No Tile Left Behind: Evaluating Multi-Tile Leaves

This article is the third post in my series “No Tile Left Behind” focusing on leaves. For a list of articles on this topic, click here.

Now that we understand the value of individual tiles, we can now compute the value of multi-tile leaves. You can estimate of the worth of a leave by simply adding the worth of two tiles together: for example, since the M is worth 1 point and the A is worth 0.5 points, the AM leave is worth about 1.5 points. However, leaves can be defined more rigorously.The first thing that is important to understand is the formal definition of leaves, and to show that the worth of an n-tile leave is equal to the average of all of the (n+1) tile leaves when their probability is accounted for. Formally written:
The value of leave L with elements {E(1), . . . E(k)} (expressed by the notation V{E(1)… E(k)} is equal to the value of the mean of all leave (L+1) with elements {P(1)E(1),…P(k+1)E(k+1} (where V represents the value function and P represents the probability function)

In English, this means that the worth of a leave is the average of that leave plus one other tile included after we take into account the probability of drawing each tile. This is shown by the following chart, which can be confirmed by the program Quackle:

Letter + 1 with AERT +:

A: 8.7B: 16.2C: 22.1D: 19.9E: 12.8F: 12.1G: 11.9H: 19.6I: 16.4
J: 4.4K: 9.5L: 19.3M: 18.7N: 20O: 10.5P: 19.6Q: 2R: 12.7
S: 34.8T: 13U: 8.5V: 11.5W: 9.8X: 12Y: 9Z: 14.2?: 45.4

Average of AERTx = 15.4 (consistent)
Standard deviation: 7.36
% of tiles higher than the mean: 41%

By this logic, we can thus find the value of every leave by finding the value of every 7 tile leave using backward induction. This would not be so complex except for the fact that there are so many different 7 tile leaves, since there are approximately 3.2 million 7 tile racks, and there is no good intuitive way of finding the value of all those leaves. Nevertheless, this is a good mathematical expression for defining leave values, as seen in the chart above. The value of any leave is an average of sums.However, as before, this understanding of leaves does not help us as players develop an approximation. Instead, we have to use heuristics to evaluate multi-tile leaves. The main heuristics that I recommend in my book involve duplication, synergy, and vowel-consonant ratio. While duplication is merely a matter of statistics, the other two are more complex.

This sort of analysis has one major practical application: during exchanges. During exchanges, we often have to decide whether an extra marginal tile is worth keeping. In an attempt to ask ourselves this question, we should create and try to approximate tables like the one above if we want to figure out whether or not to keep a certain tile on an exchange. Often times this is not so simple, as these numbers are not universal for each position: in certain cases they need to be adjusted. Adjusting these values will be covered in my next article.