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A made-up game to illustrate some points about skill and luck

Posted By: Timothy Chow
Date: Tuesday, 27 January 2015, at 3:02 a.m.

In Response To: A made-up game to illustrate some points about skill and luck (Phil Simborg)

First, I agree with you that "getting easier decisions" is not the kind of luck we are talking about, so let's set that aside.

Also, let's reiterate what I mean by "luckier." I mean that if we have a long series of matches, and for each match, we decide who had the greater total luck in that match (as measured by the bots), then we declare that player to be the luckier player in that match. That player might be a lot luckier, or a tiny bit luckier, but for the purposes of this discussion, we just say that the player was luckier in that match (ignoring by how much the player was luckier). Then the claim is, the more skilful player will be luckier in more matches.

This does not necessarily mean that the more skilful player is luckier, period. Typically, the more skilful player will be luckier in more matches, but in those matches, he will be luckier only by a small amount. In the remaining matches, he will be unluckier, and unluckier by a lot. So in the long run, his total luck will average out to zero.

Does this mean that the bots are defining "luck" incorrectly? Perhaps, but I would argue that the definition is reasonable. Part of the point of my made-up game is to strip away many of the complications of backgammon and show that the above effect is still there.

Here is one way to think about what is happening that may help. Let's imagine a game that is based on pure luck, that I'll call "coin-flip football." A ball is placed on the field. I flip a coin and depending on whether it comes up heads or tails, it gets moved one yard up or down the field. This continues until the ball reaches one end of the field. Now by definition, the luckier side always wins. If the ball starts in the center of the field then both sides are equally likely to win, and equally likely to be the luckier team.

Now consider the following twist: Let's start with the football closer to one end of the field, thereby giving one team an "unfair" advantage. The game is still determined entirely by luck, and each individual coin flip is still fair. But now, one team will win more often than the other. I would claim that it still makes sense to say that whoever wins a game is the luckier team for that game. But now, one of the teams is consistently luckier in more games, even though every single move of the game is perfectly symmetric and unbiased. Again, the amount by which the favored team is luckier in its lucky games is smaller than the amount by which it is unluckier in its unlucky games, so the total luck will average out in the long run. There will be just as many heads as tails over the course of a long sequence of games.

The final step is to throw some skill into the mix. I proposed one way, but it doesn't matter too much how exactly you do it. I claim that unequal skill amounts to starting the football nearer one end of the field. Even if you start the football in the middle of the field, it's going to tend to drift towards the direction that the more skilful team wants it to drift. So if we separate out the pure luck part from the skill part, the skill part determines how far down the field the ball gets moved, while the pure luck part will have the same properties that it does in coin-flip football; namely, one team will be luckier in more games. And of course, it will be the more skilful team that is luckier in more games.

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