Grid and Bear it. (How to beat that weird uncle who always wants to play Battleship at Battleship)

Battleship

Random guessing or ingenious strategy? Totally Could be either one.

Battleship is played on two 10 x 10 grids as shown below:

Battleship board, poorly drawn.

Each player has a grid on which they place the following ships in any configuration they wish:

1 Aircraft Carrier – 5 squares long

1 Battleship – 4 squares long

1 Submarine – 3 squares long

1 Cruiser – 3 squares long

1 Destroyer – 2 squares long

The only constrictions on placing ships are that no two ships may overlap on a square and that they must be placed horizontally or vertically (not diagonally).

For simplicity’s sake, we will indicate squares containing ships  with blue coloring:

Each player’s grid is not visible to the other player.

One possible ship configuration

The players then alternate turns ‘shooting’ at opposing ships by calling out a square location on the grid, to which the other player responds with ‘hit’ or ‘miss’.

The first player to destroy all of the opposing ships, by hitting every square that contains part of a ship, is the winner.

Battleship is a game played on several dimensions, psychological and analytical. Unlike previous games we have covered, Battleship cannot be so easily reduced to tidy analysis. The more you know about your opponent’s tendencies and play, the better off you are in the game. Like all games which contain an element of hidden moves, that is, games in which players have less than perfect information, there is an element of chance – or, if you play based on your opponent’s strategies and tendencies, skill. To begin we will discuss general strategy against an unknown opponent, after which we will proceed to a basic discussion of levels and psychological considerations.

There are several important factors to consider in basic Battleship strategy:

#1 – Grid within a Grid

The 10×10 grid structure of the game produces 100 possible squares for the ships to exist in. One way to gain an advantage in Battleship is in reducing the amount of shots it takes to destroy all of the opposing ships. A simple division of the board into alternating squares does a wonderful job of achieving this goal:

By taking your searching shots on squares that correspond only to black or white squares on the board, you cut the shots necessary to hit each ship at least once in half. It also reduces the total amount of shots necessary to sink all ships 100% of the time from 100 to 75 (if, once you hit a ship, you miss the followup in all 3 possible directions each time and the maximum number of ship squares are on squares that don’t correspond to your choice of which color squares to use.) Decreasing the number of shots you have to take by 25% will be enough to win in most cases against players that don’t realize the advantage this strategy provides.

#2 Do not place your ships touching one another

If the opposing player hits a ship, most likely they will attempt to destroy that ship by shooting at adjacent squares. By placing ships touching one another, your reduce the possibility of your opponent missing on one of those shots, as follows:

Possible misses in Pink after a hit in red.

It may not seem like much advantage to reduce the likelihood of a followup hit by a small amount, but Battleship is a game of inches and maximizing each shot. Any small advantage you can achieve in the placement of your ships will magnify itself over the course of the game.

#3 Consider Probability

With ships of lengths as described, there are only a certain (admittedly large) number of places that ships can be placed on the board. If you consider all possible arrangements for ships (as charted by Datagenetics here), the middle squares are very much more likely to contain ships than the side squares during the initial shots if the ships are placed randomly. Of course, when playing against a particular opponent, this may not be the case. This leads us to…

Psychological Considerations

Is your opponent more likely to place a ship along the side because he knows your strategy of shooting for the middle squares? Will he place two ships next to each other, knowing that you know that it’s generally not a good idea? Does your opponent always place a destroyer in a corner? Do they know you know they always place a destroyer in the corner, and will change that strategy for this game?

The better you can answer questions of this sort, the better off you are. If you can’t answer one of these questions effectively, however, it may be better not to ask them in the first place and proceed with your initial strategy.

Levels

Levels are a game theory idea based on analysis of psychological considerations. For example, placing a ship along the side because it is far from the center, which is the most probable area to contain a ship in random placement, may be level one.

For an opponent who knows that you are on this level, it may be wise to place ships in the center, to avoid your early shots at the sides of the board. In this case, that strategy would be level two.

For an opponent that knows that you know they know that ships should be placed on the corner, they may place ships on the corner anyway, expecting you to shoot for the middle since you expect them to place ships in the middle because you understand that they are supposed to place ships on the side and that you’ll be shooting there. This would be level three, and all subsequent levels of analysis get more and more absurd as you go on up the hierarchy.

The main use of level analysis in Battleship is if, as you find locations of ships, you can identify the level your opponent is operating on. This can provide another level of insight into your opponent’s strategy.

Make sure that you consider your opponent in this analysis, as placing a 4-year old who tried to put his ships in the shape of a duck on the board on level 3 can have disastrous consequences.

QUACK!

Computational Considerations

From a purely analytical point of view, it is possible with each shot to compute the probability that each square contains a ship. If this is your sort of thing and you didn’t click on the link earlier, Datagenetics did a wonderful breakdown of the algorithm required to do this at their Battleship Analysis page.

Bonus section

Perfect information: Games like chess, in which each player can see the entire board and no moves are hidden or random, are considered to be games of perfect information. These games tend to have a much higher level of analysis associated with them, as it is is possible to reliably predict the moves of your opponents at a much higher level in than games with a random or hidden element. Each of the previous games we’ve covered (Connect Four, Tic-Tac-Toe, and Dots and Boxes) have been games of perfect information. We will cover more games of perfect information in the future (checkers, chess and others), but many of the entries coming up will consider games with a random or hidden element, since these features tend to make for a richer playing experience and a richer level of strategy for many players.

Thank you so much for reading, and I hope your next foray into gridded naval warfare is successful. Don’t forget to comment if you want to see a particular game covered in this blog, if you enjoyed the article, or if you hated the article with the intensity of prolonged artillery fire.

Category : Games
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