Am I right though that in the case of e.g. Newcomb’s problem, if you use the anti-Nirvana trick (getting -infinity reward if the prediction is wrong), then you would still recover the same behavior (EDIT: if you also use best-case reasoning instead of worst-case reasoning)?
Yes
imagine that you know that the even bits in an infinite bitsequence come from a fair coin, but the odd bits come from some other agent, where you can’t model them exactly but you have some suspicion that they are a bit more likely to choose 1 over 0. Risk aversion might involve making a small bet that you’d see a 1 rather than a 0 in some specific odd bit (smaller than what EU maximization / Bayesian decision theory would recommend), but “reflecting reality” might recommend having Knightian uncertainty about the output of the agent which would mean never making a bet on the outputs of the odd bits.
I think that if you are offered a single bet, your utility is linear in money and your belief is a crisp infradistribution (i.e. a closed convex set of probability distributions) then it is always optimal to bet either as much as you can or nothing at all. But for more general infradistributions this need not be the case. For example, consider X:={0,1} and take the set of a-measures generated by 3δ0 and δ1. Suppose you start with 12 dollars and can bet any amount on any outcome at even odds. Then the optimal bet is betting 14 dollars on the outcome 1, with a value of 34 dollars.
But for more general infradistributions this need not be the case. For example, consider X:={0,1} and take the set of a-measures generated by 3δ0 and δ1. Suppose you start with 12 dollars and can bet any amount on any outcome at even odds. Then the optimal bet is betting 14 dollars on the outcome 1, with a value of 34 dollars.
I guess my question is more like: shouldn’t there be some aspect of reality that determines what my set of a-measures is? It feels like here we’re finding a set of a-measures that rationalizes my behavior, as opposed to choosing a set of a-measures based on the “facts” of the situation and then seeing what behavior that implies.
I feel like we agree on what the technical math says, and I’m confused about the philosophical implications. Maybe we should just leave the philosophy alone for a while.
IIUC your question can be reformulated as follows: a crisp infradistribution can be regarded as a claim about reality (the true distribution is inside the set), but it’s not clear how to generalize this to non-crisp. Well, if you think in terms of desiderata, then crisp says: if distribution is inside set then we have some lower bound on expected utility (and if it’s not then we don’t promise anything). On the other hand non-crisp gives a lower bound that is variable with the true distribution. We can think of non-crisp infradistirbutions as being fuzzy properties of the distribution (hence the name “crisp”). In fact, if we restrict ourselves to either of homogenous, cohomogenous or c-additive infradistributions, then we actually have a formal way to assign membership functions to infradistirbutions, i.e. literally regard them as fuzzy sets of distributions (which ofc have to satisfy some property analogous to convexity).
Yes
I think that if you are offered a single bet, your utility is linear in money and your belief is a crisp infradistribution (i.e. a closed convex set of probability distributions) then it is always optimal to bet either as much as you can or nothing at all. But for more general infradistributions this need not be the case. For example, consider X:={0,1} and take the set of a-measures generated by 3δ0 and δ1. Suppose you start with 12 dollars and can bet any amount on any outcome at even odds. Then the optimal bet is betting 14 dollars on the outcome 1, with a value of 34 dollars.
I guess my question is more like: shouldn’t there be some aspect of reality that determines what my set of a-measures is? It feels like here we’re finding a set of a-measures that rationalizes my behavior, as opposed to choosing a set of a-measures based on the “facts” of the situation and then seeing what behavior that implies.
I feel like we agree on what the technical math says, and I’m confused about the philosophical implications. Maybe we should just leave the philosophy alone for a while.
IIUC your question can be reformulated as follows: a crisp infradistribution can be regarded as a claim about reality (the true distribution is inside the set), but it’s not clear how to generalize this to non-crisp. Well, if you think in terms of desiderata, then crisp says: if distribution is inside set then we have some lower bound on expected utility (and if it’s not then we don’t promise anything). On the other hand non-crisp gives a lower bound that is variable with the true distribution. We can think of non-crisp infradistirbutions as being fuzzy properties of the distribution (hence the name “crisp”). In fact, if we restrict ourselves to either of homogenous, cohomogenous or c-additive infradistributions, then we actually have a formal way to assign membership functions to infradistirbutions, i.e. literally regard them as fuzzy sets of distributions (which ofc have to satisfy some property analogous to convexity).