According to the book Thinking Strategically by Avinash K. Dixit and Barry J. Nalebuff, a government can coerce every citizen to register for the military by threatening to punish only the first person whose name ranks highest in alphabetical order and does not register. In fact, it goes on to say, “Nothing is special about the alphabet. The key point is that the order of punishment is pre-specified. A randomly chosen and announced order of birthdates, or social security numbers, does just as well.” It seems the assumption here is common knowledge of everyone’s rationality, and they admit that this exercise would fail to incentivize total compliance in practice for this reason. However, everyone registering for the military is the Nash Equilibrium.
In one sense, I understand why the order must be pre-specified. With a uniformly random order, the probability of being punished would be the multiplicative inverse of the number of citizens who chose not to register. However, the following reasoning also seems correct.
Suppose we remodel the game (without changing the underlying physical problem) by treating the government as a player that chooses the order of citizens from which the first who has not registered will be punished, but instead of pre-specifying the order, the government and citizens choose their strategies at the same time.
The citizens could look ahead to each possible order the government could choose. For each case, the order is known by assumption, and the citizens’ best responses collectively mirror the Nash Equilibrium of the above game where the order was actually announced ahead of time; everyone should register for the military. But given that these best responses are identical in each case, they should also constitute the best strategy overall in the simultaneous game.
What is wrong with this reasoning?
Denote the cost of the punishment by $p$, the cost of registering by $r$.
In order for the plan to work, you need to have $p > r$.
If the order is known, then the person at the top will choose to register. Since rationality is common knowledge, the second person knows this, and thus knows that were he not to register he would be punished with probability 1. And so on, and so on.
If the order is unknown, or for example random, every person can believe that the probability of punishment is miniscule, 1 divided by the number of people. It can well be that the expected cost of punishment is less than the cost of registering.
This is why criminals in Gotham commit crimes: whoever meets Batman will get beaten up for sure, but since there are many crimes, only a few of them will meet Batman and they don’t know in advance who it will be, so the expected costs may be lower than the gain from their crimes.