So luck badges. Years ago I tried doing research into the likelihood of Close Call, Pretty Lucky, and Lucky Day causing enemies to miss Mario in TTYD, taking thousands of samples for each, and summarized the badges’ effects among a bunch of others in my Badge Hunting + Stacking series. Then after getting back into TTYD assembly reverse-engineering in the wake of the Palace Skip discovery last year (at that point having 2+ years actual software engineering experience), I found the exact rates of the badges, updated that post, and left it at that.
But for the 4 years since that initial post (and 5+ years since the initial statistical research), I took for granted that given evasion rates for each badge, it would be obvious how those ought to be combined into one evasion rate when stacked or combined. Subsequent recent experience with a larger community of Paper Mario players has shown that not to be the case, so let’s do a deeper dive, shall we?
Brief note – all of the principles in this post could be applied to the evasion badges in the original Paper Mario as well, except the miss probabilities are a bit different (10/101 for Pretty Lucky, 20/101 for Lucky Day, and 30/101 for Close Call). And there aren’t multiple copies of each to stack, obviously.
Calculating Evasion Rate (Independent vs. Disjoint Probability)
The crux of the issue comes down to badges’ evasion chances being independent of one another, and the fact that the math involved with that is un-intuitive compared to just adding disjoint probabilities together.
Illustrated here is a representation of what disjoint treatment of badges would look like, and the independent probabilities they actually have (assuming you have one Pretty Lucky, one Lucky Day, and one Close Call badge on in TTYD):
That’s a lot to take in, so let me elaborate in text form.
If badges represented disjoint probabilities of being missed, then you would just sum their evasion rates to get the total evasion rate; i.e. PL+LD+CC = 10 + 25 + 33% would be 68%, and the odds of getting hit would be 32%.
In actuality, though, evasion is calculated by multiplying the indepedent probabilities of the badge resulting in a hit; i.e. PL+LD+CC = 0.90 * 0.75 * 0.67 = 0.45225 = 45.225% chance of hit, or 54.775% chance of a miss.
This extends to multiple badges of the same type; i.e. each Pretty Lucky multiplies your chance of getting hit by 90%, every Lucky Day by 75%, and every Close Call by 67%; i.e. your total evasion rate is:
1 – (1 – 0.10)PL * (1 – 0.25)LD * (1 – 0.33)CC
But which of the evasion badges is the best?
(hint: it’s Close Call.)
So that settles the question of how badge evasion is calculated, but how do you compare the effectiveness of each of the three evasion badges, particularly Pretty Lucky and Lucky Day? You cannot divide the rates by their BP cost to get an idea of “evasion per BP”, since that’d essentially be falling back on the incorrect disjoint-probability model.
Instead, since you already multiply the probabilities of badges together, you need to see what number needs to be multiplied by itself N times to produce the proper hit rate for an N-BP badge. Thankfully, math has a helpful name for that concept, the radical / Nth root, but we can do one better and use logarithms to convert all the messy multiplication (for multiplying the effects from each of the types of badges), exponentiation (for getting the term for a number of a specific badge), and radicals (for getting the effect per BP) into much nicer addition, multiplication, and division respectively.
So, let’s take and compare the log base 0.5 of each of the badges’ hit probabilities, then (i.e. how many times you would have to multiply 0.5 by itself to get the hit rate). The choice of 0.5 is admittedly arbitrary, but this way we can compare it to something meaningful; Repel Capes / Dodgy Fog (or enemy Dizziness) cause a 50% chance of evasion.
|Badge||Hit Rate||log0.5(Hit Rate)|
This gives us some concrete insight into the badges’ real relative worth; Pretty Lucky is 15.2% of the effect of a Repel Cape, Lucky Day is 41.5%, and Close Call about 57.8%. But we can do one better, and divide the log-transformed rates by the BP cost to compare the badges’ worth per BP.
|Badge||Hit Rate||log0.5(Hit Rate)||log0.5(Hit Rate) ÷ BP Cost|
Yikes, no contest now; Pretty Lucky is a good deal better than Lucky Day per BP (7.6% of a Repel Cape per BP vs. 5.9%), and Close Call destroys them both.
The Power of Math!
Rephrasing the effects of the badges this way means we have another way of representing overall evasion:
1 – (0.5)(0.152 * PL + 0.415 * LD + 0.578 * CC)
While not perhaps easier to calculate this way, it does make one fact perhaps more obvious; that being, there’s no way to get 100% evasion, as no matter how high the exponent gets, nothing will make 0.5N equal 0.
Except, technically that’s not actually true; since all RNG calls for badge evasion happen back-to-back, and Paper Mario: TTYD has a finite number of RNG states, technically it could be possible to wear enough badges that there’s no string of consecutive RNG states that do not result in at least one “miss”.
If using just Close Calls, this is actually possible to achieve at 52 badges; using the formula about 0.5(0.578 * 52) = 0.5(30.0) = about a 1/1,000,000,000 chance, but in fact no successive run of 52 out of the 4,294,967,294 RNG states produces all non-misses. Not practical in purpose, as a “mere” 17 badges already results in a miss 999 out of 1,000 times, but interesting as a curiosity.
The Mega-Comparison Chart of Awesomeness
Rather than grab a calculator every time you want to get the exact evasion rate for an arbitrary combination of badges, here’s a helpful chart that compares the evasion rate for various combinations of 1-3 Pretty Luckys, 1 Lucky Day, and 1-3 Close Calls:
You can extrapolate past the end of the chart by imagining more copies of the bars at the top added to the end, bearing in mind that adding another span of the marked length halves the hit rate. (Of course, as a rule of thumb, if you’re bothering to badge hunt for more than 2 Close Calls, you’re probably going to get missed often enough (70%+) that you don’t really need to care about exactness.)
Hopefully this doesn’t end up being more confusing than the original blurb in the badges article, but I’ve said my piece at this point. Close Call master race!