To rate myself in declarer play, I'll look at each board where Jack and I declared the same contract from the same seat and average the IMP results. Obviously, when looking at an individual result, declarer-play skill is not the only determinant. Luck plays a part as well. I might take a 70% line and go down while Jack takes a 50% line and makes it, or vice versa. Luck may also take the form of each table's having a different auction that gives different information either to declarer or to the defense. But, in the long run, these factors should even out. If we look at enough boards, the average score should be a fair measure of my edge over Jack whenever I declare a hand. (When I say 'edge over Jack,' that does not necessarily imply superiority. This edge could, after all, be negative. But I certainly hope it isn't.)

Next, I'll look at each board where Jack and I defended the same contract from the same seat. I'll take the average of those results as a measure of my edge as defender.

Bidding is trickier. I toyed with various ways of deciding whether bidding was constructive or defensive. Originally, I thought of making that distinction based on the auction itself, but I realized that was wrong. "Constructive bidding," after all, involves getting to the right contract when the hand belongs to your side, and "defensive bidding" involves interfering with your opponents' ability to do the same. So the distinction really depends not on whose hand you think it is during the auction but on whose hand it

*actually*is. One objective measure of that is the balance power. Accordingly, I'll define constructive bidding as bidding on hands where we have 21 or more high-card points and defensive bidding as bidding on hands where we have 20 or fewer high-card points.

For constructive bidding, I start by taking the score on each deal where our side has 21 or more high-card points. I now normalize each score by deducting what I have determined is my edge in cardplay. If I declared, I deduct my edge in declarer play. If I defended, I deduct my edge in defense. I then take the average of these normalized scores. The result should be a fair measure of my edge in constructive bidding. I then do the same thing for hands with 20 or fewer high-card points to determine my edge in defensive bidding. If I haven't made any mistakes in my calculations, the average of my two cardplay edges plus the average of my two bidding edges should be roughly equal to my average IMP gain per board. Perhaps I should run this method by the Gargoyle quant department to make sure it makes sense. But they're busy developing trading algorithms for us, so I don't want to distract them. I did run it by Jeff Rubens. He was of the opinion that it would take an enormous number of boards for the extraneous factors to average out to zero. He may be right, but if so we should be able to tell that by calculating the 95% confidence levels.

Before I calculate the averages, let me make some predictions. I suspect that my biggest edge over Jack will be in defensive card play. The way Jack plays is to generate random deals and to select the card that works most often across all these deals. The difficulty in this method is in choosing constraints for generating the deals, particularly constraints derived from other players' decisions or, what is often more important, from their failure to make certain decisions. "Why didn't declarer attack clubs at trick two?" "Why isn't declarer trying to ruff a heart in dummy?" "Why did declarer come to his hand to play trumps instead of playing them from the dummy?" These are the kinds of questions expert defenders are always asking themselves. Jack is, so far as I know, incapable of this kind of detective work. And defensive card play is where detective work is most important. Some of my edge disappears because I have Jack as a partner, and our ability to communicate is limited. (I don't mean just in signaling. I also mean communicating simply by the act of adopting one defense over another.) If I were defending with a human partner, I would expect our edge on defense would be enormous.

I suspect my next biggest edge over Jack will be in declarer play. Again, the ability to draw both positive and negative inferences is critical. But there are a number of declarer problems where there are few inferences to draw and where you must simply choose the highest percentage action among a variety of choices. Jack may well find those problems easier than I find them, particularly if the constraints are hard to quantify.

Next would be defensive bidding. Defensive bidding requires an understanding of how alternative actions on your part might make life easier or harder for your opponents. I don't think Jack has this understanding. Constructive bidding, on the other hand, I would expect Jack to be quite good at. Generating random deals for partner consistent with his auction and determining what your percentage action is over that set of deals is something Jack should be able to do faster and better than I do. Where I might still have an edge is in determining more accurately what "consistent with his auction" means. Jack is better at positive inferences than at negative ones. I also think Jack isn't good at factoring in how partner will react to an auction you are considering. It's one thing to conclude that you have an 80% chance at a game. It's another to conclude that you should still just invite because your chance opposite hands where partner would refuse your invitation is only, say, 30%. Jack seems just to bid game in those situations. At least that's been my impression.

Thirty-two boards isn't a large sample. But let's see where we stand anyway. Here are the averages, along with their 95% confidence levels:

**Summary of Boards 1-32**

Category | Average IMPs |

Defensive play | +1.9 ±2.7 |

Declarer play | +0.8 ±1.6 |

Defensive bidding | +1.5 ±2.9 |

Constructive bidding | +0.9 ±2.7 |

Total | +2.4 ±2.0 |

As you can see from the large confidence levels, this really doesn't mean anything yet. I'll post these results again after 64 boards.

not sure I understand your math. Are we talking margin of error? Can you please explain the numbers better. Is 2.4 the actually avg imps won per board ? 2.0 being the margin of error? if thats the case how does this relate to 95% ? I am not surprised that it was the defensive stuff that won you the majority of imps . From what I remember you had significant edge in defensive bidding and in defensive carding as well. As I have mentioned before , Jack was incapable of solving what should have been double dummy problems on defense and he showed no imagination or consistency in the defensive bidding category.

ReplyDeleteOne thing though, Jack is miles ahead of BBO's version of GIB

+2.4 is my average IMP score per board (+76 divided by 32). Plus or minus 2.0 is the 95% confidence level. That means if we played an infinite number of boards (I'm getting tired even thinking about that), there is a 95% probability that my average IMP score per board would be between +.4 and +4.4. To calculate that, you multiply 1.96 times the standard deviation of the sample (5.7) and divide by the square root of the size of the sample (32). This assumes the distribution is Gaussian. See how useful an MBA is? Just don't ask me to derive that formula.

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