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Posted By: Timothy Chow
Date: Saturday, 27 February 2016, at 1:26 a.m.

In Response To: A common fallacy about luck and skill (Timothy Chow)

For the purposes of constructing artificial games that are a mixture of luck and skill, it helps to have a "game of pure skill" available. This is a surprisingly tricky thing to come up with, because one intuition is that in a game of pure skill, the more skillful player should always win; however, it's also part of almost every definition of "game" that the outcome should have some degree of uncertainty.

My favorite close approximation to a "game of pure skill" is a spelling contest, in which you're given a large number of difficult words to spell—say a thousand. You score a point for every word you spell correctly, and lose a point for every word your opponent spells correctly. (You're both given the same words but have to produce your spellings in isolation from each other.) Most people feel that this still deserves to be called a "game." Not everyone will agree that it's "pure skill" because presumably the words are selected randomly, and some people might want to use the word "luck" to describe the variation in your ability from one day to the next (e.g., did you have a bad night's sleep?). However, with a large number of words, the luck of the word selection mostly averages out, and the more skillful player (i.e., the one who knows more words) will almost always win. Certainly, it's possible to come up with a lexicon and a pair of players for which the more skillful player will win nearly 100%.

Anything that is more of a "game of pure skill" than such a spelling contest is likely to have such a predictable outcome that it won't seem like a "game." (I'd be interested if someone thinks they can find a better candidate, though.)

Now that we have a game of pure skill that we can use as a component for our construction, the rest is relatively easy. Call our contestants Alice and Bob.

1. Hold the spelling contest first. Let's assume that Alice wins, with a score of E (to remind you of the word "equity"!). With a thousand words, –1000 ≤ E ≤ 1000.

2. Draw a number line and mark the numbers from –1000 to 1000 on it. Put a counter at E.

3. If E = 1000 then declare Alice to be the winner and terminate the game.

4. Otherwise, the pure luck part of the game kicks in. Repeatedly flip a coin and move the counter right by one unit if it lands heads and left by one unit if it lands tails. Keep going until the counter hits either –1000 (whereupon Bob is declared the winner) or 1000 (whereupon Alice is declared the winner).

The key point is that if we define "luck" analogously to the way backgammon bots do, then Alice's luck will be the number of heads minus the number of tails. By definition, Alice is luckier if and only if the total number of heads exceeds the total number of tails.

Now we're basically done. We can select Alice and Bob so that Alice typically gets all 1000 words right and Bob typically gets a couple of words right. E starts very close to 1000, so the more skillful player, Alice, almost always wins. Furthermore, the luckier player always wins, except in the rare instances when the game terminates before the pure luck stage begins and neither player is luckier.

Of course, this is a contrived game that would never fly in the real world in exactly the form stated here (though I'd argue that some game shows have a similar format, albeit less extreme). But it illustrates the point that it's possible to have the skillful player win almost all the time and to have the luckier player win almost all the time.

Now let's think about what happens if we switch the order of the luck and the skill components. That is, we start in the middle of the number line, flip the coin 999 times (say), and then switch to having Alice and Bob spell words to move the counter. What will happen now is that Alice will still win almost all the time—the counter will zoom to Alice's end once she starts spelling. But now, both players are equally likely to be luckier. In particular, Alice, the inevitable winner, will be luckier half the time, so the luckier player will win 50% of the time.

This second game is also a poor candidate for a game to play in the real world, but now that we understand the principle, we can mix things up a bit. By alternating luck and skill phases appropriately (as in backgammon for example!), we can arrange for the luckier player to win X% of the time for any desired value of X between 50 and 100.

To restate the main point, tracking how often the luckier player wins should not be used as a surrogate for measuring "how much skill" there is in a game. Unless, of course, you're trying to sucker a fish into playing with you.

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