This book reports our attempt to model the outcome of jai alai matches, not horse racing orfootball orany othersport. The critical aspect of jai alai that makes it suitable forourkind of attack is its unique scoring system, which is unlike that of any other sport I am aware of. This scoring system has interesting mathematical properties that just beg the techno-geek to try to exploit it. For this reason, it is important to explain exactly how scoring in jai alai works.
As a pari-mutuel sport, jai alai has evolved to permit more than two players in a match. Typically, eight players participate in any given match.
Let’s name them 1, 2, 3, . . . , 8 to reflect their position in the original order of play. Every point is a battle between only two players, with the active by pairdetermine by theirpositions in a first-in, first-out (FIFO) queue.
Initially, player1 goes up against player2. The loserof the point goes to the end of the queue, and the winnerstays on to play the fellow at the front of the line. The first player to total (typically) seven points is declared the winnerof the match. Because seven is exactly one point less than the numberof players, this ensures that everyone gets at least one chance in every match. Various tiebreaking strategies are used to determine the place and show positions.
Let’s see some examples of how particular games might unfold. We start with Example 1, a game destined to end in a 5–1–3 trifecta. The left side of each line of the example shows the queue of players wait-ing theirturn to compete. The two players not on this queue play the next point. As always, player1 starts against player2, and everybody be-gins with 0 points. Suppose player1 beats player2 (the event reported on the centerof the first line). Afterthis event, each player’s updated score is shown on the right side of the table. Player 1 collects his first point and continues playing against the next playerin line, player3. The loser, player2, sulks his way back to the bench and to the end of the player queue.
Continuing on with this example, player1 wins his first three points be-fore falling to player 5. For the next three points nobody can hold service, with 6 beating 5, 7 beating 6, and 8 beating 7. The survivor, 8, now faces the playersitting at the top of the queue, player 2, the loserof the opening point.
Here, the scoring system gets slightly more complicated. If you stop to think about it, a problem with any queue-based scoring system is that
EXAMPLE 1. A Simulated 5–1–3 Trifecta Illustrating Spectacular Seven Scoring
Queue Point Score
1 2 3 4 5 6 winner 1 2 3 4 5 6 7 8
3 4 5 6 7 8 1–beats–2 1 0 0 0 0 0 0 0
4 5 6 7 8 2 1–beats–3 2 0 0 0 0 0 0 0
5 6 7 8 2 3 1–beats–4 3 0 0 0 0 0 0 0
6 7 8 2 3 4 5–beats–1 3 0 0 0 1 0 0 0
7 8 2 3 4 1 6–beats–5 3 0 0 0 1 1 0 0
8 2 3 4 1 5 7–beats–6 3 0 0 0 1 1 1 0
2 3 4 1 5 6 8–beats–7 3 0 0 0 1 1 1 1
3 4 1 5 6 7 8–beats–2 3 0 0 0 1 1 1 3
4 1 5 6 7 2 3–beats–8 3 0 2 0 1 1 1 3
1 5 6 7 2 8 3–beats–4 3 0 4 0 1 1 1 3
5 6 7 2 8 4 1–beats–3 5 0 4 0 1 1 1 3
6 7 2 8 4 3 5–beats–1 5 0 4 0 3 1 1 3
7 2 8 4 3 1 5–beats–6 5 0 4 0 5 1 1 3
2 8 4 3 1 6 5–beats–7 5 0 4 0 7 1 1 3
– – – – – – 5–1–3 5 0 4 0 7 1 1 3
the players whose initial post positions are at the bottom of the queue start out at a serious disadvantage because the earlier players have more opportunities to collect points. In order to reduce the disadvantage of late post positions, the SpectacularSeven scoring system increases the reward foreach winning volley afterthe seventh physical point from one to two points.
Ourillustrative game is now at the midgame division line. Player8 goes against player2 and scores on a well-placed chic-chac. Afterwinning the previous point, player 8 had a score of 1. Because the contest against 2 was the eighth physical point, it counts twice as much as before, and thus the total score forplayer8 goes from 1 to 3. Player3 comes off the queue to win the next two points, giving him a score of 2× 2 = 4.Player1,thewinner of the first three points, now steps forward and dethrones the current leader, giving him a total score of five points. Because the first player to get to seven points is the winner, player1 needs only the next point forvictory (rememberit counts for2). But number5 is alive and knocks player1 to the end of the line. Now with a total of three points, player 5 continues on to beat his next two opponents, giving him a total of seven points and the match. Player1 (with five points) and player3 (with fourpoints) stand alone forplace and show, resulting in a 5–1–3 trifecta.
We have now seen two aspects of the SpectacularSeven scoring system. First, it is ruthless. By losing a single volley, the leading player can be sent to the end of the line and may neverget anotherchance to play. Second, point doubling improves the chances for players at the bot-tom of the queue, particularly player 8. Players 1 or 2 would have to beat theirfirst seven opponents to win without evergoing back to the queue, whereas player 8 only has to win his first four volleys (the first of which counts for1 and the last three of which count fortwo points each). Because it is rare forany playerto win seven in a row, the early players are penalized, and the system is supposed to even out.
The SpectacularSeven scoring system was introduced in the United States in the 1970s to speed up the game and add more excitement for bettors. Most games last from 8 to 14 minutes, allowing for 15 matches a night with enough time to wagerin between matches. The ratio of action-to-waiting is much betterin jai alai than horse racing because each race lasts only 2 or3 minutes. The SpectacularSeven scoring system apparently emerged from a research project at the University of Miami, meaning that I have not been the first academic to be seduced by the game.
EXAMPLE 2. A Simulated 1–2–7 Trifecta Illustrating Tiebreaking for Show
Queue Point Score
1 2 3 4 5 6 winner 1 2 3 4 5 6 7 8
3 4 5 6 7 8 1–beats–2 1 0 0 0 0 0 0 0
4 5 6 7 8 2 1–beats–3 2 0 0 0 0 0 0 0
5 6 7 8 2 3 1–beats–4 3 0 0 0 0 0 0 0
6 7 8 2 3 4 5–beats–1 3 0 0 0 1 0 0 0
7 8 2 3 4 1 6–beats–5 3 0 0 0 1 1 0 0
8 2 3 4 1 5 7–beats–6 3 0 0 0 1 1 1 0
2 3 4 1 5 6 7–beats–8 3 0 0 0 1 1 2 0
3 4 1 5 6 8 2–beats–7 3 2 0 0 1 1 2 0
4 1 5 6 8 7 2–beats–3 3 4 0 0 1 1 2 0
1 5 6 8 7 3 4–beats–2 3 4 0 2 1 1 2 0
5 6 8 7 3 2 1–beats–4 5 4 0 2 1 1 2 0
6 8 7 3 2 4 1–beats–5 7 4 0 2 1 1 2 0
tiebreaker 7–beats–4 7 4 0 2 1 1 4 0
– – – – – – 1–2–7 7 4 0 2 1 1 4 0
In Example 1, place and show were easily determined because the second and third highest point totals were unique. This is not always the case. Oursecond example shows a match in which two players are tied forsecond at the moment player1 has won the match. In this case, a one-point tiebreaker suffices to determine place and show. In general, tiebreaking can be a complicated matter. Consider the final example in which four players simultaneously tie for third place. The complete rules of the SpectacularSeven describe how to resolve such complicated sce-narios. Sometimes higherpoint totals are required in SpectacularSeven matches; forexample, the target is often nine points in superfecta games, allowing win-place-show-fourth wagering. The system naturally extends to doubles play by treating each two-man team as a single two-headed player.