Thank you for pointing me to that and getting me to think critically about it. I think I agree with all the axioms.
a rational agent should act as to maximize expected value of their value function
I think this is misleading. The VNM theorem only says that there exists a function u such that a rational agent’s actions maximize E[u]. But u does not have to be “their value function.”
Consider a scenario in which there are 3 possible outcomes: A1 = enormous suffering, A2 = neutral, A3= mild joy. Let’s say my value function is v(A1)=−9,v(A2)=0, and v(A3)=1, in the intuitive sense of the word “value.”
When I work through the proof you sent in this example, I am forced to prefer pA1+(1−p)A3 for some probability p, but this probability does not have to be 0.1, so I don’t have to maximize my expected value. In reality, I would be “risk averse” and assign p=0.05 or something. See 4.1Automatic consideration of risk aversion.
More details of how I filled in the proof:
We normalize my value function so u(A1)=0 and u(A3)=1. Then we define u(A2)=q2.
Let M=1⋅A2, then M′=q2A3+(1−q2)A1, and I am indifferent between M andM′. However, nowhere did I specify what q2 is, so “there exists a function u such that I’m maximizing the expectation of it” is not that meaningful, because it does not have to align with the value I assign to the event.
Thank you for pointing me to that and getting me to think critically about it. I think I agree with all the axioms.
I think this is misleading. The VNM theorem only says that there exists a function u such that a rational agent’s actions maximize E[u]. But u does not have to be “their value function.”
Consider a scenario in which there are 3 possible outcomes: A1 = enormous suffering, A2 = neutral, A3= mild joy. Let’s say my value function is v(A1)=−9,v(A2)=0, and v(A3)=1, in the intuitive sense of the word “value.”
When I work through the proof you sent in this example, I am forced to prefer pA1+(1−p)A3 for some probability p, but this probability does not have to be 0.1, so I don’t have to maximize my expected value. In reality, I would be “risk averse” and assign p=0.05 or something. See 4.1Automatic consideration of risk aversion.
More details of how I filled in the proof:
We normalize my value function so u(A1)=0 and u(A3)=1. Then we define u(A2)=q2.
Let M=1⋅A2, then M′=q2A3+(1−q2)A1, and I am indifferent between M andM′. However, nowhere did I specify what q2 is, so “there exists a function u such that I’m maximizing the expectation of it” is not that meaningful, because it does not have to align with the value I assign to the event.