Weekly update 50

Optimization is really fun. In fact, solving optimization problems may be the funnest part of what I do. In practice, those solutions require I do a lot of programming, which is always tedious and usually frustrating (and yet somehow still fun). For instance, this week I spent several days programming a solver for a pretty simple problem (by my own experience's standards). Strangely enough, it presented a new challenge for me that took some time to overcome. Basically, I had an optimization that was, in fact, 3 different optimizations all set up hierarchically. That's not new for me (Bungee does the same kind of thing). However, 1) I needed to find the global optimum rather than a nearby local optima, 2) it's a combination of continuous and discrete problems, 3) at the most important level, I was only optimizing a single parameter, and 4) for significant ranges of said parameter, the objective function is flat.

Anyway, it turns out the solution space can be explored pretty thoroughly with a simple method, but it took me a while to realize that.

The result is that I now have this figure for my fourth chapter:

As with most figures I put on here, this is a first draft and I dont really understand it yet. I'm not even convinced I am plotting the appropriate thing. The horizontal axis is basically showing how much the regulator cares about reducing impacts, where high values mean we allow a lot of impacts and low values mean we allow very little impacts. The vertical axis shows how much it should cost across my study sites to implement a particular policy. The whole point of the figure is to show how much better we can do with cap and trade than just caps on impacts. The gray regions show something like confidence intervals about the outcomes in the cap only (light gray) and cap and trade (darker gray) scenarios when the regulator systematically over or underestimates impacts in the absence of additional regulation. The symbols show what happens when the regulator has perfect information.

The thing that makes me uneasy is that the cap and trade symbols fall under the optimal allocation (theoretically best possible outcome). The issue is either that I'm doing something wrong in how I measure costs in the cap and trade scenario, or I'm plotting the wrong thing on the vertical axis. More thinking to do...