Utility functions usually map from expected future states to utilities. You seem to be doing something else—since you have utilities in the arguments to the utility function. Put some dollars in there instead and you get:
a utility function where U(10% chance of 10 dollars) < U(1 dollar)
...which is absolutely fine and correctly represents risk aversion.
No, though I was using 10 utiles as shorthand for “an event that, were it to occur, would give you 10 utiles”. So without that shorthand it would be something like:
Let A and B be two future states and assume without loss of generality that U(A) = 0 utiles and U(B) = 10 utiles. Then if U(10% chance of B, 90% chance of A) < 1 utile.
But that would have been ugly in the context.
a utility function where U(10% chance of 10 dollars) < U(1 dollar)
This could be the same utility function that I am talking about, but it could also be one of a risk neutral agent with a diminishing marginal utility for money.
This could be the same utility function that I am talking about, but it could also be one of a risk neutral agent with a diminishing marginal utility for money.
Those are intimately-linked concepts, as I understand it:
Quantified utility models simplify the analysis of risky decisions because, under quantified utility, diminishing marginal utility implies “risk aversion”.
Utility functions usually map from expected future states to utilities. You seem to be doing something else—since you have utilities in the arguments to the utility function. Put some dollars in there instead and you get:
...which is absolutely fine and correctly represents risk aversion.
No, though I was using 10 utiles as shorthand for “an event that, were it to occur, would give you 10 utiles”. So without that shorthand it would be something like:
Let A and B be two future states and assume without loss of generality that U(A) = 0 utiles and U(B) = 10 utiles. Then if U(10% chance of B, 90% chance of A) < 1 utile.
But that would have been ugly in the context.
This could be the same utility function that I am talking about, but it could also be one of a risk neutral agent with a diminishing marginal utility for money.
Those are intimately-linked concepts, as I understand it:
http://en.wikipedia.org/wiki/Marginal_utility#Revival