By all means, straw man away. I won’t take it personally.
However, for an example — and please don’t feel compelled to restrict your answer only to this — there’s a variant of the good old chickens/grandmother example:
Let us say I prefer the nonextinction of chickens to their extinction (that is, I would choose not to murder all chickens, or any chickens, all else being equal). I also prefer my grandmother remaining alive to my grandmother dying. Finally, I prefer the deaths of arbitrary numbers of chickens, taking place with any probability, to any probability of my grandmother dying.
I believe this violates VNM. (Am I wrong there?) At least, I don’t see how one would construe these preferences as those of an agent who is acting to maximize a quantity.
That’s the conclusion of the theorem. Which of the premise do you disagree with?
Those preferences do not violate the VNM axioms. The willingness to kill chickens in order to eliminate an arbitrary small chance of the death of your grandmother makes the preferences a bit weird, but still VNM compliant.
A live chicken might quantum tunnel from China to your grandmother’s house and eat all her heart medication. But then again, a quantum tunneling chicken could save your grandmother from tripping on fallen corn-seed. The net effect of chickens on your grandmother might be unfathomably small, but it is unlikely to ever be zero. If chickens never have zero effect on your grandmother, then your preference for the non-extinction of chickens would never apply *EDIT and so the only preferences we would need to consider would be of your grandmother’s life, which could be represented with utilities (say 0 for dead and 1 for alive).
If you’re willing to tolerate a 1⁄10,000,000 increase in chance of your grandmother’s death to save chickens from extinction, you sill have VNM-rational preferences. Here’s a utility function for that:
dead-chicken-dead-grandma 0
live-chicken-dead-grandma 1
dead-chicken-live-grandma 10,000,000
live-chicken-live-grandma 10,000,001
*Or perhaps not so small. 90% of all flu deaths happen to people age 65 or older, and chickens are the main reservoir of influenza.
If you’re willing to tolerate a 1⁄1,000,000 increase in chance of your grandmother’s death to save chickens from extinction, you sill have VNM-rational preferences.
If, by some cosmic happenstance the effect of a chicken’s life or death affected your grandmother exactly zero—no gravitational effects of the Australian chicken’s beak on your grandmother’s heart, no nothin’—then the utilities get more complicated. If the smallest possible non-zero probability a chicken could kill or save your grandmother were 10^-2,321,832,934,903, then the utilities could be something like:
It seems more likely that he really isn’t VNM-compliant. Chickens are tasty and nutritious, 1⁄1,000,000 is a small number and lets face it, grandparents of adults are already old and have much more chance than that of dying every day. It would be surprising if Said is so perfectly indifferent to chickens, especially since he’s already been explicitly telling us that he isn’t VNM-compliant.
By all means, straw man away. I won’t take it personally.
However, for an example — and please don’t feel compelled to restrict your answer only to this — there’s a variant of the good old chickens/grandmother example:
Let us say I prefer the nonextinction of chickens to their extinction (that is, I would choose not to murder all chickens, or any chickens, all else being equal). I also prefer my grandmother remaining alive to my grandmother dying. Finally, I prefer the deaths of arbitrary numbers of chickens, taking place with any probability, to any probability of my grandmother dying.
I believe this violates VNM. (Am I wrong there?) At least, I don’t see how one would construe these preferences as those of an agent who is acting to maximize a quantity.
See this thread for clarification/correction.
Those preferences do not violate the VNM axioms. The willingness to kill chickens in order to eliminate an arbitrary small chance of the death of your grandmother makes the preferences a bit weird, but still VNM compliant.
A live chicken might quantum tunnel from China to your grandmother’s house and eat all her heart medication. But then again, a quantum tunneling chicken could save your grandmother from tripping on fallen corn-seed. The net effect of chickens on your grandmother might be unfathomably small, but it is unlikely to ever be zero. If chickens never have zero effect on your grandmother, then your preference for the non-extinction of chickens would never apply *EDIT and so the only preferences we would need to consider would be of your grandmother’s life, which could be represented with utilities (say 0 for dead and 1 for alive).
If you’re willing to tolerate a 1⁄10,000,000 increase in chance of your grandmother’s death to save chickens from extinction, you sill have VNM-rational preferences. Here’s a utility function for that:
dead-chicken-dead-grandma 0
live-chicken-dead-grandma 1
dead-chicken-live-grandma 10,000,000
live-chicken-live-grandma 10,000,001
*Or perhaps not so small. 90% of all flu deaths happen to people age 65 or older, and chickens are the main reservoir of influenza.
I am not willing, no. Still VNM-compliant?
Yep. But you probably don’t care about chickens.
If, by some cosmic happenstance the effect of a chicken’s life or death affected your grandmother exactly zero—no gravitational effects of the Australian chicken’s beak on your grandmother’s heart, no nothin’—then the utilities get more complicated. If the smallest possible non-zero probability a chicken could kill or save your grandmother were 10^-2,321,832,934,903, then the utilities could be something like:
dead-chicken-dead-grandma 0
live-chicken-dead-grandma 1
dead-chicken-live-grandma 10^2,321,832,934,903 + 1
live-chicken-live-grandma 10^-2,321,832,934,903 + 2
It seems more likely that he really isn’t VNM-compliant. Chickens are tasty and nutritious, 1⁄1,000,000 is a small number and lets face it, grandparents of adults are already old and have much more chance than that of dying every day. It would be surprising if Said is so perfectly indifferent to chickens, especially since he’s already been explicitly telling us that he isn’t VNM-compliant.