I never thought I’d be much of a stats person but here we are. This post is low on formalism though, so may the gods of formalism have mercy on my soul!
This post includes the result of experiments showing the value of what is talked about, and includes some simple C++ that you can find at: https://github.com/Atrix256/LootLDS
If you’ve ever played a game that involved grinding for loot, you might have looked online and found the drop rate for a specific item, only to find that if it says it drops one out of 100 times, that it takes you 200-300 runs to get it, while your friends get the drop in the first 10 runs.
What the heck is that about?!
That, my friends, is the nature of regular old random numbers – aka white noise – the kind of random numbers you get from rolling fair dice, flipping fair coins, hashing values using good hash algorithms, or using well made random number generators.
The core issue is that white noise random numbers can take on any valid value with equal probability at any time, regardless of whatever has happened before.
If you were betting on whether a fair coin would come up heads or tails after seeing 10 heads, if you say the next will be tails (because of course it will!) you will still only be right 50% of the time. If you flip the coin an infinite number of times, you will get an even number of heads or tails, but before reaching infinity, all bets are off.
This can be a problem for game designers too. They can collect statistics about how their randomized systems are behaving, analyze the results and come to the conclusion that everything is balanced and even. While that may be true when looking at global averages, the individual player experience may vary wildly and not be balanced at all.
Tangent: This is called variance and is the source of noise in raytraced rendering.
Tangent: There’s a fun related story here about the U.S. air force realizing there is no such thing as an average pilot:
In any case, is this “globally balanced but individually unbalanced” something we have to live with, or is there something we can do about it?
Luckily there is something we can do about it, and we can ensure that individual players have a more balanced, more pleasant, and more controlled experience, without sacrificing randomization.
Enter Low Discrepancy Sequences
A low discrepancy sequence is a sequence of numbers which are neither too close together nor too far apart.
If we put marks evenly spaced on a number line, the sequence of numbers at those marks would have zero discrepancy because they were evenly spaced. Low discrepancy numbers have a low discrepancy value that is greater than zero.
Examples of low discrepancy sequences that you may have heard of are: Sobol, Halton, Van Der Corput.
Some nice links of mine for learning more about low discrepancy sequences are:
Tangent: Going back to the raytraced noise example, regular sampling makes aliasing, and white noise sampling makes noise. Low discrepancy sequences sort of lay somewhere in the middle, gaining the benefits of both worlds, and actually having mystical powers of making numerical integration converge very quickly.
So what do low discrepancy sequences have to do with our problem?
If you use a low discrepancy sequence to generate 5 “random numbers” between 0 and 1, those 5 numbers will be roughly evenly spaced, which means that if you use those numbers on a loot table, the player is going to get a wider spread on the full possibilities of what the loot table has to offer.
If something has a very small percentage to drop, the player still has a low probability to get that drop, but if it’s a 1 in 100 drop, it’s more likely to happen at the 100th drop mark.
This is in constrast to white noise where the values may be clumped together and leave big holes in the 0 to 1 range, like: 0.114, 0.081, 0.093, 0.2, 0.95. There is a huge gap empty between 0.2 and 0.95, which is 75% of the possibilities!
There’s a problem with low discrepancy sequences though: They are deterministic – that is, they are predictable and not random. You get the same values from low discrepancy sequences every time.
Before admitting defeat though, there is another way to get randomization from this even though the sequences themselves are not random: You can shuffle the loot table!
Now, if you have thousands of players on an MMO server rolling numbers against a loot table, you probably just barfed in your mouth a little at my suggestion. There is a way to make a “shuffle iterator” though, so that you can get the benefits of shuffling a loot table, without actually having to keep copies of shuffled loot tables around. You’d use some unique player id (and more) as a random seed for the shuffle iterator, then could use a low discrepancy sequence to select loot. This way, each player would see different (randomized) loot drop sequences, but the loot rolls would still be low discrepancy.
Tangent: you can read more about a way to make shuffle iterators using low quality encryption in “Format Preserving Encryption” here: https://blog.demofox.org/2013/07/06/fast-lightweight-random-shuffle-functionality-fixed/
But we aren’t done yet…
Enter Randomized Low Discrepancy Sequences
The low discrepancy sequences talked about in the last section were deterministic, but what if we were able to make number sequences that had low discrepancy but were randomized?
That exists, and it’s known as “blue noise” because blue noise is random numbers which have only high frequency components (like how blue light is high frequency).
The property of both being low discrepancy, but also randomized, is great for many reasons way outside the scope of this article. For our loot drop purposes, it means that the loot will be both unpredictable, but also a player will get a balanced personalized experience, instead of only the global averages being balanced.
Tangent: Here’s a link about how to generate a blue noise sequence: https://blog.demofox.org/2017/10/20/generating-blue-noise-sample-points-with-mitchells-best-candidate-algorithm/
The other shoe dropping is that blue noise can take a long time to generate, so is computationally expensive. In graphics, it’s common to generate blue noise in advance and just use the pre-made sequence. In loot drops, that is a less viable option because it makes your sequence deterministic and then you are back to shuffling the loot table.
Not real sure the answer here, but it may involve just keeping track of the last N loot drops, and using Mitchell’s best candidate algorithm to generate the N+1th value, adding that to the loot drop RNG list and removing the oldest one. If you get creative you might find a solution that fits your needs.
Before we show experimental results, I wanted to defined a couple terms in regards to low discrepancy sequences.
- Progressive Sequence – A progressive sequence is a sequence of N values, where if you use less than N of the values, they still have the desired properties. For instance, if you make 100 blue noise distributed numbers, but only use the first 10, if it’s a progressive sequence, those first 10 will also be blue. If it isn’t a progressive sequence, you have to use all 100 before they are blue. This is also a property of deterministic low discrepancy sequences. For our loot drop purposes we NEED to use progressive sequences because other wise, the loot drops won’t be balanced until the very end, which kind of defeats the point.
- Open Sequence – An open sequence is one that you can always add more items to. If you regularly space 4 samples from 0 to 1 you are going to get 0, 0.25, 0.5, 0.75. If you want to add a 5th number you can’t! That means that this sequence is not open. Many low discrepancy sequences are open, and using Mitchell’s best candidate to generate blue noise does make an open sequence. For loot drops, we generally do want open sequences, because we usually don’t know how many times the player is going to loot something in advance.
The numbers below are from experiments done using the code that goes with this blog post. It’s ~380 lines of C++ in a single file using only standard includes. You can find it at: https://github.com/Atrix256/LootLDS
I used the following sequences:
- White Noise – Straight up vanilla RNG.
- Blue Noise – Using Mitchell’s Best Candidate to generate a progressive, open, uniform blue noise distributed number sequence.
- Golden Ratio – Starting at 0, i add the golden ratio to the previous loot drop value to get the next loot drop value. I use modulus to keep the value between 0 and 1. The golden ratio has some interesting & good properties as a low discrepancy sequence.
- Sobol – This is the low discrepancy sequence that leads in numerical integration convergence speeds.
For each sequence type, I generated 10 random loot tables which had 2 to 6 items, each item having a random roll weighting between 1 and 20. I then rolled as many loot drops as i needed to make it so the actual loot drop percentages were within 1% of what the loot drop table said they should be.
Higher numbers mean it took more loot drops for the actual loot dropped to reach the probabilities the loot table said they should be. Lower numbers mean it took fewer loot drops to reach the desired probabilities. I did 10 runs so that you can see if things are consistently good or consistently bad. Just doing a single test isn’t enough to show if something just got lucky one time, or if it’s a good choice.
Here are the results….
- White Noise: 50513, 7834, 1859, 516, 8824, 3650, 1380, 24461, 35, 12455
- Blue Noise: 72, 77, 143, 9, 129, 308, 353, 236, 176, 205
- Golden Ratio: 47, 34, 50, 76, 55, 51, 114, 77, 21, 105
- Sobol: 216, 155, 161, 77, 13, 71, 56, 75, 127, 51
It’s important to note that the loot tables themselves are generated with white noise, which is a source of variance in the results above. 10 samples isn’t a whole lot, so in a real analysis you might want to do more (100,000 or more runs?) but hopefully you should see that white noise really is not great. I was also surprised to see that Sobol didn’t do that well compared to blue noise and golden ratio. It must just do better at higher dimensions.
One last thing I wanted to mention is that this isn’t limited to loot drop tables. You can use these concepts for randomized events, procedural content generation, and basically anything else you use random numbers for in games.
The important takeaway here is that even if things look right when looking at global averages, using white noise makes the individual experience very different from that global average. Having better control over crafting a player’s individual experience is possible though, and has the possibility of giving a game a more hand crafted feel, even though you are still using RNG.
FRIENDS DON’T LET FRIENDS USE WHITE NOISE!