It seems like there might be a problem with this argument if the true vi are not just unknown, but adversarially chosen. For example, suppose the true v2 are the actual locations of a bunch of landmines, from a full set of possible landmine positions V2. We are trying to get a vehicle from A to B, and all possible paths go over some of the V2. We may know that the opponent placing the landmines only has n2 landmines to place. Furthermore, suppose each landmine only goes off with some probability p even if the vehicle drives over it. If we can mechanistically predict where the opponent placed the landmines, or even mechanistically derive a probability distribution over the landmine placements, this is no problem, we can just use that to minimize the expected probability of driving over a landmine that goes off. However, suppose we can’t predict the opponent that way, but we do know the opponent is trying to maximize the probability that the vehicle drives over a landmine that isn’t a dud. It seems like we need to use game theory here, not just probability theory, to figure out what mixed strategy the opponent would be using to maximize the probability that we drive over a landmine, and then use that game-theoretic strategy to choose a mixed strategy for which path to take. It seems like the game theory here involves a step where we look for the worst (according to our utility function) probability distribution over where the landmines are placed, because this is how the opponent will have actually chosen where to put the landmines. Doesn’t this look a lot like using μ rather than U as our utility function?
It seems like there might be a problem with this argument if the true vi are not just unknown, but adversarially chosen. For example, suppose the true v2 are the actual locations of a bunch of landmines, from a full set of possible landmine positions V2. We are trying to get a vehicle from A to B, and all possible paths go over some of the V2. We may know that the opponent placing the landmines only has n2 landmines to place. Furthermore, suppose each landmine only goes off with some probability p even if the vehicle drives over it. If we can mechanistically predict where the opponent placed the landmines, or even mechanistically derive a probability distribution over the landmine placements, this is no problem, we can just use that to minimize the expected probability of driving over a landmine that goes off. However, suppose we can’t predict the opponent that way, but we do know the opponent is trying to maximize the probability that the vehicle drives over a landmine that isn’t a dud. It seems like we need to use game theory here, not just probability theory, to figure out what mixed strategy the opponent would be using to maximize the probability that we drive over a landmine, and then use that game-theoretic strategy to choose a mixed strategy for which path to take. It seems like the game theory here involves a step where we look for the worst (according to our utility function) probability distribution over where the landmines are placed, because this is how the opponent will have actually chosen where to put the landmines. Doesn’t this look a lot like using μ rather than U as our utility function?