Is the strategy minimal? I think from a game theory standpoint the question of what the optimal Battleship strategy is is kind of interesting. Practically speaking the optimal strategy may not be hugely better than choosing coordinates at random, but it’s trivial to show that random choice can be improved upon.
It’s easy to think of a boring strategy that is clearly better than random guessing, but I’m sure you can do better and I agree it seems interesting to think about how you could do better. Surely knowing there is nothing in one space adjusts the probability of something being in the surrounding spaces up to a distance where a boat could overlap with that space.
And even the game theory perspective pretty much assumes you are playing against a blank slate when you might be able to discern things about your opponent, especially over multiple games.
Then again, no one ever fired at “under the left cushion of the couch downstairs” so I had that game on lock.
The actual article…
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My ultimate strategy was to stack boats. If they ever actually found my one superboat, I was screwed, but I only ever recall that happening once. And it was still close, as it was pretty late in the game.
Cool, thanks for posting this! Interestingly one of the implications of this analysis is that purely random placement of boats is suboptimal, because it results in the increased probability density in the center of the board. From the perspective of information theory I suppose one would want to adopt a boat placement strategy that resulted in a completely even probability density, which should maximize the amount of information that needs to be extracted.
My brother’s innovation was to use the outer edge of the board.
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.”
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I don’t think Battleship is a terrible game. I actually think that with some modifications, it would make an excellent educational game that teaches kids Cartesian coordinates and linear equations. You could have a limited number of ship searing lasers that need correct firing solutions but have dramatic results. Parabolic radar waves can be deployed to locate enemy ships. I think there’s a real opportunity here.
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It’s a tricky business in part because we also want to minimize the information gained from a hit. Evening out the probability distribution over the whole board so that the chance of hitting a boat on some initial shot is equal over the whole set of coordinate pairs should be possible, but if it results in a tendency for the boats to be closer to the edges (which it would), then it comes with a secondary cost. Once a hit occurs the search for the other parts of the boat is shorter. It would be neat if there were a competition for Battleship-playing programs/bots…
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I think it’s one of two cases: there is a Nash Equilibrium and everyone plays the same thing or it’s paper rock scissors. It’s hard to imagine it ending up more complicated than that.
Rock paper scissors has a Nash equilibrium. But anyway a multiplayer tournament can require more complex strategies, because then we have to start thinking about maximally exploitive strategies against weaker opponents.
One of these days I should learn to French-braid. When my hair was long I’d very occasionally braid it, but French-braiding always seemed beyond me. Maybe if my daughter’s hair gets long enough I can practice.
The trick is to start when they’re little, so by the time they’re old enough to have an opinion on your skill, you actually HAVE some skill.
That works for a lot more than just hair braiding, by the way!
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I actually like playing the game with my daughter. Mostly on paper, though we’ve also played it on a smartphone. And there is some strategy to keep in mind. Ie. if you get a hit then you obviously go for the spaces around- something that my daughter quickly figured out and the fact that she could win half the time, made it a good game to play. Thinking about strategy, one can try dividing it into quadrants and try to selectively cover as much area and slowly filling that in - can be outsmarted by placing ones ships in the corner or all around the sides. I think a more interesting game would be if the ships move (hard to do on paper of course) but easy with a software version. I’d rather try to play something like the old wooden ships and iron men wargame in which wind conditions factored in and could randomly change) and you tried to get into a rake position where you could fire at the ends of the enemy ships where they had little chance of defence. But I’m curious to see you show how random choice could be improved upon.
Apparently, the hirsute infant was actually an art object. I’m not sure whether that should make us feel better or not.
Awesome… it does make me feel better, though I was already considering trying to buy one to freak out my colleagues. Now I won’t bother looking.
What is this foul sorcery??? Madge is Australian!!!
I assume you mean Rock.
“Yer psyching in it”? Who the hell is that?
Did I say Madge was Australian? I meant to say Madge was really, REALLY, Australian…
That particular dialect is native to the greater Sydney area.
