I promised another Leap Frog update blog with graphs, so let's go.
As of this writing, we have 17 days of Leap Frog done. That has allowed us to watch the traditional exponential decay patterns establish themselves and to see in two different periods how the increased difficulty of adding a bonus achievement each week is changing the dynamic. The simple answer is pretty substantially. The first day of each new week, the first day that there is another achievement added has a dramatically larger impact on the number of croakers than just increasing the ratio. Day 8's drop was 91, compared with a total of 227 for the rest of the week. Day 15's drop was 61, ~50% higher than either of the days that has come since.
Elimination Spikes. Day 1: "I'm just here for the badge." Day 8: "I made it through the first week and multiple achievements looks hard." Day 15: "I made it through two weeks and while multiple achievements wasn't too bad, this three business is a bit much and these ratios are getting pretty high."
But these spikes are becoming less dramatic. 91 is a much bigger spike from 53 (Day9's drop) in terms of percentage than Day 16's drop. The exponential decays are becoming less spiky and likely converging.
I plugged the data into Excel, per usual and got our first graph using the built-in linear regression functionality. Here's what it spits out.
For those looking for some refreshers on what some of this means, check out my posts from Leap Frog 1: Xpovos' blog post - mTAhlete
and Xpovos' blog post - LeapFrog: Croaked
. So, our R^2 is looking pretty solid at 98%, so let's evaluate for 1. That is, how many days (x) do we need to get a value of 1 (y) for the sole survivor.
~86.8. Huuh. So basically we'd expect if the progression holds up that Leap Frog will end with a single survivor on day 87. The poor sucker who wins this will have earned on the day before 13 different achievements each with a ratio of 18.2 or higher. That seems improbable. I think in fact that our linear regression is inadequate. Excel is a good tool, but this is not the program's strong suit. Let's try a better regression analysis tool and see if we can't get a more accurate formula to evaluate.
Another way of trying to calculate exponential decay is to look at the half-life. That is, how long does it take 50% of what is remaining to turn into something else. In this case, how many days does it take for 50% of our frogs to croak? With 1759* frogs at the start, we hit our first day with less than 50% (<880) on Day 7 (half-life of 6-7 days). We're down to under a quarter now, so we get to check our math. Our first day with less than 25% of our original group (<440) was Day 16. So one half-life was 6-7 days and two half-lifes was 15-16 days. Therefore, we can estimate that our half-life is probably pretty close to 7 days, perhaps a touch over.
Next, how many half-lives do you need for Valve to release Half-Life 3? Wait, no. How many half-lives do you need for 1759 to croak down to 1?
Step one: Decay constant
Step two: Back to the beginning with the decay constant
So, with this half-life estimation path we get a contest that ends after 76 days. Despite being "only" 10 or 11 days shorter than the previous estimate, it becomes a lot simpler to hypothesize. Now the winner wins on only 10 achievements each at 16.0 ratio each. Still brutal, but nowhere near as bad.
But isn't that still implausbile? There are only about 200 achievements total that could be unlocked for 16.0 or better, and likely some of them will have been used even to get to that point?
Certainly. But this has always just been about random speculation. I make no assertions that the math is genuinely predicting anything. I just point out that the exponential decay model fits pretty well and predicts a much longer contest than we are currently expecting. This outpaces even my hearty 40-60 day prediction from a while back. And the fact is that math never lies. We're going to have to see a change from this pattern of decay (more decay faster as we accelerate in difficulty) or we are going to see a lengthy period of hopping.
*1759 is our current count of entrants. Due to some technical glitches we actually added contestants since the contest started and I ran my first set of numbers in this blog: Xpovos' blog post - 1758 Men Enter; 373 Don't Even Try
. That's why the numbers don't match up perfectly. But all of the math in this blog is based on 1759 initial starters.