There’s been a lot of response to the Call For Distillers, so I’m experimenting with a new post format. This post is relatively short and contains only a simple mathematical argument, with none of the examples, motivation, more examples, or context which would normally make such a post readable. My hope is that someone else will write a more understandable version.

Jacob is offering a $500 bounty on a distillation.

Goal: following the usual coherence argument setup, show that if multiple decisions are each made with different input information available, then each decision maximizes expected utility given its input information.

We’ll start with the usual coherence argument setup: a system makes a bunch of choices, aiming to be pareto-optimal across a bunch of goals (e.g. amounts of various resources) $u_{1} \dots u_{m}$ . Pareto optimality implies that, at the pareto-optimum, there exists some vector of positive reals $P_{1} \dots P_{m}$ such that the choices maximize $\sum_{i} P_{i} u_{i}$ . Note that $P$ can be freely multiplied by a constant, so without loss of generality we could either take $P$ to sum to $1$ (in which case we might think of $P$ as probabilities) or take $P_{1}$ to be $1$ where $u_{1}$ is amount of money (in which case $P$ is a marginal price vector).

When the goals are all “the same goal” across different “worlds” $X$ , and we normalize $P [X]$ to sum to $1$ , $P [X]$ is a probability distribution over worlds in the usual Bayesian sense. The system then maximizes (over its actions $A$ ) $\sum_{X} P [X] u (A, X) = E_{X} [u (A, X)]$ , i.e. it’s an “expected utility maximizer”.

That’s the usual setup in a nutshell. Now, let’s say that the system makes multiple decisions $A = A_{1} \dots A_{n}$ in a distributed fashion. Each decision is made with only limited information: $A_{i}$ receives $f_{i} (X)$ as input (and nothing else). The system then chooses the functions $A_{i} (f_{i} (X))$ to maximize $E_{X} [u (A, X)]$ .

Consider the maximization problem for just $A_{i} (f_{i}^{*})$ , i.e. the optimal action for choice i given input $f_{i}^{*}$ . Expanded out, the objective is $E_{X} [u (A, X)] = \sum_{X} u (A_{1} (f_{1} (X)), \dots, A_{i} (f_{i} (X)), \dots A_{n} (f_{n} (X)), X)$ .

Note that the only terms in that sum which actually depend on $A_{i} (f_{i}^{*})$ are those for which $f_{i} (X) = f_{i}^{*}$ . So, for purposes of choosing $A_{i} (f_{i}^{*})$ specifically, we can reduce the obje...

Coherence Arguments

Coherence Arguments