aka "how to profit off people's irrationalities".
The philosophical question of what it means to be rational is answered by the popular slogan "Rationality is systematized winning" -- so, in particular, a rational belief is one that is expected to deliver you a profit.
This is closely linked to the "practical" definition of probability -- assigning probability $p$ to observation $X$ means valuing at $p$ a unit wager on $X$ (an asset with payoff 1 if $X$ happens, an 0 otherwise) -- you are willing to buy the wager for any price less than $p$, and sell it for any price higher.
(We use money to make things simple, but more fundamentally we have a cardinal utility function whose expectation we seek to maximize -- so this is a valid abstract definition and does not depend on a specific currency or monetary system.)
However, if you accept that probability is definitionally subjective (even if updated by evidence via Bayes's theorem, your priors are still subjective), then assigning a probability $p$ to a particular event cannot itself be rational or irrational. You can't quite calculate whether someone who holds that belief will make a profit or loss, because doing so would require assigning an underlying "objective" probability to $X$.
You can only say that a person is irrational if you can prove that their beliefs always lead to them losing, regardless of the results of the observation. Such beliefs are said to be incoherent, a notion that is identical to the financial concept of arbitrage. Irrationality in this sense is about an inconsistency in beliefs about different, related observations.
Consider observations $X_1,\dots X_n$ (in the language of probability theory, these are elements of our sigma algebra). Suppose you assigned to them the probabilities $p_1,\dots p_n$.
Your beliefs are irrational iff I can come up with a list of unit wagers (called a Dutch book) to trade with you that always deliver me a profit regardless of the outcome of each observation.
For example, if $X_1\subseteq X_2$ and you assign probabilities $p_1=0.5,p_2=0.1$, then I can buy from you a unit wager on $X_2$ and sell you a unit wager on $X_1$, and always make a profit.
The axioms of probability theory can then be shown to be equivalent to forbidding a Dutch book. If the values of all bets are positive between 0 and 1, the value of the bet on the entire sample space is 1 and bets are additive, then it becomes impossible to construct a Dutch book. Conversely, a violation of any of these axioms allows a bet on something equivalent to the sample space to be traded for less than or greater than 1, allowing for a Dutch book.
Similar Dutch book arguments can be extended to conditionalization and Bayes's theorme.
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