Joshua Thong

Many criteria of algorithmic fairness have been put forth in the current literature, of which the two most important ones are arguably Equalized Odds and Calibration. However, despite their plausibility, Kleinberg et al. (2017) have shown that they are inconsistent outside of trivial cases, so at least one must be rejected in non-trivial cases. Recently, Eva (2022) propounded a weakening of Calibration, called Base Rate Tracking, though not necessarily in response to the preceding impossibility result. Regardless, even though it is weaker, Stewart et al. (2024) have shown that it is still inconsistent with Equalized Odds outside of trivial cases. In this paper, I propose a weakening of Base Rate Tracking I call Comparative Base Rate Tracking that is consistent with Equalized Odds even in non-trivial cases, yet this weakening is still quite substantial nonetheless. Importantly, I show that Comparative Base Rate Tracking can diagnose several deficient cases that cannot be diagnosed by other recently proposed criteria of fairness, like Spanning by Nielsen and Stewart (2024a) and Spacing by Eva (2024).

Consider the following claims: 1. Rational credences are real-valued. 2. A rational agent is more confident in A than in B just in case appropriate set-theoretic relations between the relevant events and/or appropriate inequalities between her numerical credences, whether conditional or not, hold. 3. If a rational agent’s conditional credence in A, given A\cup B, is greater than her conditional credence in B, given A\cup B, then she is more confident in A than in B. 4. There are two distinct and particular ways of ordering the events in a lottery over the naturals, for each of which there exists a rational agent whose comparative confidence ordering corresponds to that ordering. Versions of the first three claims have been defended by various authors, though not necessarily in conjunction, and I claim that the fourth is at least plausible. In this paper, I show that the conjunction of these four claims is inconsistent. Thus, at least one claim must be rejected.

This paper is a discussion note on Isaacs et al. (2022), who have claimed to offer a new motivation for imprecise probabilities, based on the mathematical phenomenon of non-measurability. In this note, I clarify some consequences of their proposal. In particular, I show that if their proposal is applied to a bounded 3-dimensional space, then they have to reject at least one of the following: (i) If A is at most as probable as B and B is at most as probable as C, then A is at most as probable as C. (ii) Let A ∩ C = B ∩ C = ∅. A is at most as probable as B iff (A ∪ C) is at most as probable as (B ∪ C). But rejecting either statement seems unattractive.

In a fair finite lottery with n tickets, the probability assigned to each ticket winning is 1/n and no other answer. That is, 1/n is unique. Now, consider a fair lottery over the natural numbers. What probability is assigned to each ticket winning in this lottery? Well, this probability value must be smaller than 1/n for all natural numbers n. If probabilities are real-valued, then there is only one answer: 0, as 0 is the only real and non-negative value that is smaller than 1/n for all natural numbers n. However, this answer seems counter-intuitive, as it violates Regularity: for all possible events A that are assigned a probability, A is assigned a positive probability. It is possible for ticket i to win in the second lottery. So, the set {i} should be assigned a positive probability and not 0. Consequently, in order to satisfy Regularity, probabilities must be allowed to take on `non-real' values, e.g. a positive infinitesimal x that is smaller than 1/n for all natural numbers n. So, what regular probability is assigned to {i} in the second lottery? This probability value must be positive but smaller than 1/n for all natural numbers n. Given the definition of x, x is a correct answer. But x isn't unique, because every other positive infinitesimal is a correct answer as well and there are uncountably many positive infinitesimals. After all, every positive infinitesimal is strictly between 0 and 1/n for all natural numbers n. Nothing about a fair lottery over the natural numbers uniquely determines x as the correct answer and nothing about the lottery rules out other positive infinitesimals as a correct answer as well. What probability to assign to {i} is underdetermined. In this thesis, I explore this difference between fair finite lotteries and fair infinite lotteries. The constraints governing the former uniquely determine the probabilities assigned to events in those lotteries, but this isn't true for the latter lotteries, when Regularity is a constraint on probabilities. Now, although there are uncountably many correct probability values that can be assigned to {i} in the second lottery, there are some applications of probability theory, for which at least one correct probability value must be chosen in order to accomplish that application, e.g. calculation of expected utilities. In this thesis, I spell out sufficient conditions for when a choice like that is problematic. These conditions apply to precise probabilities, imprecise probabilities, and qualitative probabilities. So, I will consider Benci et al. (2018)'s and Pruss (2021)'s suggestion of allowing probabilities to be imprecise, as well as resorting to qualitative probabilities, to overcome the choice problem. I will argue that both don't solve the problem. Furthermore, there's a hitherto unnoticed difficulty in deriving imprecise probabilities by supervaluating over precise `non-real' probability values. To avoid this difficulty, precise probability values should be real-valued. Thus, Regularity cannot be a constraint on probabilities. But as it turns out, dropping Regularity as a constraint and requiring precise probabilities to be real-valued solve the choice problem.

Rational agents seem more confident in any possible event than in an impossible event. But if rational credences are real-valued, then there are some possible events that are assigned 0 credence nonetheless. How do we differentiate these events from impossible events then when we order events? de Finetti (1975), Hájek (2012) and Easwaran (2014) suggest that when ordering events, conditional credences and subset relations are as relevant as unconditional credences. I present a counterexample to all their proposals in this paper. While their proposals order possible and impossible events correctly, they deliver the wrong verdict for disjoint possible events assigned equal positive credence.