I've been getting a bit stuck recently on how to reconcile the two seemingly-competing ideas of nondeterministic and probabilistic decision rules.
As an example:
Let $t=0$ denote the current time and let $t=T$ denote some future time.
Let $w_t$ denote our wealth at time $t$.
Assume we have a market with one risk-free asset, $B$, and a collection of various other risky assets such options, futures, equity, etc.
Let $B_t$ denote the value of our risk-free asset at time $t$. We take $B_0 \in (0,1)$ and $B_T=1$.
Say we want to decide how to allocate our wealth into these various assets, perhaps more than once, such that we optimize some functional of our future wealth (create a trading strategy).
If we try to apply only nondeterministic decision rules and use the strategy that maximizes the minimum of $w_T$, we would probably end up putting everything into the risk-free asset meaning we would have a guaranteed, but very low return as $B_0$ tends to be close to 1 in practice.
On the other end, if we apply probabilistic decision rules such as attempting to say, choose the strategy that maximizes the expected value of the logarithm of $w_T$, we would probably have something that falls much more in line with conventional balancing of risk versus reward as we would likely choose some other strategy that doesn't predominantly invest in the risk-free asset and also somewhat reflects the utility of the wealth as the more dollars you have, the "less" a single dollar is worth to you. However, assuming our chosen strategy puts little to nothing into the risk-free asset, we open ourselves up to a massive risk in the form of tail events, model risk, epistemic uncertainty, etc. since however unlikely, our risky assets could lose most/all of their value?
Is there any way to reconcile these two sets of decision rules since they do seem to individually contribute helpful attributes such as security/safety on one end and likely reward on the other? Would it just be on us to create a decision rule that is say, a weighted average of nondeterministic and probabilistic decision rules that reflects our preferences? Or are there some "optimal" criteria we can aspire to?
