In strategic situations it is often important that a person's actions not be predictable by his opponent(s). Examples abound in sports: The pitcher who "tips" his pitches is usually hit hard. Batters who are known to "sit on" one pitch usually don't last long in the major leagues. Tennis players must mix their serves to the receiver's forehand and backhand sides; if the receiver knew where the serve was coming, his returns would be far more effective. Point guards who can only go to their right don't make it in the NBA. The principle is just as important outside of sports: for example, a regular participant in sealed-bid auctions will not win many contracts if his competitors always know what his bid is going to be.

An important insight, due to the great physicist and mathemetician John von Neumann in the 1920s, is that strategic unpredictability should be systematic -- the unpredicability should actually be predictable, in a probabilistic sense, and it should depend upon the particular nature of the strategic situation in question. For example, in a tennis match there is a specific rate at which the server should mix serves to the forehand with serves to the backhand, and this "mixing rate" depends upon the particular abilities of the server and the receiver. Of course, the server must still implement his "mixing rate" randomly: if the correct rate is one-third of the time forehand and two-thirds backhand, it wouldn't work very well to just hit one to the forehand, then two to the backhand, over and over. That's just as predictable as hitting them all to the forehand. For another example, when the theory is applied to poker it tells us that in order for a player to play well he must bluff a certain proportion of the time (as good poker players already know), but it also tells us that there is a correct rate at which to bluff, and that this rate depends on one's oppponents -- and of course it tells us that the bluffing must be carried out randomly.

Many experiments designed to test this theory of "mixed strategy play" with human subjects have been carried out over the past 40 years or more. The theory has not fared well. It is our contention, however, that even if the strategic situation (the "game") in an experiment is extremely simple and the subjects understand it well, they are not experts who have developed an understanding of the strategic subtleties of playing the game, and that this may account for much of the theory's failure. Indeed, it may simply not be possible to become very skilled in the limited time frame of an experiment, and in any case the amounts of money involved may not be large enough to motivate the subjects to try to become expert quickly.

Professional sports, on the other hand, provide us with strategic competition in which the participants are very highly motivated, and in which they have devoted their lives to becoming experts at their games. While their recognition of the "right" way to mix may be only subconscious, any significant deviation from the correct rate of unpredictability is likely to be pounced upon by an opponent.

In the paper "Minimax Play at Wimbledon" we use classic Grand Slam tennis matches, together with some game theoretical results we have derived in another paper, to provide an empirical test of the mixed strategy theory. Statistical analysis of the play of John McEnroe, Bjorn Borg, Boris Becker, Pete Sampras and others establishes that their patterns of serving are consistent with the theory. Conversely, when we apply the same statistical tests to the data from experiments on mixed strategy play, we soundly reject the assumption that the experimental subjects were playing as the theory predicts. Thus, it appears that the theory may be useful for understanding and predicting strategic behavior by highly motivated and expert competitors, even if it is not successful at accounting for behavior by inexperienced laboratory subjects.