Or: Formalizing Timeless Decision Theory

Previously:

0. Decision Theories: A Less Wrong Primer
1. The Problem with Naive Decision Theory
2. Causal Decision Theory and Substitution

WARNING: The main result of this post, as it's written here, is flawed. I at first thought it was a fatal flaw, but later found a fix. I'm going to try and repair this post, either by including the tricky bits, or by handwaving and pointing you to the actual proofs if you're curious. Carry on!

Summary of Post: Have you ever wanted to know how (and whether) Timeless Decision Theory works? Using the framework from the last two posts, this post shows you explicitly how TDT can be implemented in the context of our tournament, what it does, how it strictly beats CDT on fair problems, and a bit about why this is a Big Deal. But you're seriously going to want to read the previous posts in the sequence before this one.

We've reached the frontier of decision theories, and we're ready at last to write algorithms that achieve mutual cooperation in Prisoner's Dilemma (without risk of being defected on, and without giving up the ability to defect against players who always cooperate)! After two substantial preparatory posts, it feels like it's been a long time, hasn't it?

But look at me, here, talking when there's Science to do...

Or: Causal Decision Theory and Substitution

Previously:

0. Decision Theories: A Less Wrong Primer
1. The Problem with Naive Decision Theory

Summary of Post: We explore the role of substitution in avoiding spurious counterfactuals, introduce an implementation of Causal Decision Theory and a CliqueBot, and set off in the direction of Timeless Decision Theory.

In the last post, we showed the problem with what we termed Naive Decision Theory, which attempts to prove counterfactuals directly and pick the best action: there's a possibility of spurious counterfactuals which lead to terrible decisions. We'll want to implement a decision theory that does better; one that is, by any practical definition of the words, foolproof and incapable of error...

I know you're eager to get to Timeless Decision Theory and the others. I'm sorry, but I'm afraid I can't do that just yet. This background is too important for me to allow you to skip it...

Over the next few posts, we'll create a sequence of decision theories, each of which will outperform the previous ones (the new ones will do better in some games, without doing worse in others⁰) in a wide range of plausible games.

Or: The Problem with Naive Decision Theory

Previously: Decision Theories: A Less Wrong Primer

Summary of Sequence: In the context of a tournament for computer programs, I give almost-explicit versions of causal, timeless, ambient, updateless, and several other decision theories. I explain the mathematical considerations that make decision theories tricky in general, and end with a bunch of links to the relevant recent research. This sequence is heavier on the math than the primer was, but is meant to be accessible to a fairly general audience. Understanding the basics of game theory (and Nash equilibria) will be essential. Knowing about things like Gödel numbering, quining and Löb's Theorem will help, but won't be required.

Summary of Post: I introduce a context in which we can avoid most of the usual tricky philosophical problems and formalize the decision theories of interest. Then I show the chief issue with what might be called "naive decision theory": the problem of spurious counterfactual reasoning. In future posts, we'll see how other decision theories get around that problem.

In my Decision Theory Primer, I gave an intuitive explanation of decision theories; now I'd like to give a technical explanation. The main difficulty is that in the real world, there are all sorts of complications that are extraneous to the core of decision theory. (I'll mention more of these in the last post, but an obvious one is that we can't be sure that our perception and memory match reality.)

In order to avoid such difficulties, I'll need to demonstrate decision theory in a completely artificial setting: a tournament among computer programs.