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BGonline.org Forums
Proposed quantitative definition of skill
Posted By: Timothy Chow
Date: Tuesday, 19 June 2012, at 3:31 p.m.
Here is a proposal for a mathematical definition of skill. It is closely related to Robertie's "complexity," and is also influenced by discussions with a game-design expert friend of mine.
Suppose we have a population of players of a game. We can set up a match probability table (MPT) (name chosen in order to peeve Peever) where the (i, j) entry pi,j is the probability that player i beats player j. For simplicity, let's ignore ties, so that pi,j + pj,i = 1 for all i and j. In particular, pi,i = 0.5 for all i.
The amount of skill is then defined to be the interdecile range of all the entries in the MPT.
(That is, list the entries in the MPT in order, and subtract the 10th percentile value from the 90th percentile value.)
For example, if pi,j = 0.5 for all i and j, then the skill is 0. At the other extreme, suppose that pi,j = 1 for all i < j; then the skill is 1 as long as n is big enough.
There's nothing magical about the interdecile range; some other measure of statistical dispersion might do just as well or better, but I pick this one for concreteness.
Here are some comments about this definition.
1. This definition of skill depends on the population of players, and not just on the rules of the game itself. This feature of the definition may violate your intuition about what the word "skill" means, or ought to mean. If it does, observe the following fact: if an alleged "game of skill" is far too difficult for any human to play (e.g., "calculate the 5 billionth digit of pi" with no computational aids), then the MPT will be indistinguishable from that of a game of pure chance. You might still insist that this game is a game of skill but that the players are just too unskillful, but it's at least plausible to say that for all practical purposes, there is no skill (or that skill is irrelevant) in such a game. If you take the latter point of view, then it is natural to allow skill to be population-dependent.
2. This definition of skill is also subject to probability amplification, which is a fancy way of saying that if you play a bunch of matches instead of just one match, then the probability that the more skillful player comes out on top increases. This becomes an issue if you try to compare the amount of skill involved across different games. For example, is one game of chess at classical time controls "equivalent" to a 25-point match of backgammon?
3. Only skill has been defined, and not luck. Naturally, one can simply decree that "luck + skill = 1" and define "luck" that way. There are, however, other ways of thinking about luck. One way to define luck is as systemic randomness, by which I mean, roughly speaking, randomness in the outcome of a match that is introduced into the system by the way the game is set up, and that is independent of any characteristics of the players. For example, chess lacks any systemic randomness and thus involves no luck in this sense; however, if we define skill as I have suggested and insist that "luck + skill = 1" then chess will have luck in this latter sense.
4. What is the point of the definition? The main application I can see is in the legal realm, where courts sometimes rule against "gambling games" but are sometimes sympathetic to arguments of the form, "X is not a gambling game because it involves a lot of skill." Being able to quantify what one means by such a statement may help argue one's case in the courtroom. The above definition of skill does, I believe, capture most of what proponents of skill-based games wish to capture, namely the degree of advantage that a strong player has over a weak player. In particular, for legal purposes, the above definition of skill should be superior to a definition based on the lack (or relative lack) of systemic randomness. Backgammon has systemic randomness but we'd like to argue that it is a skillful game, whereas "calculate the 5 billionth digit of pi" has no systemic randomness but is not worth trying to protect legally.
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