Imagine that you are a service provider and you have two ways of reaching customers. One way has you selling directly to customer while the other goes through a middleman and requires paying a commission. If you have a limited capacity, how much should you allocate to one channel over the other? What if you have to sell though the channel requiring a commission first so you cannot easily reallocate capacity between the channels?

That is the challenge facing many restaurants who offer reservations through OpenTable. Many restauranteurs game the system and may be less than honest on OpenTable about available capacity. Here is how **MainStreet** explains the issue (How to Tell if the Restaurant Is Lying to You, Feb 10).

Longtime OpenTable user Marcy Schackne offers testimonial validation. She checked OpenTable to book at the Palm steakhouse in Bal Harbour, Fla; it showed up full, but when she called and asked for a table, she was promptly given a reservation.

Precisely the same happens at hundreds of restaurants every night.

What gives? Dennis Lombardi, executive vice president for food services strategies at retail consulting firm WD, said that for many restaurants, the $1 per diner they pay OpenTable for a booking – on top of a fixed monthly fee – “rankles.”

They think they can book diners more cheaply themselves,” he said.

Adi Bittan, CEO of feedback service OwnerListens, with many restaurant clients, added: “For times when they expect to be full based on past experience, they do not want or need to take the OpenTable reservation. They’re taking a gamble, because they could end up with empty tables — and then the diner will walk by and see it — but it’s a calculated gamble based on probability. Since most of us, restaurant managers included, are not economists or mathematicians, we understand this dynamic intuitively but will often get those exact probabilities wrong.”

Meaning the restaurant bets that it doesn’t need OpenTable, but the empty chairs it winds up with make a mockery of their inductive capabilities.

I’m not really an economist nor a mathematician, but I have thought about this problem.

This is, in fact, a variant of a classic problem in revenue management — arguably *the* classic problem in the field — first studied by Ken Littlewood back in 1972. To see how this works, suppose that a customer who books through OpenTable provides the restaurant with a margin of O while one who phones for a reservation provides a margin of P. Suppose that the restauranteur has already decided to save K seats for phone customers. Should the restauranteur bump that up to K+1? Note that what she is doing (effectively) is giving up O dollars for the possibility of P dollars but she only gets those P dollars if demand for phone reservations is strictly greater than K. That is, she will be better off increasing the number of seats saved for callers if:

P*Probability(D > K) > O

where D represents demand for over-the-phone reservations.

Following through that logic, the restauranteur should keep increasing the level of reservations held for callers until

Probability(D > K*) = O/P,

where K* is the optimal number of seats saved for phone reservations.

Note that this should be a pretty easy policy to implement. All we need is the distribution of phoned-in reservation requests and an estimate of the margin on each customer. The former can be had from looking at historical data. The one complication may be that we need to look at the number of requests that came in after the capacity allocated to OpenTable has sold out. That is, three weeks out when there are plenty of seats available on OpenTable, some luddite may call instead of going on line. That then takes a seat away from OpenTable inventory. What we are interested in is how many people call after OpenTable says there are no seats.

As for the margins, that’s pretty easy. Unless there is some compelling evidence that callers spend differently, the only real difference in the margins is that commission paid to OpenTable so O = P – 1 and we can rewrite the fraction defining K* as (P-1)/P.

It turns out that this fraction has a nice interpretation. It gives the fraction of time that the restaurant should have to turn down a phone request. If P = $10, that means that 90% of the time (or, more likely, on 90% of weekend nights), customers are going to call and be told they are out of luck. Said another way, if P is fairly high, it is not clear that it is worth messing around and holding back seats from OpenTable since it very costly to let a table sit idle.

Of course, that depends on P. If P is low, it is different story. For example, if P $1.25, then that fraction becomes 20% and lots of capacity gets saved for callers. That gets to an important question of how we measure these margins. Restaurants can be a very low margin business. However, we are really thinking here about what we make on serving one more party. Having one more party this evening doesn’t bump the rent up or likely require extra staff in the kitchen. If a dollars worth of greens can be turned into an $8 house salad, then P is pretty high and little capacity should be saved for called-in reservations.

A final point, the discussion so far has assumed that the distribution of calls for reservations doesn’t depend on the number of reservations available for callers. If customers realize what is going on, that may not be true. Calling takes effort and hence is costly. Further, that effort may be for naught. In such a setting, one can show that it may be best to forgo saving any seats for callers.

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