Cookie Clicker Beta - Grandma-Type Boosts

We all know what I'll call Grandma-Type Boosts, boosts that you unlock by getting 15 of an object that double the effectiveness of grandmas and give you a type of grandma among the grandmas on your screen. In the beta these have a neat change:

I really like this. Something Cookie Clicker could use more of is things that interact with one another.

Well, naturally I had a spreadsheet going, and I noticed that the value I was calculating for my buildings wasn't matching their actual output. This affected only those buildings with the grandma bonus, so I was fairly sure the grandma bonus was not doing what it said.

I tried adding in the bonus before other doubling multipliers, but that didn't work. After a few experiments I resorted to looking at the code. The game computes the cookies per second of each object by adding up the number of doubling upgrades, raising two to the power of that, and multiplying that by the base value of the object. How do grandmas fit in? Well, if you have the upgrade then it multiplies the number of grandmas by 0.01 and then adds that in with the doubling upgrades. But that doesn't increase the CpS by 1% per grandma, that multiplies the CpS by 2 to the power of 0.01 per grandma.

That means that 50 grandmas don't give 50% more, they give about 41% more. On the other hand, 200 grandmas give four times as much rather than three times, and 300 grandmas give eight times instead of four times.

I don't really have a way to know whether Orteil meant for grandmas to do what the tooltip says or whether Orteil meant for the tooltip to express what grandmas do. But since grandmas give this bonus to all objects except for cursors and themselves, this basically means that the bonus from grandmas is exponential rather than linear, every 100 grandmas doubles your overall output, each grandma is a 0.7% increase.

We know that the cost of grandmas, like other buildings, increases exponentially as well, so this doesn't get too out of hand. Since each grandma gives 0.7% more cookies per second, but each one costs 15% more, the marginal value of each grandma decreases by 13.5%. That means that if it took you one minute to save up for a grandma, the next one will take 14.2% longer, about 68.5 seconds. One hundred grandmas later, all other things being equal, you'd have to wait about 407 days to save up for the next one.

So this isn't going to ultimately make a huge difference, but if you consider the difference between getting 14.2% compound interest and 15% compound interest, you'll know that as the numbers get bigger you'll start noticing a big difference. Because that difference is ultimately translated back into cookies to spend on a resource with an exponential cost, the total will only end up being a little bit higher.

Still, it makes a big difference to the relative value of grandmas and prisms as the game goes on, and that's kind of nice since grandmas are so key to the plot of Cookie Clicker.