Difference-making risk aversion (the accounts RP has considered, other than rounding/discounting) doesn’t necessarily avoid generalizations of (2), the 50-50 problem. It can
just shift the 50-50 problem to a different place, e.g. 70% good vs 30% bad being neutral in expectation but 70.0001% being extremely good in expectation, or
still have the 50-50 problem, but with unequal payoffs for good and bad, so be neutral at 50-50, but 50.0001% being extremely good in expectation.
To avoid these more general problems within standard difference-making accounts, I think you’d need to bound the differences you make from above. For example, apply a function that’s bounded above to the difference, or assume differences in value are bounded above).
On the other hand, maybe having the problem at 50-50 with equal magnitude but opposite sign payoffs is much worse, because our uninformed prior for the value of a random action is generally going to be symmetric around 0 net value.
Proofs below.
Assume you have an action with positive payoff x (compared to doing nothing) with probability p=50.0001%, and negative payoff y=-x otherwise, with x very large. Then
Holding the conditional payoffs x and -x constant, but changing the probabilities at 100% x and 0% y=-x, the act would be good overall. OTOH, it’s bad at 0% x and 100% y=-x. By Continuity (or the Intermediate Value Theorem), there has to be some p so that the act that’s x with probability p and y=-x with probability 1-p is neutral in expectation. Then we get the same problem at p, and a small probability like 0.0001% over p instead of p can make the action extremely good in expectation, if x was chosen to be large enough.
Holding the probability p=50% constant, if the negative payoff y were actually 0, and the positive payoff still x and large, the act would be good overall. It’s bad for y<0 low enough.[1] Then, by the Intermediate Value Theorem, there’s some y so that the act that’s x with probability 50% and y with probability 50% is neutral in expectation. And again, 50.0001% x and otherwise y can be extremely good in expectation, if x was chosen to be large enough.
Each can be avoided if the adjusted value of x is bounded and the bound is low enough, or x itself is bounded above with a low enough bound.
I think the same would apply to difference-making ambiguity aversion, too.
y=-x if difference-making risk averse, any y< -x if difference-making risk neutral, and generally for some y<0 if the disvalue of net harm isn’t bounded and the function is continuous.
Difference-making risk aversion (the accounts RP has considered, other than rounding/discounting) doesn’t necessarily avoid generalizations of (2), the 50-50 problem. It can
just shift the 50-50 problem to a different place, e.g. 70% good vs 30% bad being neutral in expectation but 70.0001% being extremely good in expectation, or
still have the 50-50 problem, but with unequal payoffs for good and bad, so be neutral at 50-50, but 50.0001% being extremely good in expectation.
To avoid these more general problems within standard difference-making accounts, I think you’d need to bound the differences you make from above. For example, apply a function that’s bounded above to the difference, or assume differences in value are bounded above).
On the other hand, maybe having the problem at 50-50 with equal magnitude but opposite sign payoffs is much worse, because our uninformed prior for the value of a random action is generally going to be symmetric around 0 net value.
Proofs below.
Assume you have an action with positive payoff x (compared to doing nothing) with probability p=50.0001%, and negative payoff y=-x otherwise, with x very large. Then
Holding the conditional payoffs x and -x constant, but changing the probabilities at 100% x and 0% y=-x, the act would be good overall. OTOH, it’s bad at 0% x and 100% y=-x. By Continuity (or the Intermediate Value Theorem), there has to be some p so that the act that’s x with probability p and y=-x with probability 1-p is neutral in expectation. Then we get the same problem at p, and a small probability like 0.0001% over p instead of p can make the action extremely good in expectation, if x was chosen to be large enough.
Holding the probability p=50% constant, if the negative payoff y were actually 0, and the positive payoff still x and large, the act would be good overall. It’s bad for y<0 low enough.[1] Then, by the Intermediate Value Theorem, there’s some y so that the act that’s x with probability 50% and y with probability 50% is neutral in expectation. And again, 50.0001% x and otherwise y can be extremely good in expectation, if x was chosen to be large enough.
Each can be avoided if the adjusted value of x is bounded and the bound is low enough, or x itself is bounded above with a low enough bound.
I think the same would apply to difference-making ambiguity aversion, too.
y=-x if difference-making risk averse, any y< -x if difference-making risk neutral, and generally for some y<0 if the disvalue of net harm isn’t bounded and the function is continuous.