The decision by a retailer to go with pay first/pay later models is only one small part of an overall pricing strategy/experience design problem that is generally called "yield rate management." Besides this decision, other decisions that go into YRM include sizing the floorspace, the density of packing of the tables (or product displays), sizing the menu, deciding on how many choices to offer etc., choosing the service "style" (brusque vs. "take all the time you need") and selecting which service elements to offer for free (free breadsticks/appetizers/fortune cookies/crayons/drawing paper for kids, juke boxes...), whether or not you allow bargaining, and whether you should hire expensive sales people with psychological manipulation skills or cheap ones who just handle transactions.

There are also back-end couplings to things like inventory levels and ordering frequencies from the supply chain. Pay first/pay later might end up depending indirectly on whether you buy fresh seafood in the morning from the pier, or get weekly deliveries of frozen stock.

And these days there are things like Groupon, which can throw your entire YRM model temporarily out the window.

Note: Most people who should be solving YRM as a system-design problem don't. They think of pricing design as an unrelated afterthought to qualitative/aesthetic experience design. Breadsticks and crayons for kids are FINANCIAL decisions, not merely aesthetic ones to be outsourced to the interior decorator, or worse, the owner's second cousin's wife who knows nothing about either design or pricing but is "artsy." This is one BIG reason businesses fail. I suspect a good fraction of the pay now/pay later decisions made by restaurants out there are WRONG.

The string holding this system design problem together is finance. It has to do with margins and velocities (the speed at which a product moves), and these dynamics are not unique to the restaurant business, and it is useful to understand things more broadly. Besides the pay-first end (fast food) and pay-later end (dine-in restaurants) there is also the intermediate category of coffee shops (pay as you go over several hours) and bars (pay as you go OR keep a tab open).

Coffee shops are particularly interesting because they face a particularly tough YRM problem. In Vegas where I live, there is a coffeeshop that hands out WiFi access codes with time limits based on the price of the drink you order. Right question being asked: YRM; godawful answer. The only real solution for coffee is to actually plan on offering 2 distinct experiences that feed off each other: high-velocity in-out traffic that can subsidize low-velocity sitting traffic. Starbucks grew big because they got this formula right. The take-out/dine-in mix is also why Chinese restaurants are able to offer leisurely AND cheap dine-in.

The general ideas that help understand these phenomena can even be applied to online shopping experiences (like on Amazon... in fact one of the neatest innovations in shopping is how Amazon manages to have its cake and eat it too... One-Click shopping automatically bundles together things ordered within a certain fixed time window).

It all begins with this equation:

Profits = Margins*Velocity

If margins are low, velocity must go higher to maintain the same profit.

Let's unpack this a little bit more.

Velocity is not the rate at which you pay. It is the rate at which you make firm purchase decisions. You make multiple such decisions in a given buying context. Items in a given context are usually not too different in terms of margins.

Due to the basic phenomenon of diminishing marginal returns, if a buying context has only one kind of item, you'll hit diminishing returns in as quickly as one purchase decision (you're not going to buy 3 burritos one after the other in a burrito shop). If there's more variety, you can buy other things. But the overall shopping experience also induces fatigue so there is diminishing marginal returns with respect to the immediate payofff (which comprises immediate product use value, plus the high from shopping itself). So it pays to engineer shopping contexts so there is a gap between decisions, allowing your buying desire to recover. Free stuff in between can help this recovery process. If there's immediate consumption-related satiation, you have to wait longer.

So putting these observations together, the yield rate per visit per customer is the graph of average yield over time. It will have a certain shape, a peak and a leveling off. It can be shaped using a variety of design techniques.

If it is a shared shopping context (especially with social network effects, where people imitate buying behavior etc.), it is also useful to sum over all users shopping together and dividing by floor space to get a spatial yield rate per square foot (or per seat). If the time-rate is relatively predictable and/or you have high sunk costs, you can make it a little coarser by designing against occupancy rate instead (airlines and casinos do this, since a seat costs and yields predictable amounts per unit time). An interesting example of spatial YRM is the use of seat-fillers in the entertainment business. People enjoy a show more if the theater is full, so since empty seats are not much cheaper than full seats, it makes sense to give away lossy or free tickets or even PAY people to attend, to maximize the value for the paying, profitable customers.

Finally, there is a crucial boundary condition: the arrival rate of new shoppers, who have a certain probability of staying and a certain probability of leaving (due to overcrowding, wait times, queue lengths, etc.). For an unscheduled/unreserved service, this is usually a Poisson distribution for a given time of day/day of week, with a size determined by your marketing.

So from the retailer's point of view, there is an optimization problem: design the user experience so that you kick the shopper out at just the right time in their yield curve, and preferably have that time coincide with their own preferred exit time. Even if they stay longer and make more purchase decisions, if the rate is too slow, the opportunity costs of lost sales from others might be higher.

Finally, both buyer and seller pay transaction costs when the payment is made, so it makes sense to bundle as many purchase decisions together as possible, but OTOH, some shopping contexts have more firm commitments at decisions. So if you put something in your grocery store cart, you are likely to buy it. But if you are dawdling in a coffee shop, you will likely not buy all you need up front. A clothing retailer will see a higher return rate than a grocery store.

How do you put all this complexity together?

Surprisingly, the answer is "fluid mechanics." The two basic ways of understanding yield (as a time flow of decisions and as a spatial density of decisions) are actually the two basic ways of modeling fluids flowing through systems (they are called Lagrangian and Eulerian models).

The design of the spatial (physical or virtual) shopping environment is the design of what I call a "field" and is similar to the design of plumbing. You have to think about friction, lubrication, ways to damp up flow in some parts, speed it up in other parts, adding exit valves and pressure gauges at the right places...

The design of the temporal experience (the flow) is induced by the field design. But you can start from that end and back out the field design as well, based on expected incoming rate, desired flow speed etc.

It is actually possible to put all these ideas together as a mathematical model that pops out optimal answers to questions like "should I use pay first/pay later?" given what-if inputs like arrival rate, floor space, inventory costs etc.

At one point I considered building a web-based calculator for this with nice graphs etc., and even thought of a startup idea based on it (the math models are basically similar to stuff I was working on in grad school). But then I got lazy and decided it would be too much work.