Sequential trials and dealing with risk
What if my preferences for making an extra few dollars outweigh the risk of losing the same amount? I will stop on why one's preferences might be asymmetric in a little while in this section, and there is scientific evidence that this asymmetry is ingrained in our minds for evolutionary reasons, but you are right, I have to optimize the expected value of the asymmetric function of the parameterized utility now, as follows:
Why would an asymmetric function surface in the analysis? One example is repeated bets or re-investments, also known as the Kelly Criterion problem. Although originally, the Kelly Criterion was developed for a specific case of binary outcome as in a gambling machine and the optimization of the fraction of money to bet in each round (A New Interpretation of Information Rate, Bell System Technical Journal 35 (4): 917–926, 1956), a more generic formulation as an re-investment problem involves a probabilistic distribution of possible...