Ihor Kendiukhov

They plan to kelebet, or whatever the ergodic mapping prescribes, and they are kelebeting, regardless of what the local branch structure looks like at any given moment.

This connection between ergodicity economics and resolute choice has not been articulated before.

But it is, I think, the cleanest way to see why EE can violate independence without irrationality.

Now, you may or may not accept the entire EE program, but, at the very minimum, I think the conclusion that the agent should pay attention to the dynamic of the gamble and that concrete utility function should depend on the gamble is undeniably valid.

Subheading The broader landscape

The fact that independence is the weak point of the VNM framework is reflected in the structure of the entire field of generalized decision theory, where the majority of alternative frameworks are built specifically by relaxing or replacing the independence axiom.

Rank-dependent utility, Quiggin, 1982, replaces independence with comonotonic independence, independence holds only for gambles that rank outcomes in the same order.

The result is a preference functional that includes a probability weighting function, which distorts the cumulative distribution before integrating against the utility function.

Cumulative prospect theory, Tversky and Kahneman, 1992, combines probability weighting with reference dependence and loss aversion.

It was developed to explain empirical patterns of choice under risk, and it violates independence in multiple ways.

Quadratic utility, Chu, Epstein, and Siegel, allows the preference functional to be a bilinear form in probabilities, meaning it is quadratic rather than linear in the probability measure.

This captures something akin to sensitivity to the variance of a gamble, not just its mean.

Betweenness Preferences, Deckel, 1986.

Chu, 1989, weaken independence to the requirement that if you are indifferent between two lotteries, any mixture of them is equally good.

This is strictly weaker than full independence and yields preference functionals defined by implicit functional equations rather than explicit integrals.

This convergence is not coincidental.

When multiple independent research programs, developed by different people with different motivations over several decades, all arrive at the same structural move, relax independence, it suggests that the constraint being relaxed is objectively too strong.

Allais paradox.

The Allais paradox is the oldest and most famous demonstration that people systematically violate the independence axiom.

The setup, in its simplified form, goes like this.