In going from your second bullet point to your third, you jump from a positive statement to a normative one. If one has a utility function for the states all agents, then one can extend VNM to that utility function, and try to maximize that, but that just begs the question of what utility to assign to the states of other agents.
uv(v(1), b(0)) > ub(v(0), b(1)
Besides the missing parenthesis, there’s a crucial conceptual problem with that statement. If “uv(v(1), b(0))” means “a utility function as defined by the VNM theorem”, then the statement does not follow. VNM says that a utility function exists; it does not say that the function is unique. Since uv(v(1), b(0)) is not uniquely defined, asking whether it is greater than another number doesn’t make sense.
uv(v(1), b(0)) is the map. There is an abstract object that comes from a set for which a correspondence can be set up with the real numbers, but that doesn’t mean the object is a real number. Saying “The real number that I’m using to represent the value Veronica’s utility function is greater than the real number that I’m using to represent the value of Betty’s utility function” is a vacuous statement. It’s a statement about the map, not the territory. Each agent has a comparison method, but there is no universal comparison method.
Besides the missing parenthesis, there’s a crucial conceptual problem with that statement. If “uv(v(1), b(0))” means “a utility function as defined by the VNM theorem”, then the statement does not follow. VNM says that a utility function exists; it does not say that the function is unique. Since uv(v(1), b(0)) is not uniquely defined, asking whether it is greater than another number doesn’t make sense.
No. The post says, “Betty likes apple pies, but Veronica loves them, so uv(v(1), b(0)) > ub(v(0), b(1)).” It says that Betty’s utility, in the situation where she has one pie and Veronica does not, is less than Veronica’s utility in the situation where she has one pie and Betty does not. That is a constraint that we impose on the two utility functions.
This post is incomplete, as was noted at its beginning, and you really shouldn’t be trying to figure it out now. “Incomplete” is very bad, in mathematics. I’m moving it into Drafts.
In going from your second bullet point to your third, you jump from a positive statement to a normative one. If one has a utility function for the states all agents, then one can extend VNM to that utility function, and try to maximize that, but that just begs the question of what utility to assign to the states of other agents.
Besides the missing parenthesis, there’s a crucial conceptual problem with that statement. If “uv(v(1), b(0))” means “a utility function as defined by the VNM theorem”, then the statement does not follow. VNM says that a utility function exists; it does not say that the function is unique. Since uv(v(1), b(0)) is not uniquely defined, asking whether it is greater than another number doesn’t make sense.
uv(v(1), b(0)) is the map. There is an abstract object that comes from a set for which a correspondence can be set up with the real numbers, but that doesn’t mean the object is a real number. Saying “The real number that I’m using to represent the value Veronica’s utility function is greater than the real number that I’m using to represent the value of Betty’s utility function” is a vacuous statement. It’s a statement about the map, not the territory. Each agent has a comparison method, but there is no universal comparison method.
veronica.prefers?( ((v(1), b(0)), (v(0), b(1)) ) ⇒ true
betty.prefers?( ((v(1), b(0)), (v(0), b(1)) ) ⇒ false
prefers?( ((v(1), b(0)), (v(0), b(1)) ) ⇒ undefined method `prefers?′ for main:Object (NoMethodError)
No. The post says, “Betty likes apple pies, but Veronica loves them, so uv(v(1), b(0)) > ub(v(0), b(1)).” It says that Betty’s utility, in the situation where she has one pie and Veronica does not, is less than Veronica’s utility in the situation where she has one pie and Betty does not. That is a constraint that we impose on the two utility functions.
This post is incomplete, as was noted at its beginning, and you really shouldn’t be trying to figure it out now. “Incomplete” is very bad, in mathematics. I’m moving it into Drafts.