## The point value of a passivity November 19, 2008

Posted by tomflesher in Uncategorized.
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Sports are weird. Sometimes the things that determine the winner of a contest aren’t the on-field scores, at least not directly. Clock management, penalties, and other intangibles often end up determining the winner. How can we properly value those sorts of events? I’m going to post a brief analysis of an easy case, passivity warnings in international wrestling.

The rules of international wrestling have changed over the past few years. Right now, matches are decided under a two-of-three-periods system with multiple victory conditions. At the time I was involved in refereeing, the match was longer (two three-minute periods) and the victory conditions were simpler (a pin or a ten-point differential ended the match immediately, with the winner being decided based on points at the end of the match if neither occurred). If the wrestlers were tied after the six-minute match, they wrestled an additional three-minute overtime period, and if at the end of that period they remained tied, the match was decided on criteria. For simplicity, the only criterion I’ll consider is the one that decided most tied matches – passivities, warnings for refusing to wrestle aggressively. The wrestler with fewer passivities won the match. In addition, I’m going to consider the older rules, since the passivity criterion has been eliminated by forced scoring under the new rules.

Smart coaches managed their wrestlers to avoid passivity calls, and would often tell the wrestler with fewer passivities that he had a half-point advantage. That is, if the points were equal, the wrestler with fewer passivities would win, but a technical point scored against him would put him behind again. That seemed imprecise to me, since it didn’t take into account the marginal value of additional passivities.

I decided that each marginal passivity could be evaluated using an inverse power function of two; that is, the value of a lead in passivities was Σ(1/2n), where n is the difference between the passivity totals. A wrestler ahead by one passivity is indeed ahead by half a point; a wrestler ahead by two passivities has a lead equivalent to 3/4 of a point (1/2 + 1/4), and so on. The inverse power function captures the diminishing returns of a strategy designed to maximize a lead in passivities – the sum will never reach one, and each passivity is worth less than the one before it. However, each passivity does slightly lower the probability of the trailing wrestler making up the difference and forces him to try to score technical points, which are worth more because they’re more difficult to score.

I’m sure it’s also possible to do a finance-style discount analysis of the value of a particular lead in points based on the amount of time remaining in the match. A lead would have to be discounted based on the expected number of points to be scored by each wrestler in the time remaining, since a lead at the very beginning of the match is much less safe than a lead in the final seconds. That would require data analysis to determine the expected value of the points to be scored by the leading wrestler and by the trailing wrestler given the time remaining and the point spread. Research project idea: determine a method of discounting the value of a lead in points based on the number of seconds remaining.