If fairness is one of my values, it can’t necessary be represented by such a function. (I.e., it may need to be a function from lotteries over world-histories to real numbers.)
You refer to cases such as A = “I give the last candy to Alice”, B = “I give the last candy to Bob” and you strictly prefer the lottery {50% A, 50% B} to {100% A} or {100% B}?
But remember that we’re talking about entire world histories, not just world states—If you take A0 = “I arbitrarily give the last candy to Alice”, A1 = “I flip a coin to decide whom to give the last candy to, and Alice wins”, etc., you can easily have A1 = B1 > A0 = B0, since A1 and A0 are different (one includes you flipping a coin, the other doesn’t). So a function from world histories would suffice, after all.
I’m pretty sure Nisan meant to define “world-histories” in a way to exclude utility functions like that, otherwise it’s hard to make sense of the convexity property that he assumes in his theorem. (Hopefully he will jump in and confirm or deny this.)
Yes, we should assume the agent has access to a source of uncertainty with respect to which the functions v_i are invariant.
In fact, let’s assume a kind of Cartesian dualism, so that the agent (and a single fair coin) are not part of the world. That way the agent can’t have preferences over its own decision procedure.
You refer to cases such as A = “I give the last candy to Alice”, B = “I give the last candy to Bob” and you strictly prefer the lottery {50% A, 50% B} to {100% A} or {100% B}?
But remember that we’re talking about entire world histories, not just world states—If you take A0 = “I arbitrarily give the last candy to Alice”, A1 = “I flip a coin to decide whom to give the last candy to, and Alice wins”, etc., you can easily have A1 = B1 > A0 = B0, since A1 and A0 are different (one includes you flipping a coin, the other doesn’t). So a function from world histories would suffice, after all.
I’m pretty sure Nisan meant to define “world-histories” in a way to exclude utility functions like that, otherwise it’s hard to make sense of the convexity property that he assumes in his theorem. (Hopefully he will jump in and confirm or deny this.)
Yes, we should assume the agent has access to a source of uncertainty with respect to which the functions v_i are invariant.
In fact, let’s assume a kind of Cartesian dualism, so that the agent (and a single fair coin) are not part of the world. That way the agent can’t have preferences over its own decision procedure.