RE: No bugs, just a question

[Ian Shaw]

Hi Tim, 

What an awesome explanation. Thanks a lot. 

-- Ian 

-----Original Message-----
From: Bug-gnubg <bug-gnubg-bounces+ian.shaw=riverauto.co.uk@gnu.org> On Behalf 
Of Timothy Y. Chow
Sent: 06 July 2022 14:59
To: bug-gnubg@gnu.org
Subject: Re: No bugs, just a question

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