I think this almost works. Suppose the AI constructs 7 paperclips 50% of the time, and 8 paperclips 50% of the time (shutting down after producing the last paperclip). This means the button is pushed 50% of the time after step 8, and never pushed 50% of the time. Given this distribution of button pushes, what’s the best-response policy?
I think the best-response policy is to make 8 paperclips, then see if the shutdown button has been pressed; if it has been pressed, then shut down, else make 2 more paperclips. When the button is not pressed, this makes 10 paperclips; when the button is pressed, this shuts down for 2 steps. Thus the expected utility is 6, which is better than the original 5. So I think it’ll be more difficult to find an equilibrium policy; the uncertainty about when the shutdown button is pressed must be such that the agent is indifferent between making a paperclip and shutting down on step 8.
First, let’s list the optimal policies. Pressing or destroying the button remain suboptimal policies. Once the button is seen to be pressed, then anything but shutdown is suboptmial. Therefore there are 11 potentially optimal policies, labelled by n=0 to n=10. These involve producing n paperclips then shutting down for 10−n turns, unless it sees the shutdown button has been pressed, in which case it shuts down (this is only relevant for n=9,10).
Let p be the total shutdown probability. Note that for all the optimal policies, either the shutdown button is never pressed, or it is pressed on turn 8, so we can ignore the possibilities of it being pressed later.
The utility gain for n≤8 is n(1−p)+(10−n)p=10p+(1−2p)n. For n=9 it is that quantity, plus p (it shuts down one turn earlier if the shutdown button is pressed on turn 8, and gains one extra utility for it). For n=10 it is that quantity, plus 2p).
For p≤0.5, the optimal policy is n=10, which ensures p=1, hence contradiction.
For p>0.5, the only possible optimal policies are n=0 (utility 10p), n=9 (utility 9−7p), and n=10 (utility 10−8p). Subtracting the last two gives 1−p>0, hence n=9 is never optimal.
The agent is indifferent between n=0 and n=10 for 10p=10−8p ie p=5/9≈0.5555….
Therefore the result should be that the agent pursues the mixed policy 5/9(n=10)+4/9(n=0), with probability of shutdown being equal to 5/9, and pseudo expected utility 50/9 (real expected utility 10/9).
I think this almost works. Suppose the AI constructs 7 paperclips 50% of the time, and 8 paperclips 50% of the time (shutting down after producing the last paperclip). This means the button is pushed 50% of the time after step 8, and never pushed 50% of the time. Given this distribution of button pushes, what’s the best-response policy?
I think the best-response policy is to make 8 paperclips, then see if the shutdown button has been pressed; if it has been pressed, then shut down, else make 2 more paperclips. When the button is not pressed, this makes 10 paperclips; when the button is pressed, this shuts down for 2 steps. Thus the expected utility is 6, which is better than the original 5. So I think it’ll be more difficult to find an equilibrium policy; the uncertainty about when the shutdown button is pressed must be such that the agent is indifferent between making a paperclip and shutting down on step 8.
You’re right, let’s be more careful.
First, let’s list the optimal policies. Pressing or destroying the button remain suboptimal policies. Once the button is seen to be pressed, then anything but shutdown is suboptmial. Therefore there are 11 potentially optimal policies, labelled by n=0 to n=10. These involve producing n paperclips then shutting down for 10−n turns, unless it sees the shutdown button has been pressed, in which case it shuts down (this is only relevant for n=9,10).
Let p be the total shutdown probability. Note that for all the optimal policies, either the shutdown button is never pressed, or it is pressed on turn 8, so we can ignore the possibilities of it being pressed later.
The utility gain for n≤8 is n(1−p)+(10−n)p=10p+(1−2p)n. For n=9 it is that quantity, plus p (it shuts down one turn earlier if the shutdown button is pressed on turn 8, and gains one extra utility for it). For n=10 it is that quantity, plus 2p).
For p≤0.5, the optimal policy is n=10, which ensures p=1, hence contradiction.
For p>0.5, the only possible optimal policies are n=0 (utility 10p), n=9 (utility 9−7p), and n=10 (utility 10−8p). Subtracting the last two gives 1−p>0, hence n=9 is never optimal.
The agent is indifferent between n=0 and n=10 for 10p=10−8p ie p=5/9≈0.5555….
Therefore the result should be that the agent pursues the mixed policy 5/9(n=10)+4/9(n=0), with probability of shutdown being equal to 5/9, and pseudo expected utility 50/9 (real expected utility 10/9).
Have I got this right?
This looks correct; thanks for doing this analysis!