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BGonline.org Forums
Calculation
Posted By: Timothy Chow In Response To: Calculation (Ray Kershaw)
Date: Friday, 10 March 2017, at 6:12 p.m.
I'm about to leave on a trip and won't be able to post here for a while, so let me try to summarize, even though the dust has perhaps not quite settled.
1. "Perfect play" has a clear meaning when there is an absolute cap on the cube value (e.g., in a match).
2. In a money game, if the cube value is truly unlimited, the standard way to try to define "perfect play" is by defining a player's payoff in terms of the equity or expected value of a position. The catch is that the expected value of a position is defined as an infinite sum, and this sum might not converge—it might diverge to infinity, or it might oscillate as you add more and more terms, without settling down to a specific value. In that case, the equity of the position is undefined and then the term "perfect play" has no clear meaning.
3. The symmetric position that I posted is an example of a position with undefined equity. The symmetry allows us to give a proof of undefined equity that is very close to being mathematically rigorous. The catch, however, is that it can never arise in an actual backgammon game. So you might hope that positions with undefined equity never arise in actual backgammon.
4. Bob Floyd was the first to publish a position with (allegedly) undefined equity. The argument that Floyd's position has undefined equity is not quite mathematically rigorous, but is highly plausible. Importantly, Floyd's position has the property that it could arise in an actual backgammon game, dashing the aforementioned hope that positions with undefined equity could never arise in actual play.
5. You could still hope that maybe Floyd's position, and all positions with undefined equity, cannot be reached through perfect play. Maybe we can identify a subset of positions that all have clearly defined equities, and maybe we can define "perfect play" in terms of these positions. This is what we're still debating, but I think that this is also going to turn out to be a vain hope. The trouble is that when you reach a fork in the road and have the option of stepping off into undefined-equity-land by making Play A, on what basis are you going to say that Play A is bad and you should make Play B? I think that there isn't any satisfactory answer to this objection. If I'm right about that, it means that the problem of undefined equity will propagate all the way back to the starting position, and perfect play for money games with an unlimited cube will have no rigorously defined meaning.
6. Again, none of this has much practical significance since all these theoretical difficulties disappear if we stipulate that the cube value is never allowed to exceed (for example) 1 trillion, and in practice nobody will object to such a stipulation.
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