Ihor Kendiukhov

This was valuable groundwork, and it is to Armstrong's credit that he took the question seriously at a time when the less wrong consensus was, and to a significant extent still is, that violating any VNM axiom is ipso facto irrational.

However, Armstrong reaches one conclusion that I think is wrong.

His central result is that when an agent faces many lotteries, and those lotteries are independent and have bounded variants, the agent's aggregate behavior converges to expected utility maximization even without the independence axiom.

He writes, hence the more lotteries we consider, the more we should treat them as if only their mean mattered.

So if we are not risk-loving, and expect to meet many lotteries with bounded SD in our lives, we should follow expected utility.

This is a correct result within its assumptions, but the assumptions exclude exactly the cases where abandoning independence matters most.

Armstrong's convergence argument relies on two things.

that the lotteries are independent of each other, and that they aggregate additively, so that the law of large numbers, in its standard additive form, applies to their sum.

Under these conditions, yes, the variance of the aggregate shrinks relative to the mean, and the mean dominates, which is equivalent to expected utility maximization.

But for an agent making sequential decisions where wealth compounds multiplicatively, the aggregation is not additive.

The relevant law of large numbers for multiplicative processes concerns the geometric mean, not the arithmetic mean.

And the geometric mean of a set of multiplicative gambles is determined by the time-average growth rate, the expected logarithm of the growth factor, not by the expected value.

The convergence is to the time-average, not the ensemble average.

The same line of reasoning can be applied to any non-additive, so not only multiplicative, gamble.

Subheading Scott Garabrant's Comment, 2022, Updatelessness and Independence In December 2022, Scott Garabrant left a comment beneath a post on the EUT that I consider one of the most important things written on LessWrong in the context of this question.

I want to quote the core of it and then explain why it matters for my argument.

My take is that the concept of expected utility maximization is a mistake.

As far as I know, every argument for utility assumes, or implies, that whenever you make an observation, you stop caring about the possible worlds where that observation went differently.

Von Neumann did not notice this mistake because he was too busy inventing the entire field.

The point where we discover updatelessness is the point where we are supposed to realize that all of utility theory is wrong.