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Real Backgammon
Posted By: Daniel Murphy
Date: Tuesday, 27 April 2010, at 2:18 a.m.
In Response To: Real Backgammon (Timothy Chow)
Tim, your Game A/Game B analogy is interesting, but how is it relevant? First, the two games have the same equity. Seems like we can we call them equivalent, ie. "equal in value or measure or force or effect," but never mind that. But in my question, one player is giving up equity every game. Second, the question of the probability of short and longterm success with a given payout structure is interesting, but I don't see the connection to losing equity by making errors in a backgammon game. I didn't say anyone would win or lose in one game or ten of backgammon. The errormaking player might break even. Or win all 10 games. But he's still giving away equity every game, regardless of results.
You write: "Equities tell you only what your average results will be in the long run."
But no, they don't tell you only that (and actually don't even tell you that, exactly, since your personal results might never be average). They also tell you how much of an edge you have (or have not) in every iteration of the game, and at every move in every game. Bill did say: "Obviously, better and best play influence the probability of the consequent results; however, they don't determine the results." That seems to be another part of the point you are addressing, but not one I addressed. I do agree with Bill there, of course. If it weren't true, I'd win 100% of my games :)
I think a scenario more relevant to my 0.010pergame question is this:
Imagine that I offer you a choice of two games to play. In Game A, you have a 50% chance of winning $1 and a 50% chance of losing $1. In Game B, you have 49.5% chance of winning $1 and a 50.5% chance of losing $1. After you choose a game, you play ten rounds. Which game would you like to play?
Game A players rate to win nothing, on average. Game B players rate to lose, on average, 1 cent per game, and 10 cents in games, because they are giving away 1% every game and blundering away, so to speak, 10% equity in 10 games. Surely many players forced to play Game B will win. But few would choose it over Game A.

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