On Thu, Jul 7, 2011 at 6:17 PM, Bernie Cosell <bernie@fantasyfarm.com> wrote:

2) bridge masters seemed to occasionally make counter-probabilistic plays [playing finesses and for suit splits, etc], but as a corollary to (1), he discovered that their play was actually correct for the *actual* probabilities [due to the inadequate shuffles].

I've seen this claim before, but I don't believe it. I've never seen a source for it. I've read Persi Diaconis' paper on shuffling, and it's not there. I read the bridge publications that the experts read, such as "The Bridge World", and I've never seen it there. I've talked to a friend who has represented the US in the Bermuda Bowl (the most prestigious of the world championships), and he's never heard anything like that from any experts. It also doesn't make sense from the point of view of the sort of non-randomness expected. If you play rubber bridge, many of the suits are clumped in groups of four of the same suit. If you then deal without shuffling at all, then each person gets one card from the group of four, and suit distributions will be flatter than random. But the vast majority of expert bridge is duplicate bridge, where the cards don't start clumped together in this way. Instead, we start with the 4 hands stacked one on top of the other. And almost always, the four hands are each individually shuffled before being stacked, because the protocol is to shuffle your hand before replacing it in the duplicate board. The exception is that sometimes after a session, people want to examine a previously played hand, so they pull it out of the board and sort it to examine it, and may not reshuffle after they do this. But if this was the nonrandomness being exploited, those who were doing this wouldn't just say "the hands weren't shuffled enough, so I'll expect them to be nonrandom"; they'd get information 10 times as useful if they said "We took a look at boards 14 and 18 before shuffling, so I'll expect those two to be nonrandom"; and I've never heard this as part of the claim. Also, Diaconis' paper does not show that 7 shuffles are enough for bridge. Diaconis' paper concerns itself with how many shuffles are needed to give the 52! different orderings of the deck probabilities that are sufficiently close to equal. But bridge doesn't care about the 52! orderings; it only cares about making the probabilities of the (52!)/(13!)^4 classes orderings that yield the same bridge hands sufficiently close to equal. I haven't seen any results on how many shuffles this would take, but I would strongly suspect it to be less. Also, if people wanted to take advantage of the nonrandomness of typical shuffles, I would expect poker players (and the best poker players are just as sharp mathematically as the top bridge players) to exploit this more than bridge players, for a variety of reasons, including the greater monetary rewards, and the less random shuffle (bridge players typically shuffle 4-5 times; professional poker dealers do 3 riffs and one strip). This would be fascinating if true. But I have lots of reasons not to believe it, and I've never seen a citation. Culbertson's claim that if one suit is unevenly distributed, so are the other suits, to an extent greater than the effect of the uneven distribution of open spaces left by the first suit, was widely discredited long before computer dealing. And the claim that "the queen lies over the jack" would only apply in rubber bridge, and again, anyone unscrupulous enough to use this would use the more effective principle that "the queen lies over the jack in suits in which the queen was played immediately after the jack two hands ago". Andy