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Comparing Games of Skill and Chance

Two questions:

(1) What percentage of backgammon is skill and what percentage is luck?

(2) How does backgammon compare to other games in this respect?

Question (1) comes up a lot, but it’s a bad question. Backgammon is a game that combines luck and skill, but there’s no simple way to separate the two. If I said, for instance, that backgammon was 80% luck and 20% skill, what exactly would I mean? If someone else said that backgammon was 90% luck and 10% skill, could we devise an experiment that would prove one statement was more accurate than the other? I don’t see how.

Although the ratio of luck and skill in backgammon can’t be resolved, there is an approach that lets us compare backgammon to other games of skill and chance or even pure skill. Let’s look at how this might work.

We’ll start with chess, which has developed a successful and popular rating system over the last 60 years, thanks to the work of a Wisconsin mathematician, Arpad Elo. Chess ratings range from a high of about 2800 to a theoretical low of about zero, representing a complete beginner. Chess ratings are also designed so that a 200-point rating difference between two players anywhere on the scale means that the higher-rated player has about a 75% chance of defeating the lower-rated player.

Now consider the following experiment:

(1) Take the best player in the world. (In the case of chess, it’s Magnus Carlsen of Norway.) Call him Player 1.

(2) Find someone that the best player beats 75% of the time. Call him Player 2.

(3) Call the difference between Player 1 and Player 2 one skill differential.

(4) Find someone that Player 2 can beat 75% of the time. Call him Player 3. The difference between Player 2 and Player 3 is another skill differential.

(5) Continue this process until you have taken the chain down to an absolute beginner.

(6) Count the number of skill differentials involved. This is the complexity number of the game.

In the case of chess, the complexity number is about 14. It’s a great number, because it tells us what we really want to know: How difficult is this game? How many levels to you have to climb to move from the bottom to the top?

We can apply this process to any game, although we may need to tinker with the notion of what constitutes a single game or contest. In chess, a single tournament games usually lasts four to five hours, which seems intuitively like a good time scale for comparison across different games.

For backgammon, let’s consider a contest between two players to be not a single game lasting a few minutes but a 25-point match, which lasts about as long as a tournament chess game. By looking at some of the backgammon rating systems and extrapolating downward to include beginners, we can make an intelligent guess that backgammon’s complexity number is around 8.

For other games, we have to first think of an appropriate contest as occurring between two players and lasting in the four to five hour range. In no-limit hold’em poker, it might be a heads-up contest. In Scrabble, a best-of-five game series might fit the bill. Here’s a rough chart of how some popular games might rank on the complexity number scale. (In reading the chart, keep in mind that chess, backgammon, and no-limit hold’em are the only games where I have any real experience.)

Complexity Numbers

Go 40
Chess 14
Scrabble 10
Backgammon 8
No-limit hold’em 8
Checkers 8

I can’t hazard a guess as to the proper numbers for other worthy games like Gin Rummy and Othello. Bridge is a great game but unfortunately not a two-player game.

The nice feature of this table is that it resolves at one stroke all the muddled thinking about luck, skill, and games of skill versus games of chance. Any game with no skill, like roulette, falls automatically to zero on this scale. For other games, the relevant issue is not luck versus skill but rather the interplay of skill, chance, and complexity. Even the most complex games, like chess and go, are bounded by the human capacity to fathom their depths at the board.

(This article, in slightly modified form, first appeared in Inside Backgammon, Volume 2 #1.)

Posted: September 30, 2014 | In Backgammon Generally

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