In particular, if $0$ has reason before the auction to believe that $b(m) > b(0)$, then they could increase the payment to themselves by raising the reported value of $b(0)$ (and hence the value of $B(I - \lbrace m \rbrace)$) to just under $b(m)$, and if their estimate of $b(m)$ is accurate, they don’t risk having to buy the land at a higher price than they would be willing to pay. They might be able to achieve the same thing by raising the reported value of $t$ to just under $b(m) - L(I - \lbrace m \rbrace)$, but this requires a more complicated estimate of other participants’ values, and carries more risk, so raising $b(0)$ is the preferable strategy.

But is this really a false report of $b(0)$? Perhaps not, because participant $0$ believes that if they could buy the land for the inflated value of $b(0) - t$, then they could sell it (together with the improvements) at a profit to participant $m$; their estimate of the value of the land and improvements together is genuinely raised by their estimate of the value to someone else.