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Is All Equity Equal?

Posted By: Albert Steg
Date: Thursday, 12 May 2016, at 1:44 p.m.

I have a conceptual question that I may not even frame properly, but is basically whether errors of the same magnitude are equally costly -- and whether errors within a game can be thought of as "adding up."

So, for instance, it always "feels" to me that a substantial blunder taking or dropping a cube is worse than a checker play that XG might say is the same magnitude. 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? It feels as though the bad effects of the checker error might be effaced by reply numbers, while the cube error 'persists' to the end of the game.

Or, let's say both players played perfectly but each would make a single .25 checker play blunder -- but one player made his errors somewhere in the first 3 moves, while the other always made it in one of the last 3 moves before the game reached 'gin'. Would we expect those players to perform more or less equally well? I would think so -- though it 'feels' like the later error is more determinative because there's less time to 'recover' from the blunder.

Intellectually, I feel like I'm supposed to say 'Yes. these errors should have equivalently bad effects in the long run" -- but I don't 'feel' confident it's true, so I'm hoping the experts here can help me align my brain and my gut here.

The follow-up question is a practical application. If these equity errors (cube vs. checker play, early vs. late) really are equivalent, I would expect they would also 'add up'? So, for example in a money game, let's say I have a position that is good but not a double and would be an error of -.10. (a) would my improper cube be validated if I could rely on my opponent making subsequent errors that cumulatively exceed my errors by at least .10 -- or, (b) better still, wouldn't it just be a difference in excess of .05, since the cube is on 2, doubling the weight of the equities? And of course subsequent re-cube errors would be magnified as well. If the equity losses due to checker errors and subsequent re-cube errors really do 'add up' in this way, then it would seem that quite aggressive 'incorrect' cube action against a demonstrably weaker opponent should be profitable. -- and yet I am very skeptical that this is actually the case. Giving away the cube prematurely just feels really expensive.

The most obvious place where the question is most important is in chouettes -- say I'm facing 5 players who all double me at once and it's a big .80 take, should I really drop the strongest opponent if he is skilled enough to improve the team's play by making merely a single play that is .05 better than they would have made without him because .05 * 4 other cubes = .20 ? This just seems really, really wrong to me. Obviously there is a chance that the rolls may unfold in an easy-to-play way such that an opportunity for even a .05 differential play never arises (say a bear-off double vs. an opening game double) -- but is a strong prospect of there being a single such piece of consultation really worth the full .20? In other words, should I be willing to pay 'non consultation fees' far more often than I currently do?

I've asked this of a few different players I respect and the answers haven't always been confident, and when confident, haven't always lined up. But I don't think I ever ask the question the same way twice. I'd appreciate any clarification of how one should think about this, pointers to articles, earlier threads etc. Also, I'd note I do own a copy of jake & Walter's 'Can a Fish Taste Twice as Good?' and while I admire that work a great deal, it isn't really getting at the more fundamental questions I'm trying to get at here.

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