How do I randomly equalize unequal values?

Question

Say I have multiple unequal values a, b, c, d, e. Is it possible to turn these unequal values into equal values just by using random number generation?

Example: a=100, b=140, c=200, d=2, e=1000. I want the algorithm to randomly target these sets such that the largest value is targeted most often and the smallest value is left alone for the most parts.

Areas where I've run into problems: if I just use non-unique random number generation, then value e will end up going under the other values. If I use unique number generation, then the ration between the values doesn't change even if their absolute values do. I've tried using sets where a certain range of numbers have to be hit a certain number of times before the value changes. I haven't tried using a mix of unique/non-unique random numbers yet.

I want the ratio between the values to gradually approach 1 as the algorithm runs.

Another way to think about the problem: say these values a, b, c, d, e, are all equal. If we randomly choose one, each is as likely to be chosen as any other. After we choose one, we add 1 to that value. Then we run this process again. This time, the value that was picked last time is 1-larger than any other value so it's more likely to be picked than any one other value. This creates a snowball effect where the value picked first is likely to keep getting picked and achieve runaway growth. I'm looking for the opposite of this algorithm where we start after these originally-equal values have diverged and we bring them back to the originally-equal state.

I think this process is impossible because of entropy and the inherent one-way nature of existence.

Answer 1

Well, there is a technique called Inverse Weights, where you sample items inverse proportional to their previous appearance. Each time we sample a, b, c, d or e, we update their appearance numbers and recalculate probabilities. Simple python code, I sample numbers [0...4] as a, b, c, d, e and start with what you listed as appearances. After 100,000 samples they looks to be equidistributed

import numpy as np

n = np.array([100, 140, 200, 2, 1000])

for k in range(1, 100000):

    p  = (1.0 / n) # make probabilities inverse to weights
    p /= np.sum(p) # normalization

    a = np.random.choice(5, p = p) # sampling numbers in the range [0...5)

    n[a] += 1 # update weights

print(n)

Output

[20260 20194 20290 20305 20392]