There have been various attempts to explain Kelly betting behaviour within standard decision theory. Here is one I would totally unbiasedly recommend. Geometric rationality is a frameworks which instead takes logarithmic/multiplicative/geometric maximization as the default case, so I wondered if we can create a scenario that so favours additive thinking that it will act accordingly. We can, and with very weak assumptions on the scenario, too[1].
We have a fair coin, which will be tossed twice. Before each toss, you have the opportunity to choose between [2$ if heads] and [1$ if tails]. What would a geometric agent do? It has four hypotheses for what could happen: HH, HT, TH, and TT, each with probability 1/4. Each would then get to make the decision 1/4th of the time[2], and makes it in a way it has the highest profit if that hypothesis is true. So, in a simple form it would mean that HH would bet on HH, HT on HT, etc so that as a whole you bet on heads and tails equally often.
However, HT and TH can come to a mutially beneficial agreement: If they both bet exactly on their beliefs, they expect to get 3$ if they get the decision, and 0$ if the other gets it, or 1.5$ in expectation. However, if they can agree to both bet HH, they both think they'll get 2$. This is because getting your way when you expect head is more valuable than when you expect tail. HH of course has no reason to bet anything other than HH, and TT bets TT. So on the whole, we would be betting on H in 75% of cointosses, or a 50% net bet on H[3].
If we toss the coin n>2 times, a similar thing happens: Everyone who expects more than 1/3rd heads benefits from banding together and betting all heads. Similarly anyone who expects fewer heads than that bands together and bets all tails. As n grows, the former group makes up more and more of the probability mass[4], until in the limit you make the choice agreeing with additive maximization almost certainly.
Is this the Nash bargaining solution? Yes. In fact it is the only symmetric[5] pareto-optimum within these coalitions, and those are the only coalitions that make sense if you want symmetry, because the only symmertic coalition actions are linear combinations of H, T, and everyone doing their own thing. The treat-point also doesn't matter so long as it's symmetric.
I expect similar things will happen with different probabilities and payoffs, if they vary from bet to bet, if there are some multiplicative bets in the mix, etc.
What do we make of this? First, in Scott's formulation, geometric rationality is two-tiered: arithmetic maximisation inside a hypothesis, geometric maximization between hypothesis. This drastically increases the viability of going geometric all the way down - where before, you would have been completely scope-insensitive doing that, now you aren't in most realistic situations.
Second, if you also buy my other argument, about arithmetic maximization acting Kelly-like in the right circumstances, the status of the linear/logarithmic debate has changed significantly. Now, I expect the two to give numerically similar answers in most realistic scenarios. Instead, the edgecase now is the "singular bet before the world ends", and it's based on their behaviour there that you should decide between the two.
The formulation in the original would be that they control that fraction of your resources. I've changed it to random dictatorship because that generalizes better. Due to the same mechanism argued in this post, I expect this will become resource splitting in those cases where it makes sense.
The improvement is not monotonous however, due to the cutoff being just over or under one of the fractions available at a given n.
There is also a strange case here with those hypothesis which believe in exactly 1/3rd heads - they can collectively join either coalition, or neither, (so long as they do it together) and don't expect to gain or lose anything either way.
Symmetric wrt utility facts eg HT and TH should get the same utility. This has the interesting consequence that if the payoffs were equal, any sequence of heads and tails can become a coalition action, with the other coalition chosing its opposite. This doesn't change your final action probabilities, but suggests there could be a more natural setup for this than mine.
There have been various attempts to explain Kelly betting behaviour within standard decision theory. Here is one I would totally unbiasedly recommend. Geometric rationality is a frameworks which instead takes logarithmic/multiplicative/geometric maximization as the default case, so I wondered if we can create a scenario that so favours additive thinking that it will act accordingly. We can, and with very weak assumptions on the scenario, too[1].
We have a fair coin, which will be tossed twice. Before each toss, you have the opportunity to choose between [2$ if heads] and [1$ if tails]. What would a geometric agent do? It has four hypotheses for what could happen: HH, HT, TH, and TT, each with probability 1/4. Each would then get to make the decision 1/4th of the time[2], and makes it in a way it has the highest profit if that hypothesis is true. So, in a simple form it would mean that HH would bet on HH, HT on HT, etc so that as a whole you bet on heads and tails equally often.
However, HT and TH can come to a mutially beneficial agreement: If they both bet exactly on their beliefs, they expect to get 3$ if they get the decision, and 0$ if the other gets it, or 1.5$ in expectation. However, if they can agree to both bet HH, they both think they'll get 2$. This is because getting your way when you expect head is more valuable than when you expect tail. HH of course has no reason to bet anything other than HH, and TT bets TT. So on the whole, we would be betting on H in 75% of cointosses, or a 50% net bet on H[3].
If we toss the coin n>2 times, a similar thing happens: Everyone who expects more than 1/3rd heads benefits from banding together and betting all heads. Similarly anyone who expects fewer heads than that bands together and bets all tails. As n grows, the former group makes up more and more of the probability mass[4], until in the limit you make the choice agreeing with additive maximization almost certainly.
Is this the Nash bargaining solution? Yes. In fact it is the only symmetric[5] pareto-optimum within these coalitions, and those are the only coalitions that make sense if you want symmetry, because the only symmertic coalition actions are linear combinations of H, T, and everyone doing their own thing. The treat-point also doesn't matter so long as it's symmetric.
I expect similar things will happen with different probabilities and payoffs, if they vary from bet to bet, if there are some multiplicative bets in the mix, etc.
What do we make of this? First, in Scott's formulation, geometric rationality is two-tiered: arithmetic maximisation inside a hypothesis, geometric maximization between hypothesis. This drastically increases the viability of going geometric all the way down - where before, you would have been completely scope-insensitive doing that, now you aren't in most realistic situations.
Second, if you also buy my other argument, about arithmetic maximization acting Kelly-like in the right circumstances, the status of the linear/logarithmic debate has changed significantly. Now, I expect the two to give numerically similar answers in most realistic scenarios. Instead, the edgecase now is the "singular bet before the world ends", and it's based on their behaviour there that you should decide between the two.
I wouldn't be surprised if this is already known somewhere, but I didn't find it.
The formulation in the original would be that they control that fraction of your resources. I've changed it to random dictatorship because that generalizes better. Due to the same mechanism argued in this post, I expect this will become resource splitting in those cases where it makes sense.
Due to the same mechanism argued in this post, hypotheses will agree to not even make these opposed bets if you have any risk aversion.
The improvement is not monotonous however, due to the cutoff being just over or under one of the fractions available at a given n.
There is also a strange case here with those hypothesis which believe in exactly 1/3rd heads - they can collectively join either coalition, or neither, (so long as they do it together) and don't expect to gain or lose anything either way.
Symmetric wrt utility facts eg HT and TH should get the same utility. This has the interesting consequence that if the payoffs were equal, any sequence of heads and tails can become a coalition action, with the other coalition chosing its opposite. This doesn't change your final action probabilities, but suggests there could be a more natural setup for this than mine.