21 January 2009

What about Scrabble?

Scrabble game math nimMy wife likes Scrabble™, and she suggested I should write something about it. She pointed out the book Everything Scrabble suggests that Scrabble is a math game more than it is a word game. I hadn’t really thought much about this game before, and it seemed like a difficult topic, but I agreed to think about it and write something here (then promptly went to sleep). To my great surprise, I had the basics figured out within 24 hours. Scrabble is closely related to Nim.

I thought this might make a good example of my thought process when I look at a game, so I’ll give you the gruesome details of a game dissected.

Scrabble game math nimWords: Scrabble is a complicated game, with over 100,000 words in The Official Scrabble Players Dictionary. My first thought was to consider Scrabble with fewer words, maybe even just one, or none at all, just putting down tiles in a row or column. Words are really just a kind of limit to how the tiles can be played.

Tiles: The tiles are worth differing numbers of points, and this is certainly an important part of Scrabble. However, the value of tiles players draw should be roughly equal (random, but fair) especially if we are disregarding words, so making all the tiles worth one point each is a reasonable simplification. Also, a maximum of seven tiles at a time is needed for large vocabulary of words, but without words it is arbitrary.

Scrabble game math nimBoard: After the other simplifications, all that is left to play for are the squares that give a player double or triple points for a single tile or entire play. I might even disregards point for tiles entirely and just play for these squares, and the player who can get the most of these squares will win. This was my “AHA!” moment where I started to recognize this as a variation of the game Nim. In scrabble you put tiles down, in Nim you pick pieces up. In Scrabble play to put your tile on the next (or last) double points square, in Nim you try not to take the last peice.

So Scrabble is a complicated sort of Nim with points gained for how and where you play the tiles, and a complicated set of rules for how you can play the tiles (ie: words).

My previous efforts at describing games using this process have all ended up with some sort of a probability distribution (including Markov chains and random walks) after I removed all aspects of player decision. Nim is different, and player decision is the key. There is no probability here, and there is always a winning strategy.

This suggests that this property of Nim, with the first moving player having a winning advantage (sometimes the second player), is a basic property of many games, maybe all games. In some games this might be reversed, with the advantage going to the player that moves second, but the cause in the same. This advantage might be small, or get lost in the added complexity and randomness of the full games (like Scrabble, Monopoly), but it probably never goes away entirely. In fact the first move (white) player advantage in Chess is well known. There are plenty of board games where order of movement is important. Battletech players will immediately recognize the advantage of winning the initiative (a random roll give the last player to move the advantage). How about sports? With the Superbowl fast approaching, Football overtime is a good example; given a choice, taking the ball first with the opportunity to score and win is a clear advantage.

Never played Nim? You can try it here.

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