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BGonline.org Forums
Getting the bots to tell the whole truth: a concrete proposal
Posted By: Timothy Chow
Date: Saturday, 16 July 2011, at 9:35 p.m.
Longtime BGOnliners know that I have been lobbying for a while now for bot rollouts to report something better than the pseudocubeless numbers that we currently get. In this post, I will present a proposal that I hope will be clearer and more concrete than anything I have said before. I will argue for its merits and will try to address what I see as the primary objections.
THE PROPOSAL
I propose that bot rollouts allow the user to choose between two options: "Display traditional statistics" and "Display actual statistics." The option "Display traditional statistics" will produce the displays that we are currently accustomed to. The option "Display actual statistics" will present a similar display, but the three percentages that are currently labeled "wins/gammons/backgammons" will have a slightly different meaning, as follows.
1. "Wins" will report the actual percentage of rollout trials that terminate in a win for the player on roll. Specifically, a (re)double/pass will count as a win. Trials that continue until one player bears off all checkers will of course also count as wins for that player.
2. Instead of "gammons," the second number will report the percentage of double games won. By a "double game," I mean a game that is worth twice the value of the cube at the outset of the rollout. (If we are rolling out a (re)double/take, then by "the value of the cube at the outset of the rollout," I mean the value of the cube after the (re)double/take that we are studying. For the "no double" branch of the doubling decision, of course, the value of the cube is just the undoubled value.) Gammons of course count as double games, provided that the cube doesn't get turned during the rollout trial. But also, if the cube is offered and taken once during the rollout, and then terminates in a single win or a further redouble/pass, then that counts as a "double game" for the winner.
3. Instead of "backgammons," the third number will report the percentage of higher-value games, i.e., rollout trials that garner more than twice the initial cube value for the winner. This figure certainly includes backgammons, but will also include redoubled gammons, re-redoubled single wins, etc. (My preference would be for there to be a further option, "Display extended statistics," that further breaks down this category into triple games, quadruple games, etc., but I don't insist on this.)
MERITS OF THE PROPOSAL
The major merit of what I'm calling "actual statistics" is that the numbers reported are precisely the numbers that are needed to calculate take decisions. For example, suppose that a player offers an initial double at 4a4a. Should the opponent take? The standard calculation procedure is for the taker to compute his raw take point, and then compare it to the Basic Formula:
wins + (taker's g.v.)*(taker's gammons) – (doubler's g.v.)*(doubler's gammons).
(Here "g.v." abbreviates "gammon value.") However, if you try plugging the "number of gammons" reported by current bot rollouts into the Basic Formula, then likely as not you won't be able to get the numbers to add up correctly. There are two reasons for this. The first is that the bot reports pseudocubeless gammons rather than actual gammons. The second is that we haven't taken recube vig into account.
The point I want to emphasize is that both of these problems vanish if we simply use "double games" as I have defined them above in the Basic Formula. Intuitively speaking, the mathematics doesn't care whether you got two points by winning a gammon or by winning a single game after a double/take. Two points are two points. The concept of a double game takes recube vig into account automatically. Furthermore, gone are the worries that cubeless wins might not be the same as ATS wins. "Actual statistics" aren't cubeless in any way, shape or form.
Note that this also means that there isn't really much of a learning curve here in switching from traditional statistics to actual statistics. You just treat the double games the same way that you have been treating gammons all along, except now you don't have to make any further adjustments for recube vig.
Now, there is a caveat here: I've ignored the higher-value games. For take decisions at 4a4a, there are no higher-value games, but in general there may be. However, my main point still holds. To handle such situations, the Basic Formula must be adjusted by adding more terms corresponding to triple games, quadruple games, etc. I won't deny that the calculation gets more complicated, but in principle, the "actual statistics" (or rather, the "extended statistics") are still exactly what you need to compute the take decision, and they still take recube vig into account automatically. Moreover, there really isn't any other way to compute the take decision properly; any simpler calculation that you might propose is just going to be an approximation. The complexity of the decision is intrinsic, and is not a fault of the "actual statistics." Finally, I suspect that in many cases, the percentage of higher-value games will be negligible (or at least will be a wash, i.e., will cancel out on both sides), just as backgammons are currently negligible in most cases.
Stick has often mentioned that it's important to get good at estimating gammons. More precisely, what one needs to get good at is estimating double games, since that's the number that has to be plugged into the Basic Formula. It's hard to get good at estimating a number if the bot won't report it. This is the motivation for my proposal.
OBJECTIONS TO THE PROPOSAL
One objection may simply be, "I don't like your new-fangled ideas. They're strange and complicated and I'm not used to them. Maybe you're right in some mathematical sense, but I'm more comfortable with the old way of doing things." As I've presented this proposal, I believe that this objection is easily met. I'm not advocating discarding the traditional statistics. They will still be available if you select the option, "Display traditional statistics."
The more serious objection is that actual statistics aren't amenable to variance reduction. When I first started thinking along these lines, I tacitly assumed that it would be straightforward to apply variance reduction to "actual statistics." But when I thought about it more carefully, I realized that this was not the case. It's still conceivable to me that there's some clever way to apply some kind of variance reduction, but at the moment I don't see it. This means that "actual statistics" will be much noisier than traditional statistics, and much longer rollouts will be needed to achieve comparable levels of statistical significance.
This is a significant issue, but I think it's not all that bad. First of all, as I have already emphasized, the traditional way of doing things will still be available, so we're not losing anything; all we're doing is introducing a new option. The new option will be very valuable for special positions, such as basic reference positions. For a reference position, we will typically be willing to spend more computer time rolling the thing out, so the lack of variance reduction should not be a deal-breaker.
Another objection is that the "actual statistics" as I've defined them might not make a lot of sense when there's an automatic recube (or an almost automatic recube). I'll have to think about this a bit more, but my first instinct is that if the recube really is automatic then a "double game" should mean twice the value of the cube after the autorecube, but otherwise the definition I gave above should stand. The trouble I see is that if the recube is almost, but not quite, automatic, then too many games might end up in the "higher-value" category. Not a theoretical problem but potentially a practical one.
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