Well, I’m no mathematician, but I was thinking of something like ordinal arithmetic.
If I understand it correctly, this would let us express value-systems such as —
Both snuggles and chocolate bars have positive utility, but I’d always rather have another snuggle than any number of chocolate bars. So we could say U(snuggle) = ω and U(chocolate bar) = 1. For any amount of snuggling, I’d prefer to have that amount and a chocolate bar (ω·n+1 > ω·n), but given the choice between more snuggling and more chocolate bars I’ll always pick the former, no matter how much the quantities are (ω·(n+1) > ω·n+c, for any c). A minute of snuggling is better than all the chocolate bars in the world.
This also lets us say that paperclips do have nonzero value, but there is no amount of paperclips that is as valuable as the survival of humanity. If we program this into an AI, it will know that it can’t maximize value by maximizing paperclips, even if it’s much easier to produce a lot of paperclips than to save humanity.
Edited to add: This might even let us shoehorn deontological rules into a utility-based system. To give an obviously simplified example, consider Asimov’s Three Laws of Robotics, which come with explicit rank ordering: the First Law is supposed to always trump the Second, which is supposed to always trump the third. There’s not supposed to be any amount of Second Law value (obedience to humans) that can be greater than First Law value (protecting humans).
The problem with using hyperreals for utility is that unless you also use them for probabilities only the most infinite utilities actually affect your decision.
To use your example if U(snuggle) = ω and U(chocolate bar) = 1. Then you might as well say that U(snuggle) = 1 and U(chocolate bar) = 0 since tiny probabilities of getting a snuggle will always override any considerations related to chocolate bars.
Well, I’m no mathematician, but I was thinking of something like ordinal arithmetic.
If I understand it correctly, this would let us express value-systems such as —
Both snuggles and chocolate bars have positive utility, but I’d always rather have another snuggle than any number of chocolate bars. So we could say U(snuggle) = ω and U(chocolate bar) = 1. For any amount of snuggling, I’d prefer to have that amount and a chocolate bar (ω·n+1 > ω·n), but given the choice between more snuggling and more chocolate bars I’ll always pick the former, no matter how much the quantities are (ω·(n+1) > ω·n+c, for any c). A minute of snuggling is better than all the chocolate bars in the world.
This also lets us say that paperclips do have nonzero value, but there is no amount of paperclips that is as valuable as the survival of humanity. If we program this into an AI, it will know that it can’t maximize value by maximizing paperclips, even if it’s much easier to produce a lot of paperclips than to save humanity.
Edited to add: This might even let us shoehorn deontological rules into a utility-based system. To give an obviously simplified example, consider Asimov’s Three Laws of Robotics, which come with explicit rank ordering: the First Law is supposed to always trump the Second, which is supposed to always trump the third. There’s not supposed to be any amount of Second Law value (obedience to humans) that can be greater than First Law value (protecting humans).
The problem with using hyperreals for utility is that unless you also use them for probabilities only the most infinite utilities actually affect your decision.
To use your example if U(snuggle) = ω and U(chocolate bar) = 1. Then you might as well say that U(snuggle) = 1 and U(chocolate bar) = 0 since tiny probabilities of getting a snuggle will always override any considerations related to chocolate bars.