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BGonline.org Forums
SMITH and JONES: The amazing "FARE" position
Posted By: Timothy Chow In Response To: SMITH and JONES: The amazing "FARE" position (Nack Ballard)
Date: Saturday, 4 March 2017, at 8:29 p.m.
Nack wrote:
Let's make sure we agree on terminology. As I would define it, "perfect play" is another way of saying "best play" (with the checkers or cube), or a "series of plays that are all best." The concept of "perfect" or "best" is typically arbitrated by the best bot rollout data that is practically available. You define it the same way, yes?
Technically, I would define "perfect play" as an "expectiminimax equilibrium play," with the tacit assumption that equities are defined. In something like chess or Othello, where there are no dice, we can drop the "expecti-" prefix and refer to the minimax equilibrium. This means that we're choosing the play that minimizes our opponent's payoff, assuming that our opponent is following the same policy. As this description makes clear, there is a certain amount of circularity in this definition, and to be logically precise, one must be careful to make sure that the circularity is not vicious, but eventually bottoms out. Since chess and Othello have finite game trees, one can always start at the terminal positions and work backwards to define the minimax strategy in a non-circular fashion.
In backgammon, the dice introduce an additional complication. The payoff is not just +1 or –1 because there is randomness. In a match, we can address this by defining the payoff to be the probability of winning the match. Then we can talk about making the play that maximizes the probability of winning the match, given that our opponent is following the same strategy. Proving that this definition is logically sound is a little trickier. It is helpful that one can show that starting from any backgammon position, the probability is 1 that the game will terminate no matter how the players play. We can then show that there must exist a way of assigning win probabilities to every position that satisfies the minimax condition. I have to admit, though, that I've never thought about how to prove that there is a unique way of assigning win probabilities to every position that is consistent with the minimax condition. Surely this is true, though. Assuming it is true, then we again have a clearly defined notion of "perfect play."
Money-game play with an unlimited cube poses yet another complication, which finally brings us back to the original topic. Now the only plausible candidate for "payoff" is the equity or average payoff. The problem now is that an unlimited doubling cube can, in principle, cause the equity of a position to be infinite (or perhaps more precisely, undefined).
As has been demonstrated, my FARE position can be reached from the opening position through perfect/best play. Do I understand correctly, then, it would follow from your statement (italicized above) that the concept of perfect/best play "goes out the window" in the opening position as well, and indeed in any normal-looking backgammon position?
For example, most of us take for granted that making the 5pt is the best play with an opening 31. Is that "out the window," too?
In principle, yes, if we can construct a position P1 with undefined equity, and if we can show that some other position P2 has a nonzero probability (no matter how small) of reaching P1 via (alleged) perfect play, then the equity of P2 is undefined as well. And yes, in principle, I don't know of any way to rule out the possibility that P2 is the opening position, and hence that "perfect play" strictly speaking makes no sense.
In practice, there is no need to worry about these sorts of things, because we can for example place an absolute cap on the value of the cube, such as 1024 (this is what XG does). Then there will be no problem with undefined equities. Furthermore, since the probability that the cube gets up to 1024 with perfect play is so small, this cap is not going to affect any equity estimates that we get from bot rollouts in any measurable way. It's only because in this thread, you have explicitly introduced the possibility of truly unlimited cube values that I felt prompted to mention this technicality that arises.
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