I have three agents AB and C, each with the following preferences between two outcomes a and b:
Agents A and B prefers a>b
Agent C prefers b>a
For any two lottos <L, with an x% chance of getting a, otherwise b> and <M, with an y% chance of getting a, otherwise b>:
if X>Y
A and B prefer L
C prefers M.
If X=Y, all three agents are indifferent between L and M
if X<Y:
A and B prefer M
C prefers L.
(2 is redundant given 1, but I figured it was best to spell it out.)
This satisfies the axioms of the VNM theorem.
I’ll give you a freebee here: I am declaring that agent C‘s utility function is: uC(Pa)=−2Pa as part of the problem. This is compatible with the definition of agent C’s preferences, above.
As for agents A and B, I’ll give you less of a freebee: I am declaring as part of the problem that one of the two agents, agent [redacted alpha] has the following utility function: u[Redacted Alpha](Pa)=3Pa. This is compatible with the definition of agent [redacted alpha]‘s preferences, above. I am declaring as part of the problem that the other of the two agents, agent [redacted beta] has the following utility function: : u[Redacted Beta](Pa)=Pa. This is compatible with the definition of agent [redacted beta]’s preferences, above.
Now, consider the following scenarios:
Agent [redacted alpha] and agent C are choosing between a and b:
The resulting utility function is u[Redacted Alpha](Pa)+uC(Pa)=3Pa−2Pa=Pa
The resulting optimal outcome is outcome a.
Agent [redacted beta] and agent C are choosing between a and b:
The resulting utility function is u[Redacted Beta](Pa)+uC(Pa)=Pa−2Pa=−Pa
I have three agents A B and C, each with the following preferences between two outcomes a and b:
Agents A and B prefers a>b
Agent C prefers b>a
For any two lottos <L, with an x% chance of getting a, otherwise b> and <M, with an y% chance of getting a, otherwise b>:
if X>Y
A and B prefer L
C prefers M.
If X=Y, all three agents are indifferent between L and M
if X<Y:
A and B prefer M
C prefers L.
(2 is redundant given 1, but I figured it was best to spell it out.)
This satisfies the axioms of the VNM theorem.
I’ll give you a freebee here: I am declaring that agent C‘s utility function is: uC(Pa)=−2Pa as part of the problem. This is compatible with the definition of agent C’s preferences, above.
As for agents A and B, I’ll give you less of a freebee:
I am declaring as part of the problem that one of the two agents, agent [redacted alpha] has the following utility function: u[Redacted Alpha](Pa)=3Pa. This is compatible with the definition of agent [redacted alpha]‘s preferences, above.
I am declaring as part of the problem that the other of the two agents, agent [redacted beta] has the following utility function: : u[Redacted Beta](Pa)=Pa. This is compatible with the definition of agent [redacted beta]’s preferences, above.
Now, consider the following scenarios:
Agent [redacted alpha] and agent C are choosing between a and b:
The resulting utility function is u[Redacted Alpha](Pa)+uC(Pa)=3Pa−2Pa=Pa
The resulting optimal outcome is outcome a.
Agent [redacted beta] and agent C are choosing between a and b:
The resulting utility function is u[Redacted Beta](Pa)+uC(Pa)=Pa−2Pa=−Pa
The resulting optimal outcome is outcome b.
Agent A and agent C are choosing between a and b:
Is this the same as scenario 1? Or scenario 2?
Agent B and agent C are choosing between a and b:
Is this the same as scenario 1? Or scenario 2?
Please tell me the optimal outcome for 3 and 4.