Faculty Meeting Utility Curve

Liberal arts colleges, priding themselves on faculty governance, regularly meet to do faculty-related business. I’ll pick on selective liberal arts college in the U.S., that likely pride themselves more than usual in this arena. These colleges average 2,000 students, and with a vaunted 1:10 faculty-to-student ratio, this leads to 200 faculty members. If you can’t coerce them to attend, will they show up? You need enough to form a quorum so you can take votes, but as “efficiency deteriorates” with size, inevitably there are those who feel their time is better spent doing something else than showing up to the meeting.

I’ve been thinking about how personal behavior affects larger-scale society issues. With Covid-19, can individuals be trusted to practice physical-social distancing or wearing a mask in public without coercive penalties in place? What proportion of the population should be vaccinated, assuming a vaccine is available, and should this be forced on everyone? These are complex questions. Some people will of their own free-will adopt distancing, or wear a mask, or get vaccinated. Others will not unless forced. Individuals will make their own calculus of whether it’s “worth it” to do any of those things.

How should we think about the relationship between individual choices and its effects on society, given a range of individual preferences and values, versus what a collective group may prefer or value? These are the questions posed in a 1978 classic, Micromotives and Macrobehavior, by Thomas Schelling (awarded the economics Nobel in 2005). It’s a short, engaging, and accessible book, with many interesting examples and not too much math. The last chapter has a number of “utility graphs” based on multi-person prisoner’s dilemma scenarios. I decided to make one up based off a Schelling example. My question is simpler: Should I attend the college-wide faculty meeting?

Well, it depends. On how many other people I think will attend. Let’s see how this works in my fictitious utility graph below.

The vertical axis measures utility, the expected loss or gain of an individual who decides whether to skip or attend the meeting. The reference point A, is that utility is zero, when no one attends the meeting, i.e., the meeting doesn’t happen, and everyone is just getting their own work done.

For those who choose to skip the meeting, I’ve assumed a straight line of increasing utility because as more people attend the meeting, they are getting less of their own work done, while you, the truant, are therefore relatively more productive to the group. (In this I’m following Schelling.) If you skip, your utility increases on the red line as a proportion of how many people attended.

For those who choose to attend the meeting, your utility is measured the blue sigmoidal curve. It hardly anyone attends but you show up, you’ve wasted your time because little business can be done. For example, the dashed line B indicates that if 20% attended, their utility would be below zero. The 80% who played truant are getting ahead in their work relative to you, the sap who showed up.

But then at some point, as more people attend and quorum is reached, the college can do its business and given you (ideally) care about the affairs of the college, participating and voting in the discussion is of high utility to you (and the institution, and thus you gain a boosted benefit when this happens)! For example, if 80% attended (indicated by dashed line C), those of you who attended get more accomplished overall; even more so than the 20% who chose to skip (although they are getting their own stuff done too).

But there are diminishing returns. Once the college has more than enough people to do business, your attendance in that large meeting now provides diminishing returns. It’s not so much a boost and is maybe starting to seem less productive than if you skipped out, as illustrated by point D.

So what do I do? If I think most people aren’t going to attend, I shouldn’t either. And if everyone thinks that way, we will end up at point A – zero net utility to everyone. Schelling would call this a stable equilibrium point. Imagine a Zoom meeting. Let’s say you logged in on the dot when the meeting started, you can see how many other people are logged in, and the attendance was low – in the regime where the blue curve is below the red curve. If there wasn’t social pressure to stay, and you were homo economicus, you’d log out and leave. And others seeing the total number drop, also drop out. That’s how you end up at A.

If however, I think a quorum’s worth will attend such that our meeting will successfully engage and vote in things that benefit us as a group, and I’ll get a boost from being there, then I’ll go. Imagine the same Zoom meeting. I log in on the dot. Quorum is met. The blue curve is above the red curve. I’m staying. My laggardly colleague logs in two seconds later. Sees the benefit (blue curve above red) and adds to the attendance. A few more people do this and you’ve now moved to the other stable equilibrium (according to Schelling), the higher point where the red and blue curves intersect. Once you hit that point, Prof. Slow logs in. Sees that so many people are present, there’s diminishing returns, and thereby drops out. Thus, one doesn’t ever get to 100% attendance.

All this assumes that as individuals, we immediately and correctly calculate the utility at any point in the attendance spread. Reality makes things much fuzzier. But generating the model allows you to see what happens as the curves shift or change shape as you test different conditions. Perhaps a mild coercive penalty should be added to dissuade you from skipping out, or a sweetener to induce you to attend the meeting. Or the meetings could be structured differently.

At my college, I don’t know how proportionate attendance has changed over time. I’m sure someone keeps track. I have a suspicion it’s getting lower every year. That’s because the room’s roughly the same crowded-ness, but the number of faculty has increased as the college has grown. And the blue utility curve may be flattening. At some point, it may no longer intersect the red curve. Maybe it’s already gone below and we just don’t know it or don’t care. Maybe these idealized curves are simply wrong. But as a model, I think it’s valuable to consider, and might be closer to real life than we’re willing to admit.