Does the intuitive sense of overfitting in this mechanism design context exemplify bias-variance tradeoff?

Question

Suppose the (we can say unanimous) preference of each individual in a society is to select roads for travel by placing 95% weight on the objective of minimizing travel time, and the remaining 5% weight on the objective of maximizing aesthetic pleasure derived from the roads' scenery. In response to Braess' paradox emerging with respect to these preferences, central planners destroy an optimal subset of the initial set of roads. Then, as an independent, exogenous change, the society reduces the work week from 40 hours to 20 hours, which subsequently results in preferences unanimously decreasing their weight on travel time minimization and increasing their weight on scenery quality maximization. Obviously, which roads were destroyed need no longer be an optimal decision as the preferences being targeted have shifted, but more troubling is that if the shift is large enough, we might expect the current subset of roads to perform worse than the initial set, because Braess' paradox arises from the travel time minimization component, while for the scenery quality component, destroying roads can only hurt things, by giving individuals less options to maximize over (scenery quality is taken to be independent of traffic distribution).

We could also modify the above to include preferences over more than two optimization objectives, preferences as nonlinear functions of optimization objectives such as arise from substitute and complement effects, preference changes caused endogenously by the road destruction instead of exogenously, multiple optimization objectives from which competing instances of Braess' paradox emerge calling for mutually-inconsistent road subsets, etc. We could also consider not just the poor fit of a road subset to individuals' actual preferences, but to the preferences which they would have formed had roads not been destroyed (perhaps those who never experienced the destroyed roads with the highest scenery quality don't know what they're missing).

Intuitively, I would refer to the decision to destroy roads as potentially overfitting the initial preferences, and the central planner's aim to optimize over the initial preferences while ignoring the observation that having less options for individuals to optimize over will tend to produce worse outcomes if preferences ever shift as minimizing bias while ignoring variance. Just as aggressively fitting a regression polynomial to training data may make it generalize poorly to other data, aggressively fitting a road subset to observed preferences may make the action space generalize poorly to other preferences. However, I can't see an obvious process of random sampling underlying this context, so I'm not sure if there's actually some sort of reduction or isomorphism here. The following is my attempt so far...

Bias$\ =\ \sum_{all\ individuals}(u_{NE}(G_1) - u_{NE}(G_0))$

Variance$\ =\ \sum_{all\ individuals}(v_{NE}(G_0) - v_{NE}(G_1))$

...where $u_{NE}()$ is an individual's initial Nash Equilibrium utility as a function of available roads, $v_{NE}()$ is an individual's Nash Equilibrium utility as a function of available roads after shifting preferences, $G_0$ is the initial road set, and $G_1$ is the subset of $G_0$ producing optimal Nash Equilibrium utility conditioned on preferences described by $u$ (the central planner cannot foresee $v$). For ease of conceptualization, I am assuming each Nash Equilibrium is unique conditioned on fixed preferences and available roads.

As for the random sampling piece, perhaps the fact that the central planner cannot foresee the shift from $u$ to $v$ is sufficient uncertainty to play this role? It is certainly possible to introduce the random generation of a preference shift, but this feels more forced than a natural part of the setup.

Does this scenario qualify as bias-variance tradeoff? If so, then what piece(s) of the puzzle am I missing thus far? If not, then is there a generalization of bias-variance tradeoff extending beyond contexts of random sampling which includes random sampling and this?

Answer 1

Overfitting and bias-variance tradeoff deal with what happens when applying some model to different instantiations of the same population. Here it seems we just have different populations, there is no particular reason why the traffic planning board's solution that correctly inferred that people care mostly about travel times should work well in a setting when people no longer care about travel times. Overfitting occurs when you re-apply a model in the same setting and see poor performance, but here it seems we're just applying it to a different setting.

We've trained a model to optimize traffic by minimizing traffic time. A not-overfit model should find a reasonably good solution when applied to another setting where we want to optimize traffic by minimizing traffic time. That it produces routes which are not good when considering their aesthetic value isn't really overfitting, it's just applying a model in a setting where it has no bearing. We could get the true underlying model in the original population exactly correct with no overfitting whatsoever, and it still wouldn't perform well when applied to a different setting.

It's not clear to me how you could approach the problem with no training data, you're trying to train a model to optimize the solution without a good idea of what should be considered optimal. You can probably get away with it when the preferences haven't changed much and are still "close enough" to the original population, but I don't think overfitting is the right way to think about applying a method in two settings have significant differences.