I quite like this model, it seems natural to me that to quantify hingeyness in this model would be by considering how different the probability distributions of the total utility at the end of time are. Things like how much the range contracts also seem to be decent quick approximations of this difference. There has actually been a lot of work on quantifying this difference between probability distributions, searches for statistical distance or “probability metrics” should give you results.
If we were to define hingeyness in this model using some notion of the distance between probability distributions it seems likely we would want this distance to have the properties of a metric. It’s not obvious to me which metric would be the best choice for this model though. The Wasserstein metric seems the easiest metric from the above link to implement to me.