Ihor Kendiukhov

You can bet on the color of a drawn ball.

Most people prefer betting on red, known probability of one-third, over betting on black, unknown probability, could be anything from zero to two-thirds, even though if you assign your best estimate probability of one-third to black, the expected values are identical.

This is typically treated as another irrational bias.

The probabilities are the same in expectation, so why should ambiguity matter?

Ergodicity economics provides a natural and I think quite elegant resolution, and it comes in two layers.

The first layer is a direct Jensen's inequality argument.

Under multiplicative dynamics, the time average growth rate of a repeated gamble is a concave function of the probability.

For a simple multiplicative bet with fraction f of wealth wagered, the growth rate is something like g p equals p asterisk log 1 plus f plus 1p asterisk log 1f, which is concave in p. Now consider the Ellsberg un.

the number of black balls could be 0, 1, 2, 60.

If you are maximally uncertain and average uniformly over these possibilities, the expected proportion is 30 divided by 60 equals 1 half, which matches the known probability case.

An ensemble average reasoner sees no difference.

E in P equals one half in both cases, so the expected value of the gamble is the same.

But concavity of G in P means that Jensen's inequality applies.

EGP less than GEP.

The average time-average growth rate across all possible earned compositions is strictly less than the time-average growth rate you get when the probability is known to be one-half.

Each distinct earned composition, zero black balls, one black ball, two black balls, and so on, defines a different multiplicative process with a different time-average growth rate.

You can compute all 61 of these growth rates and average them, and that average will be strictly lower than the single growth rate corresponding to the known one-half probability because you are averaging a concave function.

The gap is mathematically inevitable, and it is completely invisible to ensemble averaging.

The second layer is about strategic optimality.

Even beyond the Jensen's inequality point, an agent under multiplicative dynamics has a further reason to prefer known probabilities.