First, as others have mentioned, opponents are very much not equal. Further, timing is important: certain trades you should be much more or less likely to take near the end of the game, for example.
Second, I don’t think it’s valid to look at expected values when all you care about is rank. Expectation is very much a concept for when you care about absolute amounts.
Third, which perhaps sums everything up: I don’t see a valid notion of utility / utility maximization for board games, other than perhaps “probability of winning,” which makes this circular (“if you’re trying to win, you should make moves that increase your probability of winning”). Utility is meant to put a linear scale on satisfaction with a given state of the world. When discussing what to do in a board game, one usually presumes the objective is to win, and satisfaction derives ultimately from winning. The closest thing you usually see to a “utility” number on an intermediate state is a heuristic, as used in e.g.: chess AIs, where you might give yourself 5 points for having a pawn in a center square. If I’m remembering my undergrad correctly, these heuristics are intended to approximate log-likelihoods of victory, but they certainly lack the soundness required to think about expected utility.
Let’s switch out of Catan, and to a game that hopefully people here know but is more directly combative: Diplomacy. Pray tell me how you propose to assign a utility score to putting a navy in the Black Sea.
I have a number of issues with this post.
First, as others have mentioned, opponents are very much not equal. Further, timing is important: certain trades you should be much more or less likely to take near the end of the game, for example.
Second, I don’t think it’s valid to look at expected values when all you care about is rank. Expectation is very much a concept for when you care about absolute amounts.
Third, which perhaps sums everything up: I don’t see a valid notion of utility / utility maximization for board games, other than perhaps “probability of winning,” which makes this circular (“if you’re trying to win, you should make moves that increase your probability of winning”). Utility is meant to put a linear scale on satisfaction with a given state of the world. When discussing what to do in a board game, one usually presumes the objective is to win, and satisfaction derives ultimately from winning. The closest thing you usually see to a “utility” number on an intermediate state is a heuristic, as used in e.g.: chess AIs, where you might give yourself 5 points for having a pawn in a center square. If I’m remembering my undergrad correctly, these heuristics are intended to approximate log-likelihoods of victory, but they certainly lack the soundness required to think about expected utility.
Let’s switch out of Catan, and to a game that hopefully people here know but is more directly combative: Diplomacy. Pray tell me how you propose to assign a utility score to putting a navy in the Black Sea.