Convention dictates that, when you feel the need to pass gas, you ought to suppress it as a courtesy to others in the vicinity. But is this an efficient social convention? If the relief that derives from passing one’s own gas exceeds the discomfort that results from smelling everyone else’s, then we might be happier with a different convention – namely, one that allows unhindered flatulence.
To be more specific, let r be the relief from passing gas, and let d be the olfactory discomfort experienced by each person for each passage of gas. For simplicity, assume these parameters are the same for all persons. Suppose there are n people, each with an equal likelihood of needing to pass gas.
In these circumstances, we might observe a kind of tragedy of the commons. Absent an effective social convention against passing gas, each person rationally chooses to let loose if:
r > dBut the collective flatulence will make everyone worse off if:
nd > rIf so, then everyone will choose to cut loose, even though they'd all be better off if they all restrained themselves. So a convention that successfully induced people to restrain their flatulence would make everyone better off. Note that I can reach this conclusion without comparing one person’s happiness to another’s. Under the suppression convention, I lose satisfaction of r from having to suppress my own, but I gain nd from everyone else suppressing theirs. So if nd > r, I’m better off under the suppression convention than under a free-emission rule.
But note that the mathematical condition above depends crucially on the number of people involved. The larger is n, the more likely is suppression to be efficient. Yet for a small enough number of people, we might find that r > nd, in which case letting loose would be preferable.
Thus, the convention of suppressing flatulence makes most sense in large-number situations. Sadly, those are the situations in which enforcement of the convention is most difficult. When n = 2, finding the guilty party is easy – if you know you didn’t do it, the other guy must have. But when n = 10 (say, at a dinner party), identifying the culprit is much harder, depending on our ability to detect flatulence vectors (where did it come from? where is the scent strongest? etc.).
The dilemma resembles that of a cartel like OPEC. A collusive agreement to cut back production (thereby raising prices) is most easily sustained with a small number of producers, but typically crumbles as the number of producers rises. The difficulty of identifying and punishing cheaters makes cheating more worthwhile, leading to a breakdown of cooperation.
Flatulence-suppression and cartel-maintenance share other features as well. For instance, if individuals differ substantially in their need to pass gas (i.e., some have larger r), then some individuals will face a greater temptation to break convention and let loose. If others follow suit, the convention may disintegrate entirely (think of a frat house). Similarly, cartels are more difficult to maintain when the cartel’s members differ in size, production capacity, or cost of production, since that means some producers face a greater incentive to cheat than others. Their choice to cheat can lead to the cartel’s downfall.
The difficulty of detecting flatulators might explain the relatively high punishment associated with it. Given that flatulence is a completely natural impulse, and that others’ discomfort is usually minor and temporary, social ostracism and public shaming might seem out of proportion to the crime. Yet when the probability of conviction is low, the magnitude of punishment must be correspondingly high to provide a sufficient disincentive. Moreover, if your bad behavior can lead to a breakdown of the convention itself, then the real cost of your behavior exceeds the discomfort caused by just your own flatulence. In such circumstances, harsh punishments make good sense.