Geometric Rationality acts linearly in additive scenarios

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…