This is the third and final post on ticket pricing motivated by the new restaurant Next in Chicago and proprietors Grant Achatz and Nick Kokonas new ticket policy.   In the previous two installments I tried to use standard mechanism design theory to see what comes out when you feed in some non-standard pricing motives having to do with enhancing “consumer value.”  The two attempts that most naturally come to mind yielded insights but not a useful pricing system. Today the third time is the charm.

Things start to come in to place when we pay close attention to this part of Nick’s comment to us:

we never want to invert the value proposition so that customers are paying a premium that is disproportionate to the amount of food / quality of service they receive.

I propose to formalize this as follows.  From the restaurant’s point of view, consumer surplus is valuable but some consumers are prepared to bid even more than the true value of the service they will get.  The restaurant doesn’t count these skyscraping bids as actually reflecting consumer surplus and they don’t want to tailor their mechanism to cater to them.  In particular, the restaurant distinguishes willingness to pay from “value.”

I can think of a number of sensible reasons they would take this view.  They might know that many patrons overestimate the value of a seating at Next. Indeed the restaurant might worry that high prices by themselves artificially inflate willingness to pay.  They don’t want a bubble.  And they worry about their reputation if someone pays $1700 for a ticket, gets only$1000 worth of value and publicly gripes.  Finally they might just honestly believe that willingness to pay is a poor measure of welfare especially when comparing high versus low.

Whatever the reason, let’s run with it.  Let’s define $W(v)< v$ to be the value, as the restaurant perceives it, that would be realized by service to a patron whose willingness to pay is $v$.  One natural example would be

$W(v) = \min \{v, \bar v\}$

where $\bar v$ is some prespecified “cap.”  It would be like saying that nobody, no matter how much they say they are willing to pay, really gets a value larger than, say $\bar v = \1000$ from eating at Next.

Now let’s consider the optimal pricing mechanism for a restaurant that maximizes a weighted sum of profit and consumer’s surplus, where now consumer’s surplus is measured as the difference between $W(v)$ and whatever price is paid. The weight on profit is $\alpha$ and the weight on consumer surplus is $1- \alpha$.  After you integrate by parts you now get the following formula for virtual surplus.

$(1 - \alpha) W(v) + (2 \alpha - 1) [v - \frac{1-F(v)}{f(v)} ]$

And now we have something!  Because  if $\alpha$ is between $0$ and $1/2$ then the first term is increasing in $v$ (up to the cap $\bar v$) and the second term is decreasing.  For $\alpha$ close enough to $1/2$, the overall virtual surplus is going to be first increasing and then decreasing.  And that means that the optimal mechanism is something new.  When bids are in the low to moderate range, you use an auction to decide who gets served.  But above some level, high bidders don’t get any further advantage and they are all lumped together.

The optimal mechanism is a hybrid between an auction and a lottery.  It has no reserve price (over and above the cost of service) so there are never empty seats. It earns profits but eschews exorbitant prices.

It has clear advantages over a fixed price.  A fixed price is a blunt instrument that has to serve two conflicting purposes.  It has to be high enough to earn sufficient revenue on dates when demand is high enough to support it, but it can’t be too high that it leads to empty seats on dates when demand is lower. An auction with rationing at the top is flexible enough to deal with both tasks independently.  When demand is high the fixed price (and rationing) is in effect. When demand is low the auction takes care of adjusting the price downward to keep the restaurant full.  The revenue-enhancing effects of low prices is an under-appreciated benefit of an auction.  Finally, it’s an efficient allocation system for the middle range of prices so scalping motivations are reduced compared to a fixed price.

Incentives for scalping are not eliminated altogether because of the rationing at the top. This can be dealt with by controlling the resale market.  Indeed here is one clear message that comes out of all of this.  Whatever motivation the restaurant has for rationing sales, it is never optimal to allow unfettered resale of tickets.  That only undermines what you were trying to achieve.  Now Grant Achatz and Nick Kokonas understand that but they are forced to condone the Craigslist market because by law non-refundable tickets must be freely transferrable.

But the cure is worse than the disease.  In fact refundable tickets are your friend. The reason someone wants to return their ticket for a refund is that their willingness to pay has dropped below the price. But there is somebody else with a willingness to pay that is above the price.  We know this for sure because tickets are being rationed at that price. Granting the refund allows the restaurant to immediately re-sell it to the next guy waiting in line. Indeed, a hosted resale market would enable the restaurant to ensure that such transactions take place instantaneously through an automated system according to the same terms under which tickets were originally sold.

Someone ought to try this.