What’s a Z Really Worth?
Why efforts to assign Scrabble tiles their “real value” miss the point of the game.
Let’s examine the potential effects of the “new” values. In both Lewis’ and Deadspin’s calculations, 10 tiles decrease in value, four increase, and 12 don’t change. More tiles fall in a band of two to five points, bringing the utility of each closer together. The consensus of my math-brained Scrabble colleagues is that this would be like a dose of lithium for the game, flattening scoring and eliminating swings that keep games interesting. Big but still-reasonable values for some tiles, especially the X and Z, are good because they improve the odds of comebacks, make tile positioning a compelling strategic consideration, and give players with lesser word knowledge a slightly better chance against those who are booked up. “Except for the Q, Josh [Lewis] basically squashes the volatility,” said Eric Chaikin, co-director of the Scrabble documentary Word Wars. “His values take the fun out.”
Quackle co-writer John O’Laughlin, a software engineer at Google, said the existing inequities also confer advantages on better players, who understand the “equity value” of each tile—that is, its “worth” in points compared with the average tile. That gives them an edge in balancing scoring versus saving letters for future turns, and in knowing which letters play well with others. “If we tried to equalize the letters, this part of the game wouldn't be eliminated, but it would definitely be muted,” O’Laughlin said. “Simply playing the highest score available every turn would be a much more fruitful strategy than it currently is.”
In response to Lewis’ findings, John Chew, co-president of the North American Scrabble Players Association and a mathematics doctoral student at the University of Toronto, wrote that Scrabble has always had an “intentional” imbalance between the face value and the equity value of the letters. Whenever the game’s lexicon changes—a fifth edition of the Scrabble dictionary is due in 2014—players adapt. “The tile values were chosen to make an interesting game, not to accurately represent the statistical properties of a particular lexicon,” he wrote.
In fact, for Alfred Butts, the face value of the letters was secondary to their distribution. That’s because Butts’ original research was for a word-formation game called Lexiko, which didn’t involve a board and didn’t assign points to individual letters.* In the early 1930s, on spreadsheets containing 26 rows, one for each letter, Butts tallied tens of thousands of letters from the pages of the New York Times, the New York Herald Tribune, and the Saturday Evening Post. Then he compared the frequency with which the letters occurred, both on their own and as part of words of particular lengths. From those tabulations, he determined how many of each letter should be included in his game.
Courtesy National Scrabble® Association.
Butts manufactured and sold a few hundred sets of Lexiko from his Queens walkup, but Milton Bradley and Parker Brothers, among others, rejected it. In 1938, Butts decided to add a board. That’s when he assigned values to letters roughly corresponding with their frequency. Butts’ files, which I read while researching my book Word Freak, contain pages of spreadsheets and notes about letter frequency and tile distribution. But there’s little about how he settled on the point values for his new game, which he named Criss-Cross Words. He did experiment, though. I found one iteration in which the Z was worth nine points; the K and V six; and the B, F, and W five. And there’s the plywood evidence of my six-point X.
My conclusion: Butts coupled intuition with direct observation of the game in action—he tested it on his wife and their friends—to arrive at values that he felt balanced equity and volatility. (Not that he would have used those words.) So, for instance, while Butts might have suspected the X was logically worth six points, he understood that eight would make the game more exciting. Similarly, while Butts included a lexicographically reasonable seven S’s in Lexiko, he knew the letter was so valuable for pluralizing words that he should reduce the number in his board game, which he did, to four.