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Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: [Request for Distillation] Coherence of Distributed Decisions With Different Inputs Implies Conditioning, published by johnswentworth on April 25, 2022 on The AI Alignment Forum. 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. 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) u1.um. Pareto optimality implies that, at the pareto-optimum, there exists some vector of positive reals P1.Pm such that the choices maximize ∑iPiui. 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 P1 to be 1 where u1 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) ∑XP[X]u(A,X)=EX[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=A1.An in a distributed fashion. Each decision is made with only limited information: Ai receives fi(X) as input (and nothing else). The system then chooses the functions Ai(fi(X)) to maximize EX[u(A,X)]. Consider the maximization problem for just Ai(f∗i), i.e. the optimal action for choice i given input f∗i. Expanded out, the objective is EX[u(A,X)]=∑Xu(A1(f1(X)),.,Ai(fi(X)),.An(fn(X)),X). Note that the only terms in that sum which actually depend on Ai(f∗i) are those for which fi(X)=f∗i. So, for purposes of choosing Ai(f∗i) specifically, we can reduce the objective to ∑X:fi(X)=f∗iu(A,X) . which is equal to P[fi(X)=f∗i]E[u(A,X)|fi(X)=f∗i]. The P[fi(X)=f∗i] multiplier is always positive and does not depend on Ai, so we can drop it without changing the optimal Ai. Thus, action Ai(f∗i) maximizes the conditional expected value E[u(A,X)|fi(X)=f∗i]. Returning to the optimization problem for all of the actions simultaneously: any optimum for all actions must also be an optimum for each action individually (otherwise we could change one action to get a better result), so each action Ai(f∗i) must maximize E[u(A,X)|fi(X)=f∗i]. A few notes on this: We’ve implicitly assumed that actions do not influence which information is available to other actions (i.e. the actions are “spacelike separated”). That can be relaxed: let fi depend on both X and previous actions A