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I refer to an article recently published here on BW: "The History of Restricted Choice" by Paul Barden, and to comments made to that article by Sergio Polini and Patrick Laborde regarding the contribution made by two Frenchmen: Emile Borel and Andre Cheron.

The short history of Restricted Choice is normally told as follows: In 1958, Terrence Reese in the "Expert Game" gave this concept the name and gave a couple of examples (KQ5 opposite A1073 and then opposite A973). Reese gave credit to Alan Truscott for having presented the concept in a magazine article. Paul Barden found the relevant two articles of the "Contract Bridge Journal", published in April and July 1954, and summarized their content here on BW.

For the record it should be stated that the concept was first identified, analyzed and solved with full numerical conclusions in the seminal book, "Théorie mathématique du bridge a la portée de tous" (mathematical theory of bridge for all), by Émile Borel and André Chéron, published in Paris in March 1940. The second edition was published in 1954 (first print), translated into English by Alec Traub and published with financial assistance from C. C. Wei in 1974.

The second edition of "Théorie mathématique du bridge" has a clear description of the Restricted Choice concept giving K9xx vs A10xxx as an example in Annex 10 (Note X), which is on Bayes Rule. It's more complicated in the first edition (in general, the first edition is much more difficult to read than the second) as Note VIII has only a small section explicitly on Bayes Rule.

However, throughout the 392 pages of the first edition, Bayes turns up again and again. Borel (who in my view is one of the greatest mathematician ever - as you will see if you have interest to continue to read) was never afraid to tackle the most complex challenges. So, instead of analyzing the opponents distribution of QJxx, he tackles QJxxxx (btw he calls x cards "Class B" cards!), analyzing the position that comes up after taking A and K, seeing three B cards and the J. He then correctly gives 65% probability that the Jx was the original distribution, because with both Q and J, he had a choice while with the J alone, his choice was restricted. See Note VI, pages 325 and 326 - although the principle is also elsewhere in the book. In terms of pure bridge, this example is close to being irrelevant but highly interesting in terms of probability application.

Anyway, Borel & Cheron are clearly the founding fathers of the principles governing probability in bridge. It's a strange coincident but they both strongly affected my personal life for reasons that have nothing to do with bridge.

Borel was a brilliant mathematician (one of the intellectuals that France produced in such great numbers around the turn of the century) and during my younger days I not only fell for his work in mathematics but also because of his bravery as a senior thinker in the French Resistance movement during the Nazi occupation for which he received highest honors. He was then already in his seventies - not an easy age for fighting under cover of darkness and when penalties of the authorities were serious indeed.

His field of speciality was probability theory and its application in the widest sense. For instance, he's the father of the strong law of large numbers (which provided the very useful proof of the infinite monkey theory: a monkey on a typewriter will eventually reproduce a Shakespeare play), which was formalized in his monumental 1914 book "Le Hazard" but based on an older article. "Le Hazard", incidentally, has the oldest analyzis I've found of an application of Bayes Rule in card games.

But it's his work on game theory that is so exciting, especially because Borel was so far ahead of everyone else. Seven years before John von Neumann established game theory as an independent subject in 1928, Borel wrote a break-through paper on the subject in 1921.

A key factor in analyzing human behavior in game theory is to study two-person poker models. This was first developed by Borel analyzing a form of poker in Chapter 5, “Le jeu de poker”, of his 1938 book, "Applications aux Jeux des Hazard". Six years later, Von Neumann presented his analysis of a similar form of poker in the seminal book on game theory — Theory of Games and Economic Behavior by von Neumann and Morgenstern (1944). Section 19 of that book is devoted to certain mathematical models of poker. In his work Borel introduced very challenging variables like "bluffing" and a player making a "mistake". In my view, the Von Neumann's model was a simple copy and paste of the Borel model. However, the Borel model was eventually judged to be flawed (and someone even said the conclusions were "false"). This was for a very strange reason. In the Borel model, a player could either fold or bet. Von Neumann added "check" as an option and his model became "better". But I was pretty sure that Borel would never make an elementary mistake like that, so in the late 1970s when I was studying the subject, I made an effort to ask all elderly Frenchmen about this issue. I was consistently told that in a two-person poker in France before the war, checking was not an option. It was only introduced after the Americans freed France.

As a further matter of bridge interest, most of us know John Nash (Beautiful Mind, a Hollywood film) and the Nash equilibrium in game theory for which he eventually received the Nobel in Economics (11 persons have so far revived the Nobel for game theory work - Borel is clearly not one of them). Some clever people have correctly pointed out that defenders in Restricted Choice situations have to play completely randomly when they have the two honors. Otherwise the Nash equilibrium cannot be achieved. But guess what: already in his 1940 book, "Théorie mathématique du bridge", Borel not only pointed out the equilibrium but he also gave exact numerical values if you know the style of the defender (it's in Note VI of the "Théorie mathématique du bridge"). For instance, the robots on BBO always take their tricks with the Q having also the K. So, if you see the K, you know they don't have the Q. And amazingly many players have also such bad habits.

Not many know that Borel was active in politics between the two world wars. But once he left as member of the French Parliament in the mid 1930s (no surprise, he was the author of the legislation for the French state lottery), he focused on the probability theory in card games. In 1938 he published his Applications aux Jeux des Hazard" and it caught the interest of one of French greatest chess players, Andre Cheron. He was his countries strongest chess player in the mid and late 1920s, won the French championship again and again and played on first board for France at the 1927 Olympics. However, in the history of chess, he will be remembered for his phenomenal work on endgames. Playing chess as a kid and teenager in the 1960s, the Cheron books (four volume in German, published 1964 - 1970) were my bible on endgames. In fact, they were so well-known that they were simply called "the Cheron" (or Le Cheron or Der Cheron).

When Cheron approached Borel to work on probabilities in bridge, he was already past his best as a chess player (in his early 40s by then, which is old in chess). But Cheron had had another equally strong passion for a long time, bridge. In fact, he wrote an article in the French chess magazine called "Chess and Bridge" (spring of either 1935 or 1936 - I haven't found it again) where he argued that bridge logic enhances chess logic. Cheron also wrote a detailed book on the Culbertson system, published 1940.

It may surprise many but major chess players and bridge players were pretty close in the 1920s and 1930s and it was fashionable among chess players to play serious bridge. The three World Champions (WC) at the time, Lasker (WC 1894-1921), Capablanca (WC 1921-1927), and Alekhine (WC 1927-1935 and 1937-1946) were all active bridge players. In fact, Lasker published many books on bridge and one of them was actually pretty decent (Das verständige Kartenspiel). A lot of other top chess players took up playing bridge. It is, finally, of some interest that Alan Truscott was an excellent chess player and played for the Oxford University team right after the war. I'm pretty sure he knew Cheron and his work well, including "Théorie mathématique du bridge".

I sincerely believe that Borel and Cheron are the true founding fathers of the mathematics of bridge, including the concept of Restricted Choice, and should be recognized as such.

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