From: Matt Mahoney (email@example.com)
Date: Wed May 20 2009 - 09:50:38 MDT
--- On Wed, 5/20/09, Stuart Armstrong <firstname.lastname@example.org> wrote:
> > But even if it were, a
> satisfying solution may not be possible. Benji's solution
> distinguishes between agents that are certain of their
> mortality and agents that are certain of their immortality.
> The latter will optimize utility over an infinite time
> window, but so would an agent that was uncertain of its
> immortality. Both would chose option A ($1 per day forever)
> over the option of a single, finite payout of the agent's
> choosing. Since their behaviors are identical, we can
> conclude that there is no difference between certain and
> uncertain belief in immortality, unless there is another
> test to distinguish them.
> Replace Benji's set-up with a system with finite pay-off:
> Option A gives you (1/2)^n each day forever, where n is the day.
> Option B gives you the option of claiming, on any day n, the payoff of
> option A from say 0 up to day 2n, say. (this gives a pay off of 2 -
> Then use Dutch book type approaches to determine what relative price
> the agent would put on option A and option B.
We wish to test two hypotheses about an agent.
H1: the agent believes itself to be immortal. It puts equal value on A and B.
H2: the agent believes that with probability 1/G that it will die at time G, otherwise it is immortal, where G is Graham's number. The agent places a very slightly higher value on option B.
You offer A an B at equal prices and the agent chooses B. You raise the price of B slightly and the agent chooses A. Now you do not know which hypothesis is true. To find out, you have to find a price for B that is higher than the price for A, but the agent still chooses B. That could take an arbitrarily long time because you could replace G with a number arbitrarily large.
-- Matt Mahoney, email@example.com
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