Chapter 1: What is the Fifth Fourth Postulate of Decision Theory?
On Independence Axiom. By Iakendiakov. Published on March 8, 2026. Heading. The Fifth Fourth Postulate of Decision Theory.
In 1820, the Hungarian mathematician Farkas Boliai wrote a desperate letter to his son J. A. Knows, who had become consumed by the same problem that had haunted his father for decades. Quote. You must not attempt this approach to parallels. I know this way to the very end.
Chapter 2: How does independence relate to avoiding exploitation?
I have traversed this bottomless night, which extinguished all light and joy in my life. I entreat you, leave the science of parallels alone. Learn from my example. The problem was Euclid's fifth postulate, the parallel postulate, which states, in one of its equivalent formulations, that through any point not on a given line, there is exactly one line parallel to the given one.
For over 2,000 years, mathematicians had felt that something was off about this postulate.
Chapter 3: What is the strongest argument for the independence axiom?
The other, four, were short, crisp, self-evident. You can draw a straight line between any two points, you can extend a line indefinitely, you can draw a circle with any centre and radius, all right angles are equal.
Chapter 4: What insights does ergodicity economics provide?
The fifth postulate, by contrast, was long, complicated, and felt more like a theorem that ought to be provable from the others than a foundational assumption standing on its own. Generation after generation of mathematicians attempted to derive it from the remaining, four, and failed. Farkas Boliai begged his son to stay away. J.A.
Noes ignored his father's advice, but not in the way Farkas feared.
Chapter 5: How do the Allais and Ellsberg paradoxes illustrate rational behavior?
Instead of trying to prove the postulate, he asked a question that turned the entire enterprise upside down. What happens if the postulate is simply false? What if you can draw more than one parallel line through a point?
Rather than deriving a contradiction, which would have constituted a proof of the fifth postulate by Reduccio, he found something a perfectly consistent geometry, as internally coherent as Euclid's, just describing a different kind of space. Loboshevsky independently reached the same conclusion around the same time. The parallel postulate was not wrong, exactly, but it was not necessary.
It was one choice among several, and the other choices led to geometries that were not merely logically valid but turned out, a century later, to describe the actual physical universe better than Euclid's flat space ever could. Roughly two centuries later, people were discussing decision theories and axioms of expected utility.
Chapter 6: How has the LessWrong community engaged with decision theory?
The standard argument went roughly like this. Rational agents must maximize expected utility. The von Neumann-Morgenstern theorem proves it. If your behavior violates the axioms, you can be Dutch-booked, turned into a money pump, exploited by anyone who notices the inconsistency. You don't want to be a money pump, do you? Then you must maximize expected utility.
QED There are four axioms in the von Neumann-Morgenstern framework. Completeness, transitivity, continuity, and independence. Three of them are relatively uncontroversial.
The fourth, independence, does enormous structural work, it is the axiom that forces preferences to be linear in probabilities, which is mathematically equivalent to requiring that preferences be representable as the expected value of a utility function.
Without independence, you still have a well-defined preference functional, by Debreu's theorem, given the other axioms, you can still order outcomes, you can still make consistent choices, but you are no longer constrained to maximize expected utility specifically. Independence is the fifth postulate of decision theory.
And just as with Euclid's fifth, I believe, the resolution is not to keep trying harder to justify it but to ask.
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Chapter 7: Why should we reconsider the expected utility theory?
What happens when we drop it?
What does the resulting decision theory look like? Is it consistent? Is it useful?
Does it perhaps describe actual rational behavior better? The answer, I will argue, is yes on all three counts. Dropping independence does not lead to irrationality or exploitability. Several well-known alternatives to expected utility theory exist precisely because they relax independence, and they do so for a reason.
Ergodicity economics, in particular, offers a principled and parsimonious replacement that derives the appropriate evaluation function from the dynamics of the stochastic process the agent is embedded in, rather than postulating an ad hoc utility function and taking its expectation.
And the less wrong community's own research into updateless decision theory has been converging on the same conclusion from a completely different direction. That the most reflectively stable agents may be precisely those who violate the independence axiom. Heading. A tale of two utilities.
Before we get to the main argument, we need to clear up a terminological confusion that silently corrupts reasoning about decision theory on the most trivial level. The word utility refers to two completely different mathematical objects, and the fact that they share a name is sad. This is well known in decision theory, and you are welcome to skip this section if you know what I am talking about.
The first object is what we might call preference utility, or F1. This is the function that economists use in consumer theory to represent your subjective valuation of bundles of goods under certainty. If you are indifferent between 2 oranges, 3 apples, and 3 oranges, 2 apples, then F1 is constructed so that F1, 2, 3, equals F1, 3, 2. The crucial property of F1 is that it is ordinal.
The only thing that matters is the ranking it induces, not the numerical values it assigns. If F1 assigns 7 to bundle A and 3 to bundle B, all that means is that you prefer A to B. You could replace F1 with any monotonically increasing transformation of it, squaring it, taking its exponential, adding a million, and it would represent exactly the same preferences.
The numbers themselves carry no information beyond the ordering. The second object is von Neumann-Morgenstern utility, or F2. This is the function that appears inside the expectation operator in expected utility theory. It is constructed not from your preferences over certain bundles but from your preferences over lotteries, over probability distributions on outcomes. The VNM theorem says
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