How many times an ultra rare trophy is rarer than a common trophy?

[cb_zen]

I really need help, I'm going crazy with a mathematical problem applied to the world of trophies. My Gosh, what hole is this that I decided to dig?

 

Just for fun, some friends linked to a YouTube channel asked me to think about what a ranking would be like that could make a balance between the pure score of the trophies and their respective rarities, so channel members could play with each other looking for some points, like the old days where it sounded like a bit of joy and reward.

 

To make this particular ranking, I think about keeping the trophy score and combining it with a score that is based on the rarity of the trophies. After all, we've all seen that a score based solely on the face value of trophies leads to tons of endless shovelware. On the other hand, a score based solely on rarity has been labeled by many as elitist and exclusionary of the majority of players. (Well, it looks like both rankings meet and get lost in the same place.)

 

We all know that Playstation defines the possible rarity of trophies into five groups: common, uncommon, rare, very rare and ultra rare. It also defines that common trophies have a rarity between 100 and 50%, uncommon between 50 and 20%, rare between 20 and 10%, very rare between 10 and 5% and ultra rare those below 5% (the values after the commas do not matter , for now).

 

Therefore, the help I need comes from this question: thinking about making a ranking of players, if you wanted to translate this difference in rarity into points within a range of 0.01 to 1 (or, if you prefer, from 1 to 100) , without needing to use the entire range, how many points should each trophy be worth in its respective rarity level, regardless of whether it is bronze, silver, gold or platinum?

 

This is an attempt to translate inversely proportional quantities (right?), where the most common is worth less and the rarest is worth more, as if it were a kind of market law.

 

The additional problem with this is trying to define how many times more an ultra rare trophy is worth in relation to the common one and, from that, mainly, also trying to define whether this is a starting point or an ending point.

 

Unfortunately, the initial simplicity I saw in the problem quickly presented itself as a sweet illusion. If I consider the percentage of lowest rarity of each trophy at the extremes, I have a distance that goes from 5% to 100%, therefore 20 times. But if I consider the maximum rarity of each extreme, the distance goes from 1 to 50%, therefore 50 times. However, it can get even worse if I finally include the commas: the distance is 500 times from 0.1% to 50% or 5000 times from 0.01 to 50%.

 

Wouldn't this 5000 higher rarity be anything but a huge exaggeration? The problem is that we can give a concrete example of this distance if we consider a one-button shovelware where you obtain the platinum trophy in 1 minute in relation to the chronological nightmare that the platinum in Crypt of the NecroDancer can represent.

 

Additionally, someone with a more playful mind might want to consider an even greater rarity distance, taking the maximum rarity of the ultra rare and opposing that to the minimum rarity on the other side, which would represent a distance of 10,000 times between one and the other. 

 

However, experience with various games tends to show that things are not always so drastic and point to a more average place. There are UR trophies that can be made in minutes, and there are easy common platinums that take a few hours (remember those that are an absolute prank and that you don't understand their low rarity at the end of the day).

 

In this case, would it be a good way to take the average rarity of the zone in which each trophy is parked? A common trophy would have an average rarity of 75%, followed by the uncommon with 35%, the rare with 15%, the very rare with 7.5% and the ultra rare with 2.5%.

 

I can't tell if, in either case, I translated the rarity into points, would I have exact numbers every time? On the other hand, I have the feeling that the value of the distance between an ultra rare and a common depends more on experience than on mathematics itself.

 

If you understand the main problem and can come up with some kind of solution, I will be immensely grateful.

[PHIL_DADDY1]

Following to find out the answer...

[ruexnimbus]

There's a lot going on here, I think I understand what you're saying. If you want to give more credit to someone with a 0.05% trophy than someone with 0.07% then you must apply the 10,000 step distance. Otherwise I think you'll have to go along with what you mentioned:
 

42 minutes ago, cb_zen said:

In this case, would it be a good way to take the average rarity of the zone in which each trophy is parked? A common trophy would have an average rarity of 75%, followed by the uncommon with 35%, the rare with 15%, the very rare with 7.5% and the ultra rare with 2.5%.

You will have more repeatable values but the nuance of experience will be sacrificed to produce a more digestible score. I think it depends on what kind of scale you want to work with. Given the percentages are in the hundredths you have a maximum domain of 10,000.

 

49 minutes ago, cb_zen said:

On the other hand, I have the feeling that the value of the distance between an ultra rare and a common depends more on experience than on mathematics itself.

Experience is something qualitative, but if you want to attempt to measure it I think the 10,000 step approach is the closest you can get.
 

[meecegoatsalot]

why dont you just make a community leaderboard on truetrophies which already scores trophies by rarity. set it private or invite only then add anyone from your channel that wants to participate.

 

if you are dead set on the maths i have no idea

[Helyx]

Search the forum; there's several formulas posted.

[Jerry_Appleby]

When pondering things like this, I used to do a formula that is basically inverted %s, so I'd do 

100/trophy%, rounded down (floor function)

That number would be your actual value. 

So getting a trophy above 50% would always result in a value of 1, since 100/(50+) < ~1.9999.

 

But then as you get to rarer trophies, a trophy valued at 2% gives a value of 50, whereas a 1% trophy gives 100. Then going even smaller, getting a 0.1% trophy gives a value of 1000. One could argue that extremely rare trophies will give a really unfair advantage, but to be fair I think someone earning super tough and rare trophies deserves extra points. But you could always give a max value that any extremely rare trophy could be worth, so that a single trophy at 0.01% doesn't give 10,000 points and skew the numbers. 

 

I like my formula because every trophy is still worth points (albeit common is only a single point), but someone who plays 10 EZPZ games with say 20 trophies apiece could only earn 200 points total, whereas a single trophy under 1% gives 100+ to another person.

Edited March 12 by Jerry_Appleby
Used wrong sign < lol

[zachpicc]

I think we need a Statistician

[Pbpth]

It depends on whether you want a linear, logarithmic or stepwise reward function. In a linear function, increase in point value between 50% and 49% rarity will be equivalent to the difference between 2% and 1%. In an exponential/logarithmic function you can potentially have minimal point differences for high rarities, with increasing point rewards for very low rarities (which might capture the relationship between 1% rarity and 0.1% rarity better).

 

With that in mind, a staring point might be to use a logarithmic function applied to trophy rarity? If you normalise percentage to a value between 1 (100% achievement rate, log of which is 0) and 0.0001 (equivalent to 0.01 achievement rate, log of which would be equal to -4). If you forced the outcome to positive and multiplied the outcome by 100, you will end up with values between 0 and 400 (which is a more manageable range).
 

Most of the variation in value will come at very low rarities though. That’s why you might want to combine with a stepwise function (maybe rounding common trophies to nearest 10%, rare to nearest 5% etc) to ensure suitable point value differentiation.