My previous post suggested a strategy for making bets on uncertain propositions (e.g. the color of the sun) without having a good prior. Here I present an impossibility result for doing this in general.

Say there are $n$ agents, and $n-1$ possible colors of the sun. Before observing the sun's color, each agent is only able to pick a single color of solar panel to buy (each color solar panel only produces power for a single possible sun color). The sun is revealed, and then the universe is split among the agents who bought the right color solar panel. Assume all agents have utility linear in the share of the universe they control.

Each agent from 1 to $n-1$ believes with certainty that the sun is a color equal to the agent's index (e.g. agent 2 believes the sun is color 2). Obviously, each of these agents the right color solar panel for the sun color they predict.

The $n$th agent is agnostic about what color the sun will be, and would like to ensure that in expectation they get a significant fraction of the universe regardless of the color of the sun.

Unfortunately, agent $n$ cannot guarantee more than a $\frac{1}{2(n-1)}$ fraction of the universe in expectation across possible sun colors (whereas they "should" be able to get a $\frac{1}{n}$ fraction according to the logic in my previous post). This is because, whenever this agent picks the right color solar panel, they will only get half of the universe (sharing with the agent whose index corresponds to the sun's color). So the best agent $n$ can do is to pick a uniformly random color solar panel, yielding an expected $\frac{1}{2(n-1)}$ fraction of the universe for each possible sun color.

I'm pretty sure that allowing agents to make enforceable contracts with each other doesn't fix the problem. The issue is that agent $n$ starts out from a pretty bad bargaining position, having very little to offer the other agents.

If the agent could buy a mixture of different solar panel colors, and the universe were divided proportionally to power generation, then the agent could spend $\frac{1}{n-1}$ of its resources on each color solar panel, yielding a $\frac{1}{n}$ fraction of the universe in all possible worlds. But this does not seem realistic to assume that actions like this are always available.

I'm not sure what to do about this. It might be a deal-breaker for this whole approach, or it might turn out that there is some realistic assumption under which an agnostic agent can guarantee a $\frac{1}{n}$ expected fraction of the universe.