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Real Backgammon
Posted By: Timothy Chow
Date: Monday, 26 April 2010, at 11:45 p.m.
In Response To: Real Backgammon (Daniel Murphy)
Daniel Murphy wrote:
If I were superexpertly making only a tiny 0.010 error in the first five moves of each game of a 10 game session, would it be wrong to think that I have made the equivalent of a 0.100 blunder over the course of the session?
I agree that it isn't obvious that these aren't equivalent, but I agree with Bill that they aren't equivalent.
Forget about backgammon for the moment and 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 a 0.1% chance of winning $9990 and a 99.9% chance of losing $10. After you choose a game, you play ten rounds. Are your two choices equivalent?
Both games break even in the long run, but I don't think anyone would claim that they're "equivalent." In Game B, after ten rounds, you have over a 99% probability of being $100 behind. In Game A it's impossible for you to fall that far behind after only ten rounds.
Equities tell you only what your average results will be in the long run. But it's theoretically possible for you to accumulate a significant equity deficit or surplus yet not live long enough for the "long run" to kick in.
Does what I've said apply not just in theory but also in practice to backgammon opening plays? I don't know, but it's quite possible. When we play backgammon in real life, we are effectively doing nonvariancereduced rollouts. If a particular equity difference doesn't show up reliably until you've done an enormous number of nonvariancereduced trials, then the probability that it will make a difference over the course of your lifetime may be very small. On the other hand, an equity difference that shows up reliably after a small number of trials is more likely to make a noticeable difference during your lifetime.

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