# Random number generator seed mistakes

## Long run or broken software?

I got a call one time to take a look at randomization software that wasn’t randomizing. My first thought was that the software was working as designed, and that the users were just seeing a long run. Long sequences of the same assignment are more likely than you think. You might argue, for example, that the chances of flipping five heads in a row would be (1/2)5 = 1/32, but that underestimates the probability because a run could start at any time. The chances that the first five flips are heads would indeed be 1/32. But the probability of seeing five heads in a row any time during a series of flips is higher.

Most of the times that I’ve been asked to look at randomization software that “couldn’t be right,” the software was fine. But in this case, there wasn’t simply a long run of random results that happened to be equal. The results were truly constant. At least for some users. Some users would get different results from time to time, but others would get the same result every single time.

The problem turned out to be how the software set the seed in its random number generator. When the program started up it asked the user “Enter a number.” No indication of what kind of number or for what purpose. This number, unbeknownst to the user, was being used as the random number generator seed. Some users would enter the same number every time, and get the same randomization result, every time. Others would use more whimsy in selecting numbers and get varied output.

How do you seed a random number generator in a situation like this? A better solution would be to seed the generator with the current time, though that has drawbacks too. I write about that in another post.

## Seeding many independent processes

A more subtle problem I’ve seen with random number generator seeding is spawning multiple processes that each generate random numbers. In a well-intentioned attempt to give each process a unique seed, the developers ended up virtually assuring that many of the processes would have exactly the same seed.

If you parallelize a huge simulation by spreading out multiple copies, but two of the processes use the same seed, then their results will be identical. Throwing out the redundant simulation would reduce your number of samples, but not noticing and keeping the redundant output would be worse because it would cause you to underestimate the amount of variation.

To avoid duplicate seeds, the developers used a random number generator to assign the RNG seeds for each process. Sounds reasonable. Randomly assigned RNG seeds give you even more random goodness. Except they don’t.

The developers had run into a variation on the famous birthday problem. In a room of 23 people, there’s a 50% chance that two people share the same birthday. And with 50 people, the chances go up to 97%. It’s not certain that two people will have the same birthday until you have 367 people in the room, but the chances approach 1 faster than you might think.

Applying the analog of the birthday problem to the RNG seeds explains why the project was launching processes with the same seed. Suppose you seed each process with an unsigned 16-bit integer. That means there are 65,536 possible seeds. Now suppose you launch 1,000 processes. With 65 times as many possible seeds as processes, surely every process should get its own seed, right? Not at all. There’s a 99.95% chance that two processes will have the same seed.

In this case it would have been better to seed each process with sequential seeds: give the first process seed 1, the second seed 2, etc. The seeds don’t have to be random; they just have to be unique. If you’re using a good random number generator, the outputs of 1,000 processes seeded with 1, 2, 3, …, 1000 will be independent.

## 8 thoughts on “Random number generator seed mistakes”

1. Better to seed with lava lamp or radiation.

2. Andy MacNamara

In looking into this in the past, I came across the research of L’Ecuyer for parallel-stream PRNGs… e.g., http://www.iro.umontreal.ca/~lecuyer/myftp/papers/streams00.pdf and later work. Do you have any thoughts on this or on other things to look into for parallel PRNG streams for Monte Carlo analysis?

3. I’m with Andy – better to use radiation for your seeds.

Regarding seeding with 1, 2, 3, … my understanding is that small seeds are generally bad because they don’t have enough information. My experience is primarily with Mersenne Twister implementations. I don’t have any hard evidence 20 minutes after waking up on a Saturday morning but my recollection was the sequences generated with seeds containing mostly zero bits were inferior to those seeded with random 32 bit numbers.

There is also a problem with virtual servers.

If you start a server and suspend it after it has seeded the RNG, and always use that as the starting point for new instances, they will of course all have their RNG in the same state.

5. Vic, random seeds sound reasonable, but that’s unnecessary at best. And its a bad idea for the reasons given in the post: you’re likely to run into the birthday problem.

A high quality random number generator should produce uncorrelated output for any two different seeds, including consecutive seeds, and as far as anyone knows, the Mersenne Twister does this.

I tested this by creating a sequence of 10,000 random numbers, using seeds 1, 2, 3, … and then doing a Pearson correlation test. The p-values are uniformly distributed, just as they should be. You could see this by eye, and you can run a Kolmogorov-Smirnov goodness of fit test.

In this plot, the lower line, the blue dots, are p-values for vectors created with consecutive seeds. The upper line, the green dots, show the p-values for generating all the vectors consecutively without resetting the seed. The K-S test statistic for this picture is 0.074, with a p-value of 0.966. This means the K-S test believes the p-values are quite uniformly distributed.

This picture was created using only 10 vectors (45 correlations) so you could see individual dots. The small number of vectors explains why there are gaps in both lines. That’s to be expected. With more vectors, the gaps fill in as you’d expect, but this also means the plots become solid green and blue lines and so hard to understand. With more vectors, you can rely on the K-S test instead, and shows that the vectors created from consecutive seeds are independent, i.e. the p-values for the correlation tests are uniformly distributed.

6. Brad: Isn’t that what you’d want and expect? If you restart a virtual machine, you’d expect it to have it’s previous state. And firing up multiple copies of a machine with the same state should produce the same output.