So for the maximin we are minimizing over all joint distributions that are κ-close to our initial guess?
Yes. That’s more accurate than what I said (originally), since you use a single joint distribution for all of the options, basically a distribution over RN, for N options, and you look at distributions κ-close to that joint distribution.
If I can’t tell the options apart any more, how is the 1/n strategy better than just investing everything into a random option? Is it just about variance reduction? Or is the distance metric designed such that shifting the distributions into “bad territories” for more than one of the options requires more movement?
Hmm, good point. I was just thinking about this, too. It’s worth noting that in Proposition 3, they aren’t just saying that the 1/N distribution is optimal, but actually that in the limit as κ→∞, it’s the only distribution that’s optimal.
I think it might be variance reduction, and it might require risk-aversion, since they require the risk functionals/measures to be convex (I assume strictly), and one of the two example they use of risk measures explicitly penalizes the variance of the allocation (and I think it’s the case for the other). When you increase κ, the radius of the neighbourhood around the joint distribution, you can end up with options which are less correlated or even inversely correlated with one another, and diversification is more useful in those cases. They also allow negative allocations, too, so because the optimal allocation is positive for each, I expect that it’s primarily because of variance reduction from diversification across (roughly) uncorrelated options. I made some edits.
For donations, maybe decreasing marginal returns could replace risk-aversion for those who aren’t actually risk-averse with respect to states of the world, but I don’t think it will follow from their result, which assumes constant marginal returns.
Yes. That’s more accurate than what I said (originally), since you use a single joint distribution for all of the options, basically a distribution over RN, for N options, and you look at distributions κ-close to that joint distribution.
Hmm, good point. I was just thinking about this, too. It’s worth noting that in Proposition 3, they aren’t just saying that the 1/N distribution is optimal, but actually that in the limit as κ→∞, it’s the only distribution that’s optimal.
I think it might be variance reduction, and it might require risk-aversion, since they require the risk functionals/measures to be convex (I assume strictly), and one of the two example they use of risk measures explicitly penalizes the variance of the allocation (and I think it’s the case for the other). When you increase κ, the radius of the neighbourhood around the joint distribution, you can end up with options which are less correlated or even inversely correlated with one another, and diversification is more useful in those cases. They also allow negative allocations, too, so because the optimal allocation is positive for each, I expect that it’s primarily because of variance reduction from diversification across (roughly) uncorrelated options. I made some edits.
For donations, maybe decreasing marginal returns could replace risk-aversion for those who aren’t actually risk-averse with respect to states of the world, but I don’t think it will follow from their result, which assumes constant marginal returns.