Analysis in Scrabble - an O.R. perspective

Some years ago, I wrote a paper on dynamic programming and board games, which looked at a number of references in the O.R. literature to the topic.  One game that I didn't mention, though it had been in my mind while I wrote the paper, is Scrabble.  Each week, a columnist in The Times writes about aspects of the game, and last Saturday (7th September 2019) the problem he studied has aspects of dynamic programming.

In any position of Scrabble, the player has to decide what letters to play from the rack and where, bearing in mind what the other player might do, what letters will be left on the rack, and what letters are in the bag to be chosen.  In terms of dynamic programming, the state space is too large to consider, which is why I steered clear of it in my survey paper.  But the problem in the column is interesting, nonetheless.

Imagine that it is early in the game, so the bag of tiles is almost full.  And your rack has low value letters.  Then your choice is "between scoring as many points as possible using the majority of such letters [hoping to get some high value letters] or playing one or two letters [because you might pick up letters which can make a seven letter word and score a bonus]"  The columnist went on to present the rack with DELNRSU .  It wasn't possible to play the word nurdles (or any other anagram) on the board so a bonus wasn't possible with that rack.  So the player used the L in the hope of drawing a vowel, because the letters DENRSU combine with every vowel to form seven letter words, and with several consonants as well.  With good fortune, early in the game, there will be a place for a bonus scoring move.  (In the game analysed, the player drew a T and managed an eight-letter word using the opponent's play.)

But, as the column pointed out, in other circumstances, it may be worth clearing more of the rack.  As in all sequential analysis (in the methodology of dynamic programming) "it all depends".

refs: "Scrabble column" by Paul Gallen [The Times (2019) 7 September 2019 Times2 - page 53]
"Dynamic Programming and Board Games: A Survey" by David K Smith [European Journal of Operational Research (2007) vol 176 p1299-1318]

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