The post calls the functions f_i “utility functions”, not “expected utility functions”.
(As an aside, some terminological confusion can result from there being a “utility relation” that compares lotteries, that can be represented by a “utility function” that takes lotteries as inputs, and separately expected utility representation of utility relation (or of “utility function”) that breaks it down into a probability distribution and a “utility function” in a different sense, that takes pure outcomes as inputs.)
However, I don’t think that f_i(S_j) denotes the total utility of a state of the universe. It is just one of the terms used to compute such a total utility.
Right.
From the comments about additivity, I take f_i(S_j) to be the amount by which the utility of a universe to species i would increase if a planet following strategy j were added to it (while the strategies of all other planets remained unchanged), regardless of how or by whom that planet is added.
Or, more usefully (since we can’t actually add planets), the utility function of aliens #k that takes a collection S of strategies for each of the planets under consideration (i.e. a state of the world) is
F_k (S) = sum_p f_k(S_p)
Then, the decision problem is to maximize expected value of F_0(S) by controlling S_0, a standard game theory setting. It’s underdetermined only to the extent PD is underdetermined, in that you should still defect against CooperationBots or DefectBots, etc.
You are right, I retract my comment.
(As an aside, some terminological confusion can result from there being a “utility relation” that compares lotteries, that can be represented by a “utility function” that takes lotteries as inputs, and separately expected utility representation of utility relation (or of “utility function”) that breaks it down into a probability distribution and a “utility function” in a different sense, that takes pure outcomes as inputs.)
Right.
Or, more usefully (since we can’t actually add planets), the utility function of aliens #k that takes a collection S of strategies for each of the planets under consideration (i.e. a state of the world) is
F_k (S) = sum_p f_k(S_p)
Then, the decision problem is to maximize expected value of F_0(S) by controlling S_0, a standard game theory setting. It’s underdetermined only to the extent PD is underdetermined, in that you should still defect against CooperationBots or DefectBots, etc.