Pattern

The flaw is: Doing things randomly in this situation does not make you very much better off.

(For sure it is better than both sides stick to April all the time. But that's beyond the point.)

The best strategy is, really, to have an easily recognized pattern of alternating between June and April. Very soon, the other medical school across the harbour would get the message and follow suite and choose a different month every year - because it is to their advantage.

In other words, the two sides do not communicate by words, but by their action. It is a cooperative game, and both sides could do better if there is some kind of coordination between them.

Or, in a layman term, we are friends, not enemies.

(One major area of contemporary research in this field is on how to handle - and distinguish if possible - a delinquent behaviour and occasional careless mistake by your opponent in a cooperative game of this kind.)

It sounds very simple, isn't it? But the implication is slightly more profound than what you imagine.