It's sometimes interesting the places you can find nuggets that can be applied to game development. The Wall Street Journal has a regular column called "The Numbers Guy". I have often been interested or amused at what appears there. After all, having just finished a book entitled, "Behavioral Mathematics for Game AI", I am obviously sort of a "numbers guy" myself.

Anyway, today's column was titled Scrabble and Other Games -- on Boards, Fields, Courts and Ice -- Have Overvalued Points; Vermont Avenue Is a Steal. For the most part, the author is talking about how, with the addition of more words containing z, x, and q to the Scrabble Dictionary, it has thrown the carefully crafted point balance out of whack. Anyone who has played Scrabble has noticed that there are more of the common letters (e, a...) and only one each of the rare letters. Additionally, the point values are significantly higher for them to reward you for the difficulty in finding words to use them in. However, there are now more of those ostensibly rare words than were originally used... making it easier to use those letters... thereby making it easier to seriously cash in on those high value tiles. The contention by some players is that it has thrown the balance of the game off.

Or has it?

There are others that have pointed out that anyone can use those new words. Therefore, the balance isn't off at all. My follow-up comment would be that the scale of the scoring is different. When I played fairly high-level Scrabble for a while a number of years ago, a good round of 1-on-1 would be up in the 400s. We were using those words such as xi and suq and qat. After all, they are in the Scrabble Dictionary so why not? Now, if we had not been using those words and simply been using... well... simple words. Netting those 400+ points would have been marginally more difficult. The rounds may have been in the 350+ range instead. For both players.

However, this is when we view games as an aggregate. In a single game, reaching into the bag and pulling out the only letter x or the only letter z is, for all intents and purposes, a random event. If you pull out one of those singletons, by definition your opponent cannot. Therefore, the simple (random) act of pulling out a high-value letter gives you a significant advantage.

And that's the rub... novice players who don't know those letters look at the z and q and x as a handicap. ("What am I going to do with this damn thing?!?") Experienced people who fall asleep with the Scrabble Dictionary look at those letters as a fortuitous occurrence... and possibly a clinching one as well. The response is going to be more along the lines of "Ha! I got it and he didn't!"

So... while one could make the claim that it is only the scale of the scores that has increased, the actual result is that it has put more weight on the random factor of what is otherwise not supposed to be a random-centric game. And that I do have a problem with.

Football and Basketball

In other parts of the article, he mentions how other scores and probabilities have changed: football field goals, basketball 3-point shots, etc. Looking at the latter, if you typically hit 50% on 2-point shots, then all you need to do is hit 33%+ on 3-point shots to make the decision to always go for them. After all, making 1 in 3 3-pointers is the same as 1 in 2 2-point shots. His comment was that, in college basketball especially where the 3-point range was shorter, many players could do that.

This is an interesting item to note as a game developer. If you were designing a game and deciding how far back to place the 3-point arc, you would specifically want to look at those very statistics. That is, at what distance does the probability of success merit increasing the payoff? If it is too close, most people will shoot from beyond it and it will take away the inside game. If it is too far away, no one will bother and will, instead, try to get as close as possible. It's a delicate balance.

Never Mind Boardwalk...

Another quick mention in the article is Monopoly. This one amused me. Back in about 1993, I whipped up a monster spreadsheet that determined that the best bet for buying and building was the light blue properties... the 2nd set on the 1st leg. This article confirmed that some dude did the same math and determined the same thing. Big deal. The difference was, I based mine on the probabilities of landing on various squares, various monopolies, etc., the cost to buy, the cost to build, and the income that you would receive. Apparently this other guy used a Monte Carlo method (32 billion random rolls) to come up with his answer. Wimp. Anyone can run a Monte Carlo simulation and get the right answer. However, you learn a lot more about the "why" of the situation by looking at the relationships of the various components. (This is a form of multi-attribute utility theory that I cover extensively in my book.)

Anyway, many of us think of game balance in the realm of RTS, TBS, or RPG games. However, even games where every unit is the same (e.g. checkers) can be an excercise in mathematical balancing. How much weight do I put on a given option? How important is it? Especially compared to other types of factors that might not be easily relatable? These are all questions that are very game-specific and can only be addressed by pondering the exact relationship in question. Therefore, there are no hints... and really, just like where to put the 3-point line, there may not even be right answers. 

(This article originally appeared on my blog, IAonAI under the title Rebalancing Scrabble and Other Games.)