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BGonline.org Forums
Is All Equity Equal?
Posted By: Timothy Chow In Response To: Is All Equity Equal? (Albert Steg)
Date: Friday, 13 May 2016, at 12:02 a.m.
To keep the discussion focused, let's restrict ourselves to money play rather than match play. Furthermore, let us agree to use unnormalized equity. That is, if the cube is on 4, then a checker-play error that the bots report as being a –0.100 error should be thought of as a –0.400 error. Though ah_clem claims that "nobody" does this, that is not true. There are times when unnormalized equity is appropriate to consider, and this is one of them.
The crucial fact, which is typically misunderstood by almost everyone in the backgammon community, is that the "equity" reported by the bots applies only to equilibrium play (often misleadingly referred to as "perfect play"). That is, it is calculated under the assumption that both players will play just like the bots do. It is therefore at best a rough approximation to the amount of money that you expect to win (or lose), given that position. To make sure we keep the two concepts distinct, let's introduce the terms equilibrium equity and actual equity.
Equilibrium equity is a clear concept, if we set aside pedantic worries about possibly unbounded cube values or the fact that the bots don't understand certain crazy positions. Actual equity is a fuzzier concept. If we try to spell out what it means, we find ourselves imagining playing out the exact same position infinitely many times with exactly the same players with exactly the same fatigue levels and memories and emotional states and ages and lighting conditions, etc., and averaging over all the outcomes. Of course, taken literally, this is nonsense. However, that's the kind of thing we implicitly have in mind when we talk about "the probability that my opponent will drop the double." If we say that this probability is 30% then we're imagining infinitely many parallel universes, with the opponent dropping in 30% of them and taking in 70% of them.
Actual equity becomes a little less nonsensical if we can contruct plausible mathematical models of the players. By making a lot of observations of the players under lots of conditions, we might be able to abstract from these observations and create a robot that plays very much like the player. The robot players can then be programmed to play out positions many times, and the results can be averaged, leading to an approximation of actual equity.
In the real world, technology has not yet advanced to the point where sophisticated models of individual players can be reliably constructed. We have to rely on our intuition, for the most part. But I spell all this out in detail because these implicit assumptions about equity are rarely made explicit, and making them explicit is necessary if we are to develop well-founded answers to your questions.
One of your first questions concerns players that play like bots except that they make a single error. The main remark I would make here is that there is a distinction between making a single error in one particular rollout trial and playing like a bot under all conceivable circumstances except one. The amount of actual equity lost by an error is an average over all possible future outcomes, and if some of those hypothetical outcomes involve deviations from the equilibrium, then they will affect the actual equity, even if no deviations from equilibrium happen to materialize in the single rollout trial that you happen to observe.
You wrote:
So, for instance, if a prop was played where both players played 'perfectly' with the exception that along the way one player would make a .20 checker play blunder while the other made a .20 initial cube-decision error, would we expect over time for the two players to break even?
You need to say a little more for this to make sense—something along the lines of, Player A plays in such a way that the expected loss in equilibrium equity per game is 0.20 from checker plays and Player B plays in such a way that the expected loss in equilibrium equity per game is 0.20 from cube errors. In such a situation, the two players would indeed break even. However, your intuition that the situation is not quite symmetric does have some merit. Opportunities to make cube errors and opportunities to make checker-play errors are not equally distributed. If you were to record a bunch of matches between these two hypothetical players, the typical game might involve multiple checker-play errors by A, with very few indistinguishable-from-equilibrium games, whereas B might play a number of indistinguishable-from-equilibrium games, interspersed with games with (say) a long sequence of missed doubles and the occasional gigantic take/pass error. A game in which both players make exactly one non-equilibrium play of size 0.20 might be quite rare.
Furthermore, if we turn the situation around, and instead of hypothesizing from the outset that A and B throw away 0.2 equilibrium equity per game on average, we try to infer a model for A and B after observing a single game between them in which A made just one non-equilibrium checker play and B made just one non-equilibrium cube play, then it's not clear that we would model A and B as playing that way every single time. So it's a pretty shaky inference, from just one observation, that everything would average out in the long run.
Anyway, the point is that the questions you're asking are much more subtle than they seem at first glance. Accurately analyzing them is a non-trivial task, and that may explain why you, and others you have asked, are unsure about them.
I won't try to answer at this time your questions about practical chouette play because that's also a complicated issue. I'll just note that the additivity of actual equity is probably the least bad of your assumptions. What I suspect is troubling you is the difference between the short run and the long run. It is quite common to be faced with a choice between Play A and Play B where Play A offers higher actual equity but the most likely outcome of Play A is worse than the most likely outcome of Play B. If you're in it for the long haul then you do well to aim for equity maximization, but if you don't play very often then you may wish to maximize your probability of coming out ahead instead.
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