Entropy Defined

When making an investment or decision, there are two important variables: worth and risk.  Balancing these factors is essential when trying to make monetary, professional, personal, and game decisions.  For example, a play that opens a triple-triple alley is very risky since it could result in a huge score, but it typically has little effect on worth since triple-triples are fairly unlikely.  In Scrabble, risk is assessed in terms of entropy.

The concept of entropy is prevalent in many other games.  Sports have “prevent” defenses, many games advocate trading pieces or health when ahead, and trailing players in nearly any game or sport become aggressive, sometimes to the point of recklessness.  However, the idea is not so prevalent in Scrabble.  Scrabble has only been around for about 65 years, and for much of the time, it was perceived as a word game instead of a strategy game. Game strategy has only played a major role in Scrabble over the last decade, and as a result, concepts such as entropy are fairly new.

Entropy measures the ability for a trailing player to overtake their opponent in score. In statistical terms, entropy can be equated to the variance of score difference between you and your opponent, or the likelihood of large scoring swings to occur. When behind, entropy is desirable as it increases the likelihood of catching up (as well as your chances of getting blown out). When ahead, entropy is undesirable since it allows your opponent to win if he is fortunate enough to draw well. (When the score is tied, entropy is neither desirable nor undesirable.)

When making decisions about entropy, there are a number of factors that you must take into consideration, including:

  • Game score: When behind, you should usually increase entropy, while you should usually decrease entropy when you are ahead.  Increasing or decreasing entropy also depends on your propensity for either you or your opponent to outscore a bonus.
  • Stage of game: As the game progresses, entropy becomes significantly more important and easier to control as there is less time for random chance to affect the outcome of the game and more options available to exhaust open bonus lines.
  • Long term ramifications: It is important to observe the situation and determine if you can significantly affect entropy.  If there are two bonus lines open and you want to block one of them, will your opponent be able to viably create bonus lines on future turns?  If so, it is probably not worth reducing entropy for one turn: the benefits of entropy are only substantial if they last.
  • Valuation: Entropy should be thought of as another metric alongside points and leave.  If two plays have comparable valuations, but one player has a clear lead, you should make the play with a preferable level of entropy.

Entropy is usually divided into two categories: rack entropy and board entropy.  Board entropy assesses the ability for the score to change on a specific board.  High entropy boards have few scoring spots but many bonus lines.  Board entropy can vary greatly both in the long and short term: bonus lines can be thought of as “currency” for short term entropy while space and board dynamics are the “currency” for long term entropy.  More about board entropy will be discussed in a future article.

Rack entropy describes the variability of one’s remaining tiles to score well.  While certain leaves consistently score well (such as CHOY), other leaves have both significant upside and downside (a leave such as AEI can easily lead to a bonus or an exchange) leading to a high entropy.  For example, a fishing leave such as IORSTU has significant bonus potential, but with a bad draw (such as an I or R) the rack has limited bonus or scoring potential.

There are primarily two gaps when it comes to rack entropy: the gap between exchanging and scoring, and the gap between scoring and bonuses.  A leave such as AIIO or GLNTW has a high entropy because a good draw can make the rack score well (or even occasionally result in a bonus) while a bad draw forces you to exchange.  Meanwhile, a leave such as EILNT is a good leave but is usually relegated to ~20 point plays when it fails to result in a bonus.  A leave such as AEIST crosses all three thresholds, possibly resulting in an exchange, medium scoring play, or a bonus.