Re: No bugs, just a question

[Timothy Y. Chow]

There is a subtle point about luck that is not well understood even by 
some professional mathematicians.

In a series of games (or matches, but for simplicity let me focus on 
games), one must distinguish between

1. counting the number of games in which I was luckier, and

2. determining who was luckier overall on a roll-by-roll basis.

The distinction is subtle, but very important, so let me belabor it a bit. 
For #2, what we do is to examine each roll individually, note how lucky it 
was, and add up the luck over all rolls in all games.  For #1, what we do 
is to examine each game in turn and, *restricting our attention to only 
the rolls in that game*, determine who had more total luck.  If I had more 
total luck in that game, then I declare that game to be a "game in which I 
was luckier"; otherwise, it's a game in which I was unluckier.

Let me emphasize that in #1, we ignore *how much luckier* I was in that 
game.  I could be massively luckier, or just barely luckier, but as long 
as I'm luckier, I declare that game to be one in which "I was luckier." 
Similarly, I ignore whether I was massively unluckier or just barely 
unluckier when declaring a game to be one in which I was unluckier.

Now here is the crucial observation.  It is entirely possible, and in fact 
common, for two players to be equally lucky in sense #2, and yet for there 
to be a highly lopsided count in sense #1.  That is, I might be unluckier 
in many more games, and yet equally lucky on a roll-by-roll basis.  This 
sounds like a contradiction, but it is not; for example, maybe I am 
unluckier in 90% of the games, but in each of those games, my net luck is 
-0.1 per game, whereas in the 10% of the games in which I am luckier, my 
net luck is +0.9 per game.  Then I am much unluckier in sense #1 but 
equally lucky in sense #2.

"Okay," you might grudgingly concede, "that's *possible*, but surely 
that's highly *unlikely*."  If backgammon were purely a game of luck, then 
you're right; it would be highly unlikely.  However, this is where skill 
comes in:

   The more skillful player will (almost always) be luckier in sense #1.

This fact is highly counterintuitive to most people.  After all, the more 
skillful player is just as likely as the less skillful player to be 
luckier in sense #2.  There is no correlation between skill and "luck #2," 
so how could there be a correlation between skill and "luck #1"?  A full 
mathematical proof is complicated, but here is the main idea: If I'm more 
skillful, then I need less luck to win the game.  In most games, I'll get 
the small margin of luck that I need to win the game, and only rarely will 
I suffer the long string of bad luck that will cause me to lose.

Here's a much simpler game to illustrate the idea: "Unfair Football." 
There's a ball on a field and it moves left and right randomly until it 
crosses one of the two goal lines.  The ball's random motion is symmetric; 
it is just as likely to move left as it is to move right.  But the game is 
unfair *because the ball doesn't start in the middle.*  The ball starts 
closer to one of the goal posts.  Obviously, the team with the unfair 
advantage will win more often, even though the motion of the ball is 
"fair" in some sense.  Although the analogy with backgammon is not 
perfect, it is pretty good; having more skill is analogous to having an 
unfair advantage in Unfair Football.  In both games, if you examine luck 
on a move-by-move basis, it is unbiased; nevertheless, one side 
consistently wins more often.

In particular, in Unfair Football, the luckier player (in sense #1) 
*always* wins.  In backgammon, the luckier player (in sense #1) does not 
always win, but we expect that the luckier player (in sense #1) will 
almost always win, for basically the same reasons.

This doesn't mean that skill is irrelevant in backgammon, any more than 
the bias in Unfair Football is irrelevant.  The skillful player will win 
more often, *because greater skill causes greater luck in sense #1* (even 
though it cannot affect luck in sense #2).  The beauty of backgammon as a 
gambling game lies precisely in this seeming paradox.  The "mark" will 
notice that the "shark" only wins when the shark is luckier, and reasons 
that luck must even out in the end.  So the mark keeps playing and keeps 
losing, because the mark does not understand the difference between luck 
#1 and luck #2.

Tim