Links/Notes: Random walks, processes and human notion of randomness

Notes about random walks, stochastic processses, and stuff.

From IRC: often if some kind of ‘randomness’ is desired in real user applications (e.g. games, music player shuffle), we don’t actually want a true random process (for example, uniformly distributed value). True randomness is counter-intuitive. Humans don’t get randomness, and would consider dice that doesn’t “remember” it’s past states weird or even unfair. (See: Gambler’s fallacy.)

The casinos specialize in taking advantage of people’s poor grasp of “true randomness” and relieving people from their extra cash. But what if we want to generate suitably random outcomes that humans find believable or enjoyable because they correspond to human expections how “random outcomes” look like?

For example, in a game where $A$ happens in the round $n$ with probability $p$ (e.g. enemy player spawning at location $(x,y)$), the chance of $A$ happening in the round $n+1$ should be less than $p$.

(This idea is not originally mine.)

Question. How to model this kind of “randomness”? Has it a name?

Possibly related concepts:

• stochastic process

  • one possible general modelling framework

  • Markov chain

  • Wiener process

• random walk

  • self-interacting random walk

• martingale

  • game / stochastic process, where knowledge of the past doesn’t affect present expectation

  • in particular, $E(X_{n+1} \mid X_1, \dots, X_n) = E(X_n)$

• what if one wants to model the “randomness generating procedure” as a state machine / automata?

  • probabilistic automaton