On the price of pizza

I’ve got lots of interesting things to blog about. I’ve been running some interesting perceptual experiments, I’m working on adapting and extending some data analysis methods, I’ve been teaching a class on statistical signal processing, and I’ve read a number of essays and articles in which thought-provoking, seemingly wrong things were written.

So, naturally, this will be about the cost of pizza at a local pizzeria.

Sometimes we get pizza from a local place called Dewey’s. Normally, we get a couple of large pizzas. One time recently, though, I decided to get three medium pizzas, thinking that this would give us more or less the same amount of food while allowing us to maximize the variety of toppings. It occurred to me then (though after I had placed the order) that I should figure out which pizza size is the best deal. Tonight, we got Dewey’s again, and I remembered to work this out before ordering.

On the “create your own” page, they give the sizes (in diameter), the base price for each size, and the cost per topping for regular and gourmet toppings.

The sizes are 11″, 13″, and 17″.

The base prices are $8.95, $13.45, and $15.95.

Each regular topping costs $1.50, $1.75, or $2.00, and each gourmet topping costs $1.75, $2.00, or $2.25.

If r indicates the radius, b indicates the base price, n indicates the number of toppings, and c indicates the cost per topping, the cost of a pizza expressed as the number of square inches per dollar is then given by \displaystyle{\frac{\pi r^2}{b+nc}}, which is a non-linear function of n, and the cost expressed as the number of dollars per square inch is the reciprocal of this, \displaystyle{\frac{b+nc}{\pi r^2}=\frac{b}{\pi r^2} + \frac{c}{\pi r^2}n}, which is a linear function of n.

This difference is very nearly interesting.

Anyway, I wrote a short Python script to calculate and plot costs for each size and topping type from 0 to 10 toppings.

Here’s the graph for square inches per dollar:

unit_per_dollar

And here’s the graph for dollars per square inch:

dollar_per_unit

The most obvious result here is that large pizzas are by far the best deal. You get way more pizza per dollar (or, if you prefer, you spend fewer dollars per unit pizza) with a large than with a small or medium.

It’s also kind of interesting that medium pizzas are the worst deal if you’re getting 2 or fewer toppings, but that small pizzas are the worst deal if you’re getting 3 or more toppings (comparing within topping class).

Finally, given the multiplicative role that each additional topping plays, it’s not surprising that the more toppings you get, the bigger the difference between regular and gourmet toppings.

So, it looks like it’ll be only large pizzas from Dewey’s for us in the future.

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