If we consider the relation between utility functions and probability distributions, it gets even more literal. An utility function over
X could be viewed as a target probability distribution over
X, and maximizing expected utility is equivalent to minimizing cross-entropy between this target distribution and the real distribution.
This view can be a bit misleading, since it makes it sound like EU-maxing is like minimising H(u,p): making the real distribution similar to the target distribution.
But actually it’s like minimising H(p,u): putting as much probability as possible on the mode of the target distribution.
(Although interestingly geometric EU-maximisingis actually equivalent to minimising H(u,p)/making the real distribution similar to the target.)
When distributing probability over outcomes, both arithmetic and geometric maximisation want to put as much probability as possible on the highest payoff outcome. It’s when distributing payoffs over outcomes (e.g. deciding what bets to make) that geometric maximisation wants to distribution-match them to your probabilities.
This view can be a bit misleading, since it makes it sound like EU-maxing is like minimising H(u,p): making the real distribution similar to the target distribution.
But actually it’s like minimising H(p,u): putting as much probability as possible on the mode of the target distribution.
(Although interestingly geometric EU-maximising is actually equivalent to minimising H(u,p)/making the real distribution similar to the target.)
EDIT: Last line is wrong, see below.
Mind elaborating on that? I’d played around with geometric EU maximization, but haven’t gotten a result this clean.
Sorry, on reflection I had that wrong.
When distributing probability over outcomes, both arithmetic and geometric maximisation want to put as much probability as possible on the highest payoff outcome. It’s when distributing payoffs over outcomes (e.g. deciding what bets to make) that geometric maximisation wants to distribution-match them to your probabilities.