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.

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April 17, 2011 at 10:50 pm

David MillerIt sounds like you want to maximize consumer surplus in a dominant strategy mechanism without giving away money ex post. If so, and there are two bidders with independent values uniformly distributed on [0,1], you want to use a Shao & Zhou (2008) mechanism. Here’s one (there are several variations). Ask both bidders whether their values are above or below 0.5. If either one is above, the price is 0.5, but if both are below the price is zero. If only one is above, he gets it. If both are above or both are below, flip a coin. The price is paid not to the restaurant, but to the losing bidder.

April 17, 2011 at 11:12 pm

jeffInteresting. The restriction I am imposing is that a bidder with zero value gets zero utility. So the mechanism you describe is ruled out. In any case it’s probably not what the restaurant is looking for. But I didn’t know about it, thanks for posting it.

April 18, 2011 at 12:17 am

David MillerYou’re right, probably not what the restaurant is looking for. It doesn’t literally make sense for the restaurant to maximize consumer surplus anyway, as you pointed out in your post. I’m looking forward to your theory of the “value proposition.”

April 18, 2011 at 7:07 am

Marc DelaneyJeff, I’ve been thinking of this one for the last hour now and I wonder if you’d mind to spend a minute or so pointing out where I’m going wrong…

Are you maximising the consumer surplus from the point of view of just one diner (the one who values the experience more) or total consumer surplus of all diners in our world (i.e., both of them)?

The reason I ask is if just from the point of view of the “hungrier” guy, then is it correct to multiply his surplus by a probability of winning in the auction scenario? By definition we’re focusing on the one who bids more, and thus he’s the winner with certainty. I think from his point of view, his surplus in this case is simply is v-v/2 = v/2, which rather puts us back to the same surplus as before for him. What am I missing?

I tried to take this further by summing the combined consumer surplus for both diners in each scenario, for which I’ll introduce u for the other guy, where v>u.

Then we have in the lottery version an average consumer surplus of v/2+u/2 per meal.

In the Vickrey auction system we have a surplus of v-u at all times, taken by one guy.

So is (v+u)/2 > v-u?

Well v is the larger of the two numbers, and given both come from a uniform distribution then we can say that v/2=u. Substituting in, v/2+v/4 = 0.75v for rationing, and v-v/2=0.75v for the auction. So once again, I can’t get an overall improvement. Apologies if I’m making some basic error…

April 18, 2011 at 8:56 am

jeffYes, with an auction you do want to calculate the consumer surplus of the winner and since the winner is the guy with the highest value you want to compute the consumer surplus of the person with the highest value. But you need to compute the probability distribution of that highest value.

In fact, that’s what my calculation is doing. You first fix a bidder with value v and compute what his surplus would be *if* he had the highest value. That is v/2. But then you need to multiply it by the probability that he has the highest value. you get v^2/2.

To conclude the calculation you would want to take the expectation of this with respect to v to get the expected consumer surplus of bidder 1 and then multiply by 2 to get the total expected consumer surplus (because there are two bidders.) But you don’t need that last step because for every possible realization of v each bidders consumer surplus is lower than it would be in the rationing mechanism.

April 18, 2011 at 4:50 pm

jeffAnd here is a direct calculation. We know that the highest bidder will win and obtain surplus (v-v/2) = v/2 when his value is v. We can calculate the probability distribution of “the larger of the two values.” It’s called the first order statistic. When the individual values are distributed uniformly then the density of the first order statistic is 2v (see this helpful overview which calls it the second order statistic for some reason. )

That is 2v is the probability density of the event that the highest value is equal to v. So we take the expectation of v/2 with respect to this density, i.e. we integrate from 0 to 1 the value v/2 against the density 2v. In latex: \int_0^1 v^2 dv. The indefinite integral is v^3/3, and so we get a total consumer surplus of 1/3.

That is worse than the total consumer surplus of 1/2 that we get from randomly choosing someone and giving them a ticket.

April 18, 2011 at 10:11 pm

Glen WeylJeff,

I’m not sure your rationing mechanism is generally superior to some sort of an auction. I think it depends on the distribution of valuations and whether they are log-concave or log-convex. Bulow and Klemperer have a paper (available on Paul’s website) on a very closely related topic.

Glen

April 18, 2011 at 10:40 pm

jeffglen, i believe this boils down to increasing or decreasing hazard rate? see my follow on post, or the addendum to this one which you may not have seen because i added it simultaneously with your comment. i make exactly that point.

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Joshua GansI was reading this post at a cafe just now and out of the blue the owner gave me a free cookie. Explain that!

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jeffMy explanation is that he wanted to give it to me, but only you were there.

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