# DRMacIver's Notebook

A Boltzmann Agent with Very Bad Judgement

A Boltzmann Agent with Very Bad Judgement

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 $P_1 \implies \ldots \implies P_n$. 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 $B(x) = 1 + \ldots + x^n$. 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 $x \approx 2$, which leds to the expected number of propositions believed being $\approx 9$. 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.