Cheap Talk

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 to be the value, as the restaurant perceives it, that would be realized by service to a patron whose willingness to pay is .  One natural example would be

where 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 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 and whatever price is paid. The weight on profit is and the weight on consumer surplus is .  After you integrate by parts you now get the following formula for virtual surplus.

And now we have something!  Because  if is between and then the first term is increasing in (up to the cap ) and the second term is decreasing.  For close enough to , 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.

Restaurants, touring musicians, and sports franchises are not out to gouge every last penny out of their patrons.  They want patrons to enjoy their craft but also to come away feeling like they didn’t pay an arm and a leg.  Yesterday I tried to formalize this motivation as maximizing consumer surplus but that didn’t give a useful answer. Maximizing consumer surplus means either complete rationing (and zero profit) or going all the way to an auction (a more general argument why appears below.)  So today I will try something different.

Presumably the restaurant cares about profits too.  So it makes sense to study the mechanism that maximizes a weighted sum of profits and consumer’s surplus. We can do that.  Standard optimal mechanism design proceeds by a sequence of mathematical tricks to derive a measure of a consumer’s value called virtual surplus.  Virtual surplus allows you to treat any selling mechanism you can imagine as if it worked like this

1. Consumers submit “bids”
2. Based on the bids received the seller computes the virtual surplus of each consumer.
3. The consumer with the highest virtual surplus is served.

If you write down the optimal mechanism design problem where the seller puts weight on profits and weight on consumer surplus, and you do all the integration by parts, you get this formula for virtual surplus.

where is the consumer’s willingness to pay, is the proportion of consumers with willingness to pay less than and is the corresponding probability density function.   That last ratio is called the (inverse) hazard rate.

As usual, just staring down this formula tells you just about everything you want to know about how to design the pricing system.  One very important thing to know is what to do when virtual surplus is a decreasing function of . If we have a decreasing virtual surplus then we learn that it’s at least as important to serve the low valuation buyers as those with high valuations (see point 3 above.)

But here’s a key observation: its impossible to sell to low valuation buyers and not also to high valuation buyers because whatever price the former will agree to pay the latter will pay too.  So a decreasing virtual surplus means that you do the next best thing: you treat high and low value types the same. This is how rationing becomes part of an optimal mechanism.

For example, suppose the weight on profit is equal to . That brings us back to yesterday’s problem of just maximizing consumer surplus. And our formula now tells us why complete rationing is optimal because it tells us that virtual surplus is just equal to the hazard rate which is typically monotonically decreasing. Intuitively here’s what the virtual surplus is telling us when we are trying to maximize consumer surplus. If we are faced with two bidders and one has a higher valuation than the other, then to try to discriminate would require that we set a price in between the two. That’s too costly for us because it would cut into the consumer surplus of the eventual winner.

So that’s how we get the answer I discussed yesterday.  Before going on I would like to elaborate on yesterday’s post based on correspondence I had with a few commenters, especially David Miller and Kane Sweeney. Their comments highlight two assumptions that are used to get the rationing conclusion:  monotone hazard rate, and no payments to non-buyers.  It gets a little more technical than usual so I am going to put it here in an addendum to yesterday (scroll down for the addendum.)

Now back to the general case we are looking at today, we can consider other values of

An important benchmark case is when virtual surplus reduces to just , now monotonically increasing.  That says that a seller who puts equal weight on profits and consumer surplus will always allocate to the highest bidder because his virtual surplus is higher.  An auction does the job, in fact a second price auction is optimal.  The seller is implementing the efficient outcome.

More interesting is when is between and . In general then the shape of the virtual surplus will depend on the distribution , but the general tendency will be toward either complete rationing or an efficient auction.  To illustrate, suppose that willingness to pay is distributed uniformly from to . Then virtual suplus reduces to

which is either decreasing over the whole range of (when ), implying complete rationing or increasing over the whole range (when ), prescribing an auction.

Finally when virtual surplus is the difference between an increasing function and a decreasing function and so it is increasing over the whole range and this means that an auction is optimal (now typically with a reserve price above cost so that in return for higher profits the restaurant lives with empty tables and inefficiency.  This is not something any restaurant would choose if it can at all avoid it.)

What do we conclude from this?  Maximizing a weighted sum of consumer surplus and profit yields again yields one of two possible mechanisms: complete rationing or an auction.  Neither of these mechanisms seem to fit what Nick Kokonas was looking for in his comment to us and so we have to go back to the drawing board again.

Tomorrow I will take a closer look and extract a more refined version of Nick’s objective that will in fact produce a new kind of mechanism that may just fit the bill.

Addendum: Check out these related papers by Bulow and Klemperer (dcd: glen weyl) and by Daniele Condorelli.

Last week, in response to our proposal for how to run a ticket market, Nick Kokonas of Next Restaurant wrote something interesting.

Simply, 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. Right now we have it as a great bargain for those who can buy tickets. Ideally, we keep it a great value and stay full.

Economists are not used to that kind of thinking and certainly not accustomed to putting such objectives into our models, but we should.  Many sellers share Nick’s view and the economist’s job is to show the best way to achieve a principal’s objective, whatever it may be.  We certainly have the tools to do it.

Here’s an interesting observation to start with.  Suppose that we interpret Nick as wanting to maximize consumer surplus.  What pricing mechanism does that? A fixed price has the advantage of giving high consumer surplus when willingness to pay is high.  The key disadvantage is rationing:  a fixed price has no way of ensuring that the guy with a high value and therefore high consumer surplus gets served ahead of a guy with a low value.

By contrast an auction always serves the guy with the highest value and that translates to higher consumer surplus at any given price.  But the competition of an auction will lead to higher prices.  So which effect dominates?

Here’s a little example. Suppose you have two bidders and each has a willingness to pay that is distributed according the uniform distribution on the interval .  Let’s net out the cost of service and hence take that to be zero.

If you use a rationing system, each bidder has a 50-50 chance of winning and paying nothing (i.e. paying the cost of service.)  So a bidder whose value for service is will have expected consumer surplus equal to .

If instead you use an auction, what happens?  First, the highest bidder will win so that a bidder with value wins with probability .  (That’s just the probability that his opponent had a lower value.)  For bidders with high values that is going to be higher than the 50-50 probability from the rationing system. That’s the benefit of an auction.

However he is going to have to pay for it and his expected payment is . (The simplest way to see this is to consider a second-price auction where he pays his opponent’s bid.  His opponent has a dominant strategy to bid his value, and with the uniform distribution that value will be on average conditional on being below .)  So his consumer surplus is only

because when he wins his surplus is his value minus his expected payment , and he wins with probability .

So in this example we see that, from the point of view of consumer surplus, the benefits of the efficiency of an auction are more than offset by the cost of higher prices.  But this is just one example and an auction is just one of many ways we could think of improving upon rationing.

However, it turns out that the best mechanism for maximizing consumer surplus is always complete rationing (I will prove this as a part of a more general demonstration tomorrow.)  Set price equal to marginal cost and use a lottery (or a queue) to allocate among those willing to pay the price.  (I assume that the restaurant is not going to just give away money.)

What this tells us is that maximizing consumer surplus can’t be what Nick Kokonas wants.  Because with the consumer surplus maximizing mechanism, the restaurant just breaks even.  And in this analysis we are leaving out all of the usual problems with rationing such as scalping, encouraging bidders with near-zero willingness to pay to submit bids, etc.

So tomorrow I will take a second stab at the question in search of a good theory of pricing that takes into account the “value proposition” motivation.

Addendum:  I received comments from David Miller and Kane Sweeney that will allow me to elaborate on some details.  It gets a little more technical than the rest of these posts so you might want to skip over this if you are not acquainted with the theory.

David Miller reminded me of a very interesting paper by Ran Shao and Lin Zhou.  (See also this related paper by the same authors.) They demonstrate a mechanism that achieves a higher consumer surplus than the complete rationing mechanism and indeed that achieves the highest consumer surplus among all dominant-strategy, individually rational mechanisms.

Before going into the details of their mechanism let me point out the difference between the question I am posing and the one they answer.   In formal terms I am imposing an additional constraint, namely that the restaurant will not give money to any consumer who does not obtain a ticket.  The restaurant can give tickets away but it won’t write a check to those not lucky enough to get freebies.  This is the right restriction for the restaurant application for two reasons.  First if the restaurant wants to maximize consumer surplus its because it wants to make people happy about the food they eat, not happy about walking away with no food but a payday.  Second, as a practical matter a mechanism that gives away money is just going to attract non-serious bidders who are looking for a handout.

In fact Shao and Zhou are starting from a related but conceptually different motivation: the classical problem of bilateral trade between two agents.  In the most natural interpretation of their model the two bidders are really two agents negotiating the sale of an object that one of them already owns.  Then it makes sense for one of the agents to walk away with no “ticket” but a paycheck.  It means that he sold the object to the other guy.

Ok with all that background here is their mechanism in its simplest form.  Agent 1 is provisionally allocated the ticket (so he becomes the seller in the bilateral negotiation.) Agent 2 is given the option to buy from agent 1 at a fixed price.  If his value is above that price he buys and pays to agent 1.  Otherwise agent 1 keeps the ticket and no money changes hands.  (David in his comment described a symmetric version of the mechanism which you can think of as representing a random choice of who will be provisionally allocated the ticket.  In our correspondence we figured out that the payment scheme for the symmetric version should be a little different, it’s an exercise to figure out how.  But I didn’t let him edit his comment. Ha Ha Ha!!!)

In the uniform case the price should be set at 50 cents and this gives a total surplus of 5/8, outperforming complete rationing. Its instructive to understand how this is accomplished.  As I pointed out, an auction takes away consumer surplus from high-valuation types.  But in the Shao-Zhou framework there is an upside to this.  Because the money extracted will be used to pay off the other agent, raising his consumer surplus.  So you want to at least use some auction elements in the mechanism.

One common theme in my analysis and theirs is in fact a deep and under-appreciated result.  You never want to “burn money.”  Using an auction is worse than complete rationing because the screening benefits of pricing is outweighed by the surplus lost due to the payments to the seller.  Using the Shao-Zhou mechanism is optimal precisely becuase it finds a clever way to redirect those payments so no money is burned.  By the way this is also an important theme in David Miller’s work on dynamic mechanisms. See here and here.

Finally, we can verify that the Shao-Zhou mechanism would no longer be optimal if we adapted it to satisfy the constraint that the loser doesnt receive any money.  It’s easy to do this based on the revenue equivalence theorem.  In the Shao-Zhou mechanism an agent with zero value gets expected utility equal to 1/8 due to the payments he receives. We can subtract utility of 1/8 from all types and obtain an incentive-compatible mechanism with the same allocation rule.  This would be just enough to satisfy my constraint.  And then the total surplus will be 5/8-2/8 = 3/8 which is less than the 1/2 of the complete rationing mechanism.  That’s another expression of the losses associated with using even the very crude screening in the Shao-Zhou mechanism.

Next let me tell you about my correspondence with Kane Sweeney.  He constructed a simple example where an auction outperforms rationing.  It works like this.  Suppose that each bidder either had a very low willingness to pay, say 50 cents, or a very high willingness to pay, say $1,000.  If you ration then expected surplus is about $500. Instead you could do the following.  Run a second-price auction with the following modification to the rules.  If both bid $1000 then toss a coin and give the ticket to the winner at a price of $1.  This mechanism gives an expected surplus of about $750.

Basically this type of example shows that the monotone hazard rate assumption is important for the superiority of rationing.  To see this, suppose that we smooth out the distribution of values so that types between 50 cents and $1000 have very small positive probability.  Then the hazard rate is first increasing around 50 cents and then decreasing from 50 cents all the way to $1000.  So you want to pool all the types above 50 cents but you want to screen out the 50-cent types.  That’s what Kane’s mechanism is doing.

I would interpret Kane’s mechanism as delivering a slightly nuanced version of the rationing message.  You want to screen out the non-serious bidders but ration among all of the serious bidders.