On decision-prediction fixed points

by jollybard1 min read4th Dec 201910 comments

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Fixed Point Theorems
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It seems like for embedded (reflexive, Löbian, etc) LDT agents, there ought to be a fixed point thing between decision and prediction.

Indeed, embedded agents can predict things about their own actions; but by modeling themselves sufficiently well, this should be (in the limit) equivalent to making a decision, as they will be modeling their own thoughts. Conversely, once you have decided, if you do not suffer from akrasia, then you have accurately predicted your next action. (aside: this is the source of the illusion of free will.)

This is related to the class of "metaphysical truths": truths of the form . Whenever an embedded agent believes one of those, then it must (by Löb's theorem) eventually believe . But there are lots of such truths (perhaps each different religion offers a different set of metaphysical truths), which might then lead to spurious, or even contradictory beliefs!

The key word was "eventually", assuming LDT agents are logical inductors of some kind; in the meantime, the agent may choose its beliefs. Isn't this weird? Beliefs shouldn't be arbitrary!

But you can imagine, as an (imperfect) example, the paradox of self-confidence: if you think you are competent, then you could believe in your ability to self-improve, which will encourage your to foster your own competence; on the other hand thinking that you are incompetent may lead to not believing in your self-improvement ability, leading to a downward spiral.

Each one of these are decision-belief fixed points. Each are, in way (causally?), both true and rational.

I feel like LDT will end up being a reflexive fixed point of this sort (reminiscent of the logical induction fixed point), with the catch that there are many such fixed points. The true decision an LDT agent must make is then choosing the most effective of these fixed points.

(I'm not entirely convined of this yet since I still have no idea what logical counterfactuals will look like)

The moral of the story for us humans is that:

  • akrasia should not exist, not if you can predict yourself well enough;
  • sometimes beliefs are arbitrary. choose the most productive ones, you'll end up believing them all anyway.

EDIT: I feel this is rather important, so I would appreciate getting feedback on the presentation.

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Someone who knows exactly what they will do can still suffer from akrasia, by wishing they would do something else. I'd say that if the model of yourself saying "I'll do whatever I wish I would" beats every other model you try and build of yourself, that looks like free will. The other was around, you can observe akrasia.

I don't think that's right. If you know exactly what you are going to do, that leaves no room for counterfactuals, not if you're an LDT agent. Physically, there is no such thing as a counterfactual, especially not a logical one; so if your beliefs match the physical world perfectly, then the world looks deterministic, including your own behavior. I don't think counterfactual reasoning makes sense without uncertainty.

As a human who has an intuitive understanding of counterfactuals, if I know exactly what a tic tac toe or chess program would do, I can still ask what would happen if it chose a particular action instead. The same goes if the agent of interest is myself.

I see what you mean, but

if I know exactly what a tic tac toe or chess program would do,

if you were this logically omniscient, then supposing that the program did something else would imply that your system is inconsistent, which means everything is provable.

There needs to be boundedness somewhere, either in the number of deductions you can make, or in the certainty of your logical beliefs. This is what I mean by uncertainty being necessary for logical counterfactuals.

Akrasia is the name we give the fact that the part of ourselves that communicates about X, and the part that actually does X have slightly different goals. The communicating part is always winging about how the other part is being lazy.

Perhaps, but that's not quite how I see it. I'm saying akrasia is failure to predict yourself, that is when there's a disconnect between your predictions and your actions.

I'm modeling humans as two agents that share a skull. One of those agents wants to do stuff and writes blog posts, the other likes lying in bed and has at least partial control of your actions. The part of you that does the talking can really say that it wants to do X, but it isn't in control.

Even if you can predict this whole thing, that still doesn't stop it happening.

Maybe I ought to give a slightly more practical description.

Your akrasia is part of the world and failing to navigate around it is epistemic failure.

Right, so that's not a decision-prediction fixed point; a correct LDT algorithm would, by its very definition, choose the optimal decision, so predicting its behavior would lead to the optimal decision.

Donald Hobson appears to believe that determinism implies you do not have a choice.

Instead of a) Beliefs -> Reality, it's b) Reality -> Beliefs. B can be broken or fixed, but fixing A...

a correct LDT algorithm would

How does a correct LDT algorithm turn 2 agents into 1?