From: Matt Mahoney (matmahoney@yahoo.com)
Date: Mon May 18 2009 - 19:23:07 MDT
--- On Mon, 5/18/09, Peter de Blanc <peter@spaceandgames.com> wrote:
> Matt Mahoney wrote:
> > I used the AIXI model to make the question precise. I
> defined an agent that believes it is immortal as one that
> has the goal of maximizing accumulated reward r(t) for t
> from now to infinity. If instead an agent believes it will
> die at time T, then it would rationally have the goal of
> maximizing accumulated reward summed from t to T. For a
> known environment, it would behave the same regardless of
> r(t) for all t > T.
>
> Matt, I think the basic problem here is using an unbounded
> utility function. Since you'll get infinite expected
> utilities, there's no way to choose between actions. You'd
> have this problem whether or not you believe in immortality,
> as long as your utility function is unbounded.
That's true. But if you choose finitely bounded utility functions, you can't make the distinction either because SUM(t=0..infinity, r(t)) is defined to be lim(T->infinity) SUM(t=0..T, r(t)). If the limit exists, then an immortality-believing agent will always make the same choice as an agent that expects to die after time T if T is sufficiently large.
Thus, any test for immortality belief would have to involve choices among utilities where the limit does not exist. These need not be infinite, for example, r(t) = cos(wt) where the agent may choose any w such that 0 < w < 2*pi. An immortality-believing agent will not have a preference, but a mortality-believing agent will prefer values very close to 0. However, this test fails because our original assumption is that the agent knows the environment and is able to compute the optimal choice. For the mortality-believing agent, the optimal choice is the smallest positive real number, which it cannot compute.
We could restrict the choices of w to a large but finite set. But in this case, for any value of w chosen by an immortality-believing agent, I believe it would be possible to describe a mortality-believing agent that would make the same choice.
-- Matt Mahoney, matmahoney@yahoo.com
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