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A made-up game to illustrate some points about skill and luck
Posted By: Timothy Chow
Date: Monday, 26 January 2015, at 5:28 p.m.
This is a version of something I recently posted to rec.games.backgammon. I invented an imaginary game that bears some resemblance to backgammon but is simpler to analyze. My goal was to understand better the phenomenon that the more skilful backgammon players are also luckier, according to the usual definition of "luck" in backgammon.
It is commonly stated that the explanation for this phenomenon is that skilful players "create their own luck" or "create opportunities for luck." I am not sure exactly what this means, but I gather that the idea is that a skilful play produces future positions in which more rolls play well. In other words, the key fact, it is alleged, is that what checker play you make now has a material impact on what kinds of checker-play decisions you will face in future, and will also influence the equity distribution of your future rolls.
Murat, of rec.games.backgammon fame, has long expressed skepticism about this line of reasoning. Paraphrasing, I think his point is that according to the standard definitions, a perfect neural net just makes its play based on a static evaluation of the position. Thus it does not seem that the influence of the current play on future plays and rolls should have anything to do with its "skill."
A possible rejoinder to Murat is that the imaginary perfect neural net implicitly takes into account all possible future game plans at once. Nevertheless, in what follows, I'll offer some support for Murat's position, in the sense that I am going to describe an imaginary game in which the exercise of skill on a certain turn has absolutely no influence on future rolls or tests of skill, either for me or my opponent, except insofar as unskilful play affects my equity—yet in this game, skilful players are consistently luckier. The conclusions of the following analysis should not be surprising to experts such as Maik Stiebler, but it may give pause to those who too glibly say that "skill creates luck."
My imaginary game is played on a line 10 units long, marked with the numbers from -5 (on the far left) to +5 (on the far right). The game is played with a single marker, which you can think of as representing a football. The marker starts in the middle at 0. There are two players, Red and Blue. Red's goal is to get the marker to -5 and Blue's goal is to get the marker to +5.
Red and Blue flip a (fair) coin to decide who goes first. From then on, they alternate turns. When it is Red's turn, she first flips the coin. If it comes up heads, she moves the marker one unit to the right; if it comes up tails, she moves the marker one unit to the left. If as a result, the marker lands on +5 or -5, the game ends. Otherwise, Red then has to stand at the free-throw line of a basketball court and attempt to shoot a basketball into the basket. If she succeeds, the marker stays where it is; if she fails, then the marker is moved one unit to the right, i.e., towards Blue's goal of +5. Again, if the marker lands on +5 then the game ends (with a win for Blue). On Blue's turn, he does exactly the same thing, except that if he misses his free throw, then the marker is moved one unit to the left, towards Red's goal of -5.
I'm going to make the simplifying assumption that each player's basketball skill can be represented by a single number, namely the probability of making the basket, and that each free-throw attempt is independent of everything else (so no fatigue or desperation or streak shooting or anything like that).
Now let us make some observations about this game.
1. If Red is more skilful than Blue, i.e., if Red's free-throw percentage exceeds Blue's, then Red will win more often.
2. Every coin flip by either player is equally likely to land either heads or tails. If we define Red's 'net luck' to be the number of heads minus the number of tails, then the average net luck will be zero. Similarly for Blue.
3. One important way that this game differs from backgammon is that there isn't any direct interaction between the luck part and the skill part. That is, the initial coin flip that I make when it's my turn does not influence my basketball shooting, nor does my opponent's shooting performance have any influence over mine. There is no way to formulate a "game plan" that spans several moves. All you can do on your turn is to do your best to make the basket, in isolation.
4. If Blue wins, the sum of all the moves due to coin flips (+1 or -1) and all the moves due to missed free throws (+1 for Red, -1 for Blue) must sum to +5. Similarly if Red wins, this sum must be -5.
5. Now for the key observation. Suppose at the end of a game, we say that "Red was luckier" if the sum of the coin-flip moves for both sides was negative, and similarly we say that "Blue was luckier" if the sum of the coin-flip moves for both sides was positive. Then the more skilful player will be luckier more often.
For example, I just ran a simulation of 100000 games, with Red going first 50000 times and Blue going first 50000 times. I gave Red a 95% success rate at each free throw and Blue a 90% success rate.
R won 55643 times.
B won 44357 times.
R was luckier 54911 times.
B was luckier 44637 times.
They were equally lucky 452 times.
280 times, R won despite being unluckier.
4 times, B won despite being unluckier.
So we see that the skilful player is luckier more often despite there being no direct influence of the performance of each individual skilled task on the future (other than the effect on the position of the football). There are very few games won by the unluckier player.
So we see that a large part of the reason that skilful players are luckier is simply the random-walk nature of backgammon, together with the definition of "luckier" as having greater total luck over the course of the game, without regard to the magnitude of that luck. If we take into account the magnitude of the luck, then it all balances out. Red is luckier in more games, but the amount by which she is luckier is less than the amount by which she is unluckier in her unlucky games.
Someone on r.g.b. asked about the effect of lengthening the "football field" from (-5,+5) to (-100,+100). This certainly increases the effect of skill, so as partial compensation, I dialed down Red's free-throw percentage from 95% to 92%, while keeping Blue at 90%. Simulation results:
R won 86426 times.
B won 13574 times.
R was luckier 65356 times.
B was luckier 34290 times.
They were equally lucky 244 times.
20716 times, R won despite being unluckier.
0 times, B won despite being unluckier.
So now R has a substantial chance of winning despite being unluckier, but is still luckier the majority of the time.
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