David R. MacIver's Blog

A Boltzmann Agent with Very Bad Judgement

This post was originally published at notebook.drmaciver.com.

As per previous post, it can make sense when looking at a set of consistent propositions to consider agents as Boltzmann samplers over the set of valid consistent beliefs, with their reliability measured by the expected number of true beliefs.

A thing I hadn’t previously realised is that this can cause an agent that is on average reliable to be reliably wrong for some propositions.

Consider a chain of propositions of the form (P_1 P_n). There are exactly (n + 1) possible consistent beliefs for this sampler (each defined by the first (P_i) that the agent believes), so the Boltzmann generating function is (B(x) = 1 + + x^n). Suppose (n = 10). Some simple maths (by which I mean I used sympy) shows that this agent ends up believing (P_1) with probability at least half only when (x ), which leds to the expected number of propositions believed being (). So in order to achieve (50%) reliability on the base proposition we have to achieve (90%) overall reliability!

This isn’t very surprising in some sense, but probably puts a bound on how good we can expect judgement aggregation to be in this case.