Missing the point: Point Spacing
How much is a point worth?
This at first probably seems like quite a bizarre question. However, it is quite relevant. An extra 10 points is extremely important when you’re only up 15: but when you’re up by 150, it doesn’t really mean all that much. This seems intuitive to everyone: points are more valuable the closer the score.
To most folks, this is somewhat intuitive. This fact is the heart of entropy. However, there are times when this is not true, particularly when the game is close, but not too close. In some games you’re just barely ahead and you need to keep scoring to maintain the lead, and in other games you’re ahead or behind by a bingo and as long as neither player bingos, the game is mostly decided. But in between those two outcomes is this awkward gray area: an area that becomes more dramatic as the game nears its conclusion.
To further understand, let’s take a look at the Scrabble distribution.
Now let’s take a look at another graph: the difference between any two random points on this distribution.
As you can see, the bimodal nature of the Scrabble distribution leads to a somewhat trimodal difference in scores. And the more significant the bimodality of the Scrabble distribution, the more severe the trimodality in the graph above. This is why scoring spots are not a bad thing when you are ahead: it serves to level out the graph above and make it less trimodal. This is a good thing when you are ahead.
However, when the score difference is in between these trimodal humps, score matters far less. This is the definition of point spacing: when you are between these humps, especially near the end of the game, points matter far less than other factors because of the trimodality of these humps.