How can we construct a useful measure of player skill from published statistics? We can’t just use a player’s win or in-the-money percentage, or even some combination of the two, as this measure. We need some number we can stick in ourMonte Carlo simulation to tell us what the probability is that a given player(orteam) P1beats anotherplayer(orteam) P2. The numberof game wins is, by itself, fairly meaningless in terms of point wins. This is like using total wealth to estimate yearly salary. Sure, there is some correlation, but retired people and professional heirs have large wealth with no earned income, whereas compulsive gamblers can have large salaries but no wealth. We needed a way to estimate the point-win percentages accurately to enable our system to accumulate wealth.
Ideally, frontons would publish statistics on each player’s point-win percentage, and thus we could use them directly. In fact, such statistics are kept but not published. The assignments of players to post positions are made by the fronton’s player-manager, who traditionally handicaps the best players by putting them in the less desirable post positions. Player-managers keep track of point-win statistics to help in scheduling compet-itive matches, but unfortunately they keep these statistics to themselves.
So what information do we have at our disposal? Via the Internet, we have the schedules and results forall the games each playerhas played for the past year. For each player in every game, we know that player’s initial post position and whetherhe finished first, second, orthird. Unfortunately, the published results don’t give the rest of the order, that is, the scores of players who don’t win money.
It would be impractical for us to watch and tabulate the points scored by real jai alai players. But we can watch simulated players and see what happens to them. We played 1,000,000 simulated jai alai games and kept track of what every player scored from each starting position. We broke these down into a table of the points scored by simulated players (each of whom has equal likelihood of scoring) according to what post position they started in and where they finished. What can we observe?
■ Winners score more points on average than second- or third-place finishers, as one would expect. But the percentage of points scored by winners varies significantly, depending on their initial post position.
Percentage of Points Won by Post Position and Outcome over 1,000,000 Random Games
Winner Placer Shower Other
Position % won total % won total % won total % won total 1 78.77% 966,964 63.52% 935,382 51.76% 774,133 27.26% 1,751,142 2 78.58% 950,603 63.29% 918,901 51.65% 766,861 26.73% 1,759,289 3 79.58% 797,630 66.16% 844,317 54.63% 729,834 27.91% 1,858,455 4 81.59% 700,431 67.75% 668,502 57.64% 665,485 29.45% 1,923,185 5 82.94% 574,348 70.32% 559,439 60.72% 610,894 30.83% 1,935,471 6 85.84% 545,218 70.18% 391,145 63.09% 509,161 31.80% 1,852,344 7 87.21% 471,896 72.05% 411,541 63.89% 414,517 31.76% 1,764,061 8 89.44% 528,948 70.41% 309,496 62.75% 346,252 29.96% 1,513,216 all 82.08% 5,536,038 66.88% 5,038,723 57.16% 4,817,137 29.48% 14,357,163
A playerstarting from the eighth post position has to win almost 90%
of his points, on average, to win the game. Life is considerably easier for players starting in the first two positions, who can win only 79% of theirpoints and expect to win the match.
■ The first two players can expect to be in the money even by winning only 51.7% of theirpoints. This means they can lose almost as often as they win, whereas post positions 6 or 7 must win almost 2/3 of their points to show.
■ Winners from post positions 1 and 2 each play about twice as many total points as winners from post position 7. This is because there are almost twice as many winners from position 1/2 as there are from position 7.
In addition, the points played by a typical position 1/2 winnerare larger because they can accumulate a few more points at single scoring before each volley counts fortwo.
Interesting. But what can we do with it? Foreach playerin every pub-lished game, we know which of the slots in the table he fell in. Thus, we could average the appropriate point-win percentages to get the number we need. Suppose Monolingual had played three games, winning in position 8, placing in position 7, and losing in position 6. His point-win average would be (89.44 + 72.05 + 31.8)/3 = 64.43%. Suppose Bilingual had the same win-loss record but always started from position 2. His point-win average would be (78.58 + 63.29 + 26.73) = 56.2%, which is not nearly as much to brag about. We should be careful, however. Averaging averages yields a meaningless number whenever the denominators are different.
Your average speed on a cross-country car trip is not the average of your speeds in each state because you may travel far more miles in Texas than in Delaware. Thus, we should weigh each component average appropriately before averaging, using distance-traveled-per-state in the car example and expected-number-of-points-played-per-outcome for our jai alai problem.
This tells us how often each playershould win points overthe course of the simulation but doesn’t completely resolve all issues. How can we compare a new pelotari who has only played three games with a workhorse who has played 400? Some form of compensation is needed. How can we account forperformance in singles matches versus doubles matches?
A stiff may accumulate a decent win total only because of being paired with terrific partners; alone such players are like lambs to the slaughter.
This must also be compensated for.
Yet another consideration is the trade-off between serving versus re-ceiving. In many racket sports, the server has a considerable advantage overthe receiver. Certain tennis players, such as Goran Ivanisevic, have rocket serves but are relatively clueless if someone manages to hit the ball back to them. This effect in jai alai is not so pronounced. In fact, it ap-pears that the receiving playerhas a small advantage overthe server. This is suggested by the minordifferences between the actual results of post positions 1 and 2 in the table on page 53. Unless you prefer a blue uniform to a red one, the only real difference between starting in 1 or 2 is that player 1 initially serves to player 2. Because post position 2 is slightly more suc-cessful in real life than player 1, presumably this server bias plays a small role in the outcome.