Towards my next paper, I've been thinking a lot about decision-making where one has uncertainty but not quantified probabilities or even necessarily a total ordering of possible outcomes by plausibility. Most recently, I've tried to formalize the notion of when, without quantified probabilities, one lottery may be said to be fairer than another, and of a simple rule for selecting the fairer of two coins (as in my previous paper I have made considerable use of orderings of coins by entropy).

Yester-day, in the context of such ponderings, I arrived at some interesting, simple complementarity rules. Consider two actions, each paired with one considered outcome. Between these, various plausibility relations may obtain — the first pair may be more plausible than the second, the second may be more plausible than the first, they may be equally plausible, their relative plausibility may be unknown, or the relation may be a union of two or three of the aforementioned (eg one pair may be more-or-equally plausible). In any case, whatever that relation, the same relation will hold if we reverse the order of the pairs and take the logical complement of the outcomes. Here's an example of the formal expression of one of these rules where I'm using the same notation that I did in in my entry of 19 August, and represents the relationship of the left side being more plausible than the right side.

(Common-sense examples are easy to generate. For example: If it is more likely that the Beet Weasel will bite than that the Woman of Interest will stay home, then it is more likely that she will depart than that he will refrain from biting. Or: If we don't know whether a given nickle is more likely to come-up heads than is a given quarter, then we don't know whether the quarter is more likely to come-up tails than is the nickle.)

In the context of an irreflexive, antisymmetric, transitive relation, one can identify closeness without measurement. For example, if A is more plausible than B and B is more plausible than C, then B is closer both to A and to C than they are each to the other.

This abstraction of closeness, along with the principles of complementarity, allow one to identify when one coin is more fair than another, without having any quantification of fairness, so long as one can order the plausibilities of outcomes across coins. One simple rule is to pick the coin whose most likely outcome is less likely than the most likely outcome of other coins; an equivalent rule would be to pick the coin whose least likely outcome is more likely than the least likely outcomes of other coins.

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BTW, the aforementioned previous paper is still in the hands of the editors of the journal to which I submitted it a bit less than four months ago. I've not had any word from them. But, while this journal did not provide a time-frame, other journals give frames such as six months. (A friend recently had one of her submissions rejected at just before the three-month mark.) It is at least somewhat plausible that, by the time that said previous paper is published somewhere, I will have this next paper ready to submit to a journal.

Tags: decision theory, papers

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