The silliest way to save $4 per week, part 2

To recap

TwoSix weeks ago, I laid out a subway puzzle I came across recently. The goal of the puzzle is to ride the subway as cheaply as possible. The rules of the puzzle are as follows:

1. The normal subway fare is $2.75
2. The Cash App card lets you ride the subway for $1.75 every six hours (but if you need to use it more than once within six hours the additional rides are full price, $2.75)
3. The Commuter Benefits card lets you ride the subway at a discount equal to your marginal tax rate
4. When you ride the subway twelve times in one week with a given card, all future rides on that card are free

The start of a solution

When I first started thinking about this puzzle, solutions for a few edge cases came quickly:

• If you take twelve or fewer rides per week, you should just use the Cash App card boost whenever it’s available and use the Commuter Benefits card when the boost is not available
• If you never ride the subway more than once within six hours (or even, never ride the subway more than once within six hours in your first twelve rides, since the cost of rides past twelve is irrelevant) you should always use the Cash App card
• If your taxable income is more than $170,050, your marginal income tax rate in NYC is ~42.2% or higher. This is greater than the Cash App boost’s ~36.4% discount, and there are no restrictions on using the Commuter Benefits card (besides that all purchases must be for transit), so you should always use the Commuter Benefits card
  • Aside, this makes the Commuter Benefits card a progressive benefit. That is to say, it saves you more the higher your income, which is kind of silly

If any of the above applies to you, your personal solution is simple. If you ride the subway more than twelve times per week, sometimes take the subway more than once within six hours, and make less than $170,051, your solution is more complicated.

Heat maps

At one point, when not thinking about this puzzle at all, a vision came to me. I had a vague idea that this could be represented as a heat map, where one axis represents the number of Cash App rides taken per week and the other axis represents the number of Commuter Benefits rides taken in a week. I didn’t have an idea of how it was going to work, but I saw something like this in my head:

A few days later I set about making it, then I refined it over the following few days with the help of some friends, and out came this:

Here is the general premise of the heat map:

1. In this puzzle there are a few things we know
   a. The price of a subway ride
   b. The Cash App boost discount
   c. The Cash App boost cooldown (the minimum amount of time between boosted rides)
   d. Our marginal income tax rate
   e. The maximum number of paid rides on a single card
2. And a few things we don’t know
   a. How many times we will ride the subway in a given week
   b. How many times we will ride the subway within the Cash App boost cooldown window in a given week (“unboosted situations”)
3. Another way to phrase the key question of the puzzle is “What should we do in each unboosted situation, pay with our Cash App card or pay with our Commuter Benefits card?”
4. 2b is more pertinent to the actual question at hand, so let’s estimate 2a, and see how much money different answers to 2b cost.

Please check out and make a copy of the full sheet if you want to play around with it. Once you make a copy, put your information in column B and the heat map should update to tell you how much using a Cash App card or Commuter Benefits card will cost given a certain number of unboosted situations.

There are three tabs in the sheet, which serve slightly different purposes. I think it’s best to go through a concrete example to illustrate their usage.

The default setting on the sheet has our median New Yorker’s marginal tax rate of %21.79 and estimates 18 weekly subway rides, so that’s what we’ll look at for this example. Here is the heat map from the first tab with some points of interest marked:

The way to read the “heat” in this heat map, is that green costs less and red costs more for a given number of unboosted situations. The number of unboosted situations stays static across the ⟋ diagonals. To illustrate this, I’ve drawn a line along the six-unboosted-situation diagonal. The cheapest option is to use the Cash App card six times, followed by using the Cash App card five times and the Commuter Benefits card once, followed by using the Cash App card four times and the Commuter Benefits card twice, etc…

We can see that running into zero unboosted situations yields the lowest price on the whole sheet, in the top left corner. $21 is just $1.75 * 12, but it’s good to confirm our expectation that this is the cheapest way to ride.

There is something a bit misleading about this heat map, which is highlighted in the triangle on row 1 column 18. We can see that, starting at column 18, all of the costs are the same: $25.81. This is because these cells represent only using the Commuter Benefits card, and $25.81 = $2.75 * 0.7821 * 12. Going back to our six-unboosted-situation diagonal, if we can pay $25.81 using only the Commuter Benefits card, is it fair to say that $27.00 is the best price for six unboosted situations? No, but this map is too dumb for that. For a really clear illustration of what we should actually do, let’s take a look at the second tab:

Here, all of the cells are capped to the price of using the Commuter Benefits card twelve times. The guidance is really clear: if we run into four or fewer unboosted situations we should use the Cash App card all of the time, if we run into five or more we should use the Commuter Benefits card all of the time.

Unfortunately, this does not make for a very pretty visualization, but I think it most effectively solves the puzzle. Again, feel free to copy the sheet and put in your own information to see your personal solution.

The silliest way to save $4 per week

The best case scenario for our median New Yorker is to run into zero unboosted situations and spend $21. The worst case is to run into five or more unboosted situations and spend $25.81. So we come back to the title. Is it worth it to go through all of the trouble of puzzling this out and making spreadsheets to save $4.81 per week in the best case? Probably not. But it was fun. Thanks for joining me.

Addendum

I really did have fun making the spreadsheet and solving the puzzle, but I didn’t have too much fun writing about it (hence both of the posts being late). I think part of the reason why I didn’t have as much fun writing about it is because I had solved the puzzle ahead of writing the posts. I had already done a bunch of writing and talking to people to solve the puzzle, but that was before I started this site, so I got kind of sick of going through it again on here. Part of the premise for bunchanonsense laid out in post 1 was that these posts would be written in the process of solving problems, not after the fact. I intend to do that going forward. Expect future posts to be shorter and more work-in-progress. Hope that’s not annoying.

Happy new year :)