I think there is a problem with the notion of “purely random” placement of boats. It’s not like choosing heads or tails. You could create a list of all possible placements and select uniformly between them, you could place each ship randomly one at a time, which would favour some placements over others and would depend on the order the ships were placed, or, I’m sure, you could concoct other systems.
There may actually not be a Nash Equilibrium for the placement/guessing strategy. If you realize that having your ships clustered in the middle makes it easier to solve and create a placement strategy that avoids that then you have reduced the entropy of your placement strategy, meaning that the hunting strategy can be modified to lower the probability of making shots that are less likely under your strategy. Then you chance how you place again and they change how they shoot again, etc.
I wouldn’t be surprised if there was a single placement strategy that minimized the information given if your opponent knows the strategy, but if there is, that would be the one I would be most tempted to call “purely random.”
