Seth Roberts is dead .

I was considering the Shangri-La diet, but now I'm nervous.

Wow, this is great work--congratulations! If it pans out, it bridges a really fundamental gap.

I'm still digesting the idea, and perhaps I'm jumping the gun here, but I'm trying to envision a UDT (or TDT) agent using the sense of subjective probability you define. It seems to me that an agent can get into trouble even if its subjective probability meets the coherence criterion. If that's right, some additional criterion would have to be required. (Maybe that's what you already intend? Or maybe the following is just muddled.)

Let's try invoking a coherent P in the case of a simple decision problem for a UDT agent. First, define G <--> P("G") < 0.1. Then consider the 5&10 problem:

• If the agent chooses A, payoff is 10 if ~G, 0 if G.

• If the agent chooses B, payoff is 5.

And suppose the agent can prove the foregoing. Then unless I'm mistaken, there's a coherent P with the following assignments:

P(G) = 0.1

P(Agent()=A) = 0

P(Agent()=B) = 1

P(G | Agent()=B) = P(G) = 0.1

And P assigns 1 to each of the following:

P("Agent()=A") < epsilon

P("Agent()=B") > 1-epsilon

P("G & Agent()=B") / P("Agent()=B") = 0.1 +- epsilon

P("G & Agent()=A") / P("Agent()=A") > 0.5

The last inequality is consistent with the agent indeed choosing B, because the postulated conditional probability of G makes the expected payoff given A less than the payoff given B.

Is that P actually incoherent for reasons I'm overlooking? If not, then we'd need something beyond coherence to tell us which P a UDT agent should use, correct?

(edit: formatting)

Wow, this is great work--congratulations! If it pans out, it bridges a really fundamental gap.

I'm still digesting the idea, and perhaps I'm jumping the gun here, but I'm trying to envision a UDT (or TDT) agent using the sense of subjective probability you define. It seems to me that an agent can get into trouble even if its subjective probability meets the coherence criterion. If that's right, some additional criterion would have to be required. (Maybe that's what you already intend? Or maybe the following is just muddled.)

Let's try invoking a coherent P in the case of a simple decision problem for a UDT agent. First, define G <--> P("G") < 0.1. Then consider the 5&10 problem:

• If the agent chooses A, payoff is 10 if ~G, 0 if G.

• If the agent chooses B, payoff is 5.

And suppose the agent can prove the foregoing. Then unless I'm mistaken, there's a coherent P with the following assignments:

P(G) = 0.1

P(Agent()=A) = 0

P(Agent()=B) = 1

P(G | Agent()=B) = P(G) = 0.1

And P assigns 1 to each of the following:

P("Agent()=A") < epsilon

P("Agent()=B") > 1-epsilon

P("G & Agent()=B") / P("Agent()=B") = 0.1 +- epsilon

P("G & Agent()=A") / P("Agent()=A") > 0.5

The last inequality is consistent with the agent indeed choosing B, because the postulated conditional probability of G makes the expected payoff given A less than the payoff given B.

Is that P actually incoherent for reasons I'm overlooking? If not, then we'd need something beyond coherence to tell us which P a UDT agent should use, correct?

(edit: formatting)

Thanks!

In April 2010 Gary Drescher proposed the "Agent simulates predictor" problem, or ASP, that shows how agents with lots of computational power sometimes fare worse than agents with limited resources.

Just to give due credit: Wei Dai and others had already discussed Prisoner's Dilemma scenarios that exhibit a similar problem, which I then distilled into the ASP problem.

Interesting. This would seem to return it to the class of decision-determined problems, and for an illuminating reason - the algorithm is only run with one set of information - just like how in Newcomb's problem the algorithm has only one set of information no matter the contents of the boxes.

This way of thinking makes Vladimir's position more intuitive. To put words in his mouth, instead of being not decision determined, the "unfixed" version is merely two-decision determined, and then left undefined for half the bloody problem.

By "tit for tat" I am referring to the notable strategy in the iterated prisoner's dilemma. Agents using this strategy will keep cooperating as long as the other person cooperates, but if the other person defects then they will defect too. It's an excellent strategy by many measures, beating out more complicated strategies, and we probably have something like it built into our heads.

By analogy, a "tit for tat" strategy in Newcomb's problem with transparent boxes would be to one-box if the Predictor "cooperates," and two-box if the Predictor "defects."

But what does the Predictor see when it looks into the future of an agent with this strategy? Either way it chooses, it will have chosen correctly, so the Predictor needs some other, non-decision-determined criterion to decide.

Alternately you could think of it as making the decision-type of the agent undefined (at the time the Predictor is filling the boxes), thus making it impossible for the problem to have any well-defined decision-determined statement.

1) 2TDT-1CDT.

How is this not resolved? (My comment and the following Eliezer's comment; I didn't re-read the rest of the discussion.)

2) "Agent simulates predictor"

This basically says that the predictor is a rock, doesn't depend on agent's decision, which makes the agent lose because of the way problem statement argues into stipulating (outside of predictor's own decision process) that this must be a two-boxing rock rather than a one-boxing rock.

3) "The stupid winner paradox"

Same as (2). We stipulate the weak player to be a $9 rock. Nothing to be surprised about.

4) "A/B/~CON"

Requires ability to reason under logical uncertainty, comparing theories of consequences and not just specific possible utilities following from specific possible actions. Under any reasonable axioms for valuation of sets of consequences, action B wins.

5) The general case of agents playing a non-zero-sum game against each other, knowing each other's source code.

Without good understanding of reasoning under logical uncertainty, this one remains out.

If there are multiple translations, then either the translations are all mathematically equivalent, in the sense that they agree on the output for every combination of inputs, or the problem is underspecified. (This seems like it ought to be the definition for the word underspecified. It's also worth noting that all game-theory problems are underspecified in this sense, since they contain an opponent you know little about.)

Now, if two world programs were mathematically equivalent but a decision theory gave them different answers, then that would be a serious problem with the decision theory. And this does, in fact, happen with some decision theories; in particular, it happens to theories that work by trying to decompose the world program into parts, when those parts are related in a way that the decision theory doesn't know how to handle. If you treat the world-program as an opaque object, though, then all mathematically equivalent formulations of it should give the same answer.

It seems to me that the world-program is part of the problem description, not the analysis. It's equally tricky whether it's given in English or in a computer program; Wei Dai just translated it faithfully, preserving the strange properties it had to begin with.