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Chapters in anthologies, etc. Hans B. Awards: The Gerlev Award for personal efforts to promote freedom of expression in sports. Received the Gerlev Award on behalf of Play the Game for its fight for democracy, transparency and freedom of expression in world sport. Understands Norwegian, Swedish and some Italian and Portuguese.

The website www. By continuing to use this site, you consent to the use of cookies. You can find out more about our use of cookies and personal data in our privacy policy. Received the Gerlev Award on behalf of Play the Game for its fight for democracy, transparency and freedom of expression in world sport Sports Journalist of the Year, awarded by Danish Association of Sports Journalists. Play the Game Articles. For a system like ours, which relies on making lots of small bets instead of a few big ones, fixes can be written off as a cost of doing business.

Spend too much time worrying about fixes and you turn into a conspiracy theorist and then a nut case. I have the serenity to accept the things I cannot change. Other Betting Systems Betting systems have existed for as long as gambling has. A betting system is either bogus or clever, depending upon whether it is based on a sufficiently deep understanding of the given game so that there is some method to the madness. Gambling systems, even bogus ones, are always interesting to hear about because they say something about how people perceive or misper- ceive probability.

Walk into the casino and bet a dollar on black. If it wins, boldly pocket your earnings. If it wins, you are back to where you started. After each loss, keep doubling up. Inevitably, you are going to win sometime, and at that point you are all caught up.

Now you can start again from the beginning. Nothing really, so long as you have an infinitely deep pocket and are playing on a table without a betting limit. If your table has a betting limit or you are not able to print money, you will eventually reach a point at which the house will not let you bet as much as you need in order to play by this system.

At this point you will have been completely wiped out. This doubling or Martingale system offers you a high probabil- ity of small returns in exchange for a small possibility of becoming homeless. Casinos are more than happy to let you take this chance. After all, Donald Trump has a much deeper pocket than either you or I have. Set aside enough money for a ticket on the next plane to South America. Bet the rest on one spin of the roulette wheel at even money.

If you win, return the principal and retire on the rest. Otherwise, use the plane ticket. Mathematically, the key to making this work is being bold enough to wager all the money on a single bet rather than making multiple smaller bets. You pay more tax each time you re- bet the winnings, thus lowering your chances of a big killing. Lottery numbers are selected by draw- ing numbered balls from a jar, or some equivalent method. As we will see, poor random number generators certainly exist; I will talk more about this in Chapter 3.

During the Vietnam War, the U. A total of balls, each bearing one possible birthdate, were tossed into a jar, and unlucky year-olds were mustered into the army if their birthdate was selected. It was fixed for the next year, which was presumably small consolation to those left marching in the rice paddies. Although each lottery combination is just as likely to come in as any other, there is one formally justifiable criterion you can use in picking lottery numbers. For this reason, playing any ticket with a simple pattern of numbers is likely to be a mistake, for someone else might stumble across the same simple pattern.

I would avoid such patterns as 2—4— 6—8—10—12 and even such numerical sequences as the primes 2—3—5— 7—11—13 or the Fibonacci numbers 1—2—3—5—8—13 because there are just too many mathematicians out there for you to keep the prize to yourself. There are probably too many of whatever-you-are-interested-in as well; thus, stick to truly random sequences of numbers unless you like to share. Indeed, my favorite idea for a movie would be to have one of the very simple and popular patterns of lottery numbers come up a winner; say, the numbers resulting from filling in the entire top row on the ticket form.

This will not be enough to get members of the star-studded ensemble cast out of the trouble they got into the instant they thought they became millionaires. The key decision for any player is whether to accept an additional unknown card from the house. This card will increase your point total, which is good, unless it takes you over 21, which is bad. However, a sufficiently clever player does know something about the hand he or she will be dealt.

Suppose in the previous hand the player saw that all four aces had been dealt out. If the cards had not been reshuf- fled, all of those aces would have been sitting in the discard pile. If it is assumed that only one deck of cards is being dealt from, there is no possibility of seeing an ace in the next hand, and a clever player can bet accordingly. By keeping track of what cards he or she has seen card counting and properly interpreting the results, the player knows the true odds that each card will show up and thus can adjust strategy ac- cordingly.

Card counters theoretically have an inherent advantage of up to 1. Equipped with computer-generated counting charts and a fair amount of chutzpah, Thorp took on the casinos. Most states permit casinos to expel any player they want, and it is usually fairly easy for a casino to detect and ex- pel a successful card counter. Even without expulsion, casinos have made things more difficult for card counters by increasing the num- ber of decks in play at one time.

If 10 decks are in play, seeing 4 aces means that there are still 36 aces to go, greatly decreasing the potential advantage of counting. For these reasons, the most successful card counters are the ones who write books that less successful players buy. Thorp himself was driven out of casino gambling in Wall Street, where he was reduced to running a hedge fund worth hundreds of millions of dollars. Still, almost every mathematically oriented gambler has been intrigued by card counting at one point or another.

A few years ago, the American Physical Society had its annual convention in Las Vegas, during which the combination conference hotel and casino took a serious financial hit. The hotel rented out rooms to the confer- ence at below cost, planning to make the difference back and more from the gambling losses of conference goers.

However, the physicists just would not gamble. They knew that the only way to win was not to play the game. But another group of physicists did once develop a sound way to beat the game of roulette. A roulette wheel consists of two parts, a mov- ing inner wheel and a stationary outer wheel. Things rattle around for several seconds before the ball drops down into its slot, and peo- ple are allowed to bet over this interval.

However, in theory, the win- ning number is preordained from the speed of the ball, the speed of the wheel, and the starting position of each. All you have to do is measure these quantities to sufficient accuracy and work through the physics. Finger or toe presses at reference points on the wheel were used to enter the observed speed of the ball. It was necessary to conceal this computer carefully; otherwise, casinos would have been certain to ban the players the moment they started winning.

Did it work? Yes, although they never quite made the big score in roulette. Like Thorp, the principals behind this scheme were even- tually driven to Wall Street, building systems to bet on stocks and commodities instead of following the bouncing ball. Their later ad- ventures are reported in the sequel, The Predictors.

The bigger a jackpot, the more that people want to play. The pool grows very large whenever a few weeks go by without a winner. The interesting aspect of large pools is that any wager, no matter how small the probability of success, can yield positive expected re- turns given a sufficiently high payoff. If nobody guesses right for a sufficiently long time, the potential payoff for a winning ticket can overcome the vanishingly small odds of winning.

For any lottery, there exists a pool size sufficient to en- sure a positive expected return if only a given number of tickets are sold. But once it pays to buy one lottery ticket, then it pays to buy all of them. This has not escaped the attention of large syndicates that place bets totaling millions of dollars on all possible combinations, thus ensuring themselves a winning ticket. State lottery agents frown on such betting syndicates, not because they lose money the cost of the large pool has been paid by the lack of winners over the previous few weeks but because printing millions of tickets ties up agents throughout the state and discourages the rest of the betting public.

Still, these syndicates like a discouraged public. The only danger they face is other bettors who also pick the winning num- bers, for the pool must be shared with these other parties. Given an estimate of how many tickets will be bought by the public, this risk can be accurately measured by the syndicate to determine whether to go for it. Syndicate betting has also occurred in jai alai in a big way. Palm Beach Jai-Alai ran an accumulated Pick-6 pool that paid off only if a bettor correctly picked the winners of six designated matches.

This amount was more than it would have cost to buy one of every possible ticket. I found myself in high school taking a course in computer programming and got myself hooked. It was very empowering to be able tell a machine what to do and have it do exactly what I asked. All I had to do was figure out what to ask it.

I Was a High School Bookie During my sophomore year of high school, I got the idea of writing a program that would predict the outcome of professional football games. It seemed clear to me that writing a program that accurately predicted the outcome of foot- ball games could have significant value and would be a very cool thing to do besides.

In retrospect, the program I came up with now seems hopelessly crude. It first read in the statistics for teams x and y; stats such as the total number of points scored this year, the total number of points allowed, and the number of games played so far. The cham- pion Cowboys had scored points and given up , whereas the peren- nial doormat Saints had scored and given up points, each team having played 10 games. I would then adjust these numbers up or down in response to 15 other factors, such as yards for and against and home field advantage, round the numbers appropriately, and call what was left my predicted score for the game.

This computer program, Clyde, was my first attempt to build a mathe- matical model of some aspect of the real world. This model had a certain amount of logic going for it. Good teams score more points than they allow, whereas bad teams allow more points than they score. If team x plays a team y that has given up a lot of points, then x should score more points against y than it does against teams with better defenses.

Similarly, the more points team x has scored against the rest of the league, the more points it is likely to score against y. Suppose team x has been playing all stiffs thus far in the season, whereas team y has been playing the best teams in the league. Team y might be a much better team than x even though its record so far is poor.

This model also ignores any injuries a team is suffering from, whether the weather is hot or cold, and whether the team is hot or cold. It disregards all the factors that make sports inherently unpredictable. And yet, even such a simple model can do a reasonable job of pre- dicting the outcome of football games. As an audacious year-old, I wrote to our local newspaper, The New Brunswick Home News, explaining that I had a computer program to pre- dict football games and offering them the exclusive opportunity to publish my predictions each week.

Remember that this was back in , well be- fore personal computers had registered on the public consciousness. In those days, the idea of a high school kid actually using a computer had considerable gee-whiz novelty value. To appreciate how much times have changed, check out the article the paper published about Clyde and me.

I got the job. Clyde predicted the outcome of each game in the Na- tional Football League. It was very cool seeing my name in print each week and monitoring the football scores each Sunday to see how we were doing. As I recall, Clyde and I finished the season with the seemingly impressive record of — As I recall, we all finished within a few games of each other, although most of the sportswriters finished with better records than the computer.

Instead, the Inquirer included me among 10 amateur and professional prognosticators. Each week we had to predict the outcomes of four games against the point spread. The point spread in football is a way of handicapping stronger teams for betting purposes. Think back to the Cowboys and Saints football game described earlier. It would be impossible to find a bookie who would let you bet on the Cowboys to win at even-money odds because any Saints victory required a miracle substantial enough to get canonized in the first place.

Instead, the bookies would publish a point spread like Cowboys by 14 points. If you bet on the Cowboys, they had to win by at least 14 points for you to win the bet. The Saints could lose the game by 10 points and still leave their betting fans cheering. The point spread is designed to make each game a 50—50 proposition and hence makes predicting the outcome of games much harder.

We did somewhat better on the game we selected as our best bet of the week, finishing 12—8 and in second place among the touts. Clyde finished his career with 4 years of picking the results of University of Virginia football games for our student newspaper, The Cavalier Daily. Our results were about the same as with the pros.

We went 35—19—1, correct on the easy games and wrong on the hard ones. Randy was a linebacker, 6 feet 6 inches and pounds. One day I asked him what he thought of Clyde in the newspaper, not letting on that I was the man behind the program. Back to Jai Alai Every other winter or so our family migrated down to Florida for fun in the sun and a night at the fronton. This mixed record impressed upon me the benefits of finding winners for ourselves.

The more jai alai I watched, the more it became apparent to me that the Spectacular Seven scoring system exerted a profound effect on the outcome of jai alai matches. Even a cursory look at the statistics revealed that certain positions were far easier to win from than others. It was simply not the case that good teams would usually beat bad ones, because the arbitrarily chosen position from which you started in the queue made a big difference in how many chances you had to score the required points.

If a good team got a bad starting position, its chances of winning might be less than that of a bad team in a good starting position. A good team in a good starting position had a real advantage; them that has, gets. Modern business ethics teach us that, whenever you see an unfair situation, you should exploit it for as much personal gain as possible.

How could I exploit the biases of the Spectacular Seven scoring system? The simple ideas underlying my football program were simply not sufficient for such a complex reality. However, I could get a handle on the situation using the powerful technique of Monte Carlo simulation. Monte Carlo Simulations Simulations provide insight into complex problems. Simulation is used in economics, engineering, and the physical sciences because it is often im- possible to experiment on the real thing. Economists cannot play with the U.

Rather, they will make a computer model and study the effects of such spending on it. The significance of the simulation results depends on the accuracy of the model as well as how correctly the model has been turned into a computer program. There are a wide variety of computer simulation techniques, but we will employ a curious method known as Monte Carlo simulation.

However, this connection is even deeper because the whole idea of Monte Carlo simulation is to mimic random games of chance. Suppose we want to compute the odds of winning a particularly exotic bet in roulette, such as having the ball land in a prime-numbered slot either 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, or 31 three out of the next four times we spin the wheel. The most naive approach would be to watch a roulette wheel in action for a spell, keeping track of how often we win.

If we watched for trials and won 91 times in this interval, the odds should be about 1 in To get a more accurate estimate, we could simply watch the game for a longer period. Now suppose instead of watching a real roulette wheel in action we sim- ulate the random behavior of the wheel with a computer. We can conduct the same experiments in a computer program instead of a casino, and the fraction of simulated wins to simulated chances gives us the approximate odds — provided our roulette wheel simulation is accurate.

Monte Carlo simulations are used for far more important applications than just modeling gambling. A classical application of the technique is in mathematical integration, which if you ever took calculus you may recall is a fancy term for computing the geometric area of a region. Calculus-based methods for integration require fancy mathematical tech- niques to do the job. Monte Carlo techniques enable you to estimate areas using nothing more than a dart board.

Suppose someone gave you this Picasso-like painting consisting of oddly shaped red, yellow, and blue swirls, and asked you to compute what fraction of the area of the painting is red. Computing the area of the entire picture is easy because it is rectangular and the area of a rect- angle is simply its height times its width. But what can we do to figure out the area of the weirdly shaped blobs? Now suppose that I start blindly throwing darts in the general direction of the painting.

Some are going to clang off the wall, but others will occasionally hit the painting and stick there. If I throw the darts randomly at the painting, there will be no particular bias towards hitting a particular part of the painting — red is no more a dart-attractor than yellow is a dart-repeller.

Thus, if more darts hit the red region than the yellow region, what can we conclude? There had to be more space for darts to hit the red region that the yellow one, and hence the red area has to be bigger than the yellow area.

Suppose after we have hit the painting with darts, 46 hit red, 12 hit blue, 13 hit yellow, and 29 hit the white area within the frame. We can conclude that roughly half of the painting is red. If we need a more accurate measure, we can keep throwing more darts at the painting until we have sufficient confidence in our results.

Picasso Measuring area using Monte Carlo integration. Random points are distributed among the colored regions of the painting roughly according to the fraction of area beneath. It would be much better to perform the entire experiment with a computer.

Suppose we scan in an image of the painting and ran- domly select points from this image. Each random point represents the tip of a virtual dart. By counting the number of selected points of each color, we can estimate the area of each region to as high a degree of accuracy as we are willing to wait for. I hope these examples have made the idea of Monte Carlo simulation clear. Basically, we must perform enough random trials on a computer to get a good estimate of what is likely to happen in real life.

I will explain where the random numbers come from later. Building the Simulation A simulation is a special type of mathematical model — one that aims to replicate some form of real-world phenomenon instead of just predicting it. The simulated football game would start with a simulated kickoff and advance through a simu- lated set of downs, during which the simulated team would either score or give up the simulated ball.

After predicting what might happen on every single play in the simulated game, the program could produce a predicted final score as the outcome of the simulated game. By contrast, my football-picking program Clyde was a statistical model, not a simulation.

Clyde really knew almost nothing that was specific to football. Indeed, the basic technique of averaging the points for-and- against to get a score should work just as well to predict baseball and basketball scores. The key to building an accurate statistical model is pick- ing the right statistical factors and weighing them appropriately. Building an accurate football-game simulator would be an immensely challenging task.

Jai alai is, by contrast, a much simpler game to simulate. What events in a jai alai match must we simulate? All players start the game with scores of zero points each. Each point in the match involves two players: one who will win and one who will lose. The loser will go to the end of the line, whereas the winner will add to his point total and await the next point unless he has accumulated enough points to claim the match. This sequence of events can be described by the following program flow structure, or algorithm: Initialize the current players to 1 and 2.

Initialize the point total for each player to zero. So long as the current winner has less than 7 points: Play a simulated point between the two current players. Add one or if beyond the seventh point, two to the total of the simulated point winner. Put the simulated point loser at the end of the queue. Get the next player off the front of the queue. End So long as. Identify the current point winner as the winner of the match. The only step in this algorithm that needs more elaboration is that of simulating a point between two players.

If the purpose of our simulation is to see how biases in the scoring system affect the outcome of the match, it makes the most sense to consider the case in which all players are equally skillful. To give every player a 50—50 chance of winning each point he is involved in, we can flip a simulated coin to determine who wins and who loses. Any reasonable computer programmer would be able to take a flow description such as this and turn it into a program in the language of his or her choice.

This algorithm describes how to simulate one particular game, but by running it 1,, times and keeping track of how often each possible outcome occurs, we would get a pretty good idea of what happens during a typical jai alai match. Breaking Ties The Spectacular Seven scoring system, as thus far described, uniquely determines a winner for every game. However, because of place and show betting, a second- and third-place finisher must also be established.

It is very important to break ties under the same scoring system used by the frontons themselves. To show how complicated these rules are, we present for your amusement the official state of Florida rules governing the elimination games in jai alai: 1. After a winner has been declared, play-off rules to decide place, show and fourth positions vary according to the number of points scored by the participating players or teams, and shall be played according to the players or teams rotation position not post posi- tion , i.

When there still remain five or seven players or teams, all of which are tied without a point to their credit, the play-off shall be for a goal of one point less than the number of post positions represented in the play-off. When there still remain five or seven players or teams, all of which are tied without a point to their credit, the play-off shall be contin- ued until the player or team reaches the number of points desig- nated for the game.

In case of two ties, after a Winner has been declared official, and there are still two players or teams tied with the same number of points, the place position shall be awarded to the player or team making the next point, and show position goes to the loser of said point. In games where a fourth position is required, if: a Two ties remain after win and place have been determined, the show position shall be awarded to the player or team making the next point, and fourth position shall go to the loser of said point.

In case of three ties, after a Winner has been declared official, place, show and fourth positions shall be decided among the three players or teams, with the same number of points, through elimination, according to rotating position. In case of four or six ties, after a Winner has been declared official and there remain four or six players or teams, tied for place, show or fourth, or all three, play-off shall be through elimination according to their rotating position. The first two players or teams will play the first point.

The next two players or teams will play for one point, and the remaining in case of 6 players or teams will also play for one point. Winners of the above points will play additional points to decide place, show and fourth position, as the case may require. In games where a fourth position is required and no place, show or fourth position has been determined and there remain four ties, the losing players or teams of the elimination play-off return and play a final point to determine fourth.

If at any time during a play-off a player or team reaches the des- ignated number of points the game calls for, said player, or team, shall immediately be awarded the place, show or fourth position, as the case may be and the remaining players or teams shall forfeit the right to play for said position.

As you can see, the complete tiebreaking rules are insanely compli- cated. You need a lawyer, not a computer scientist to understand them. The least interesting but most time-consuming part of writing this sim- ulation was making sure that I implemented all of the tiebreaking rules correctly. I wanted the simulation to accurately reflect reality, even those rare realities of seven players ending in a tie which happens roughly once every games.

When you are simulating a million games, you must assume that any- thing that can happen will happen. By including the complete set of tiebreaking rules in the program, I knew my simulation would be ready for anything. Simulation Results I implemented the jai alai simulation in my favorite programming lan- guage at that time Pascal and ran it on 1,, jai alai games. Today, it would take no more than a few minutes of computer time to complete the run; back in the mids it probably took a few hours.

Table 3. What insights can we draw from this table? Either of the initial players is almost twice as likely to be first, second, or third than the poor shlub in position 7. The reason is that, for player 8 to do well, he must jump at the first chance he gets. The best player 8 can get on a quick run and not win is 5, which often is not enough for a place or show.

Them as has, gets. This is as it should be because 1 and 2 are players of identical skill, both of whom start the game on the court instead of in the queue. Because players 1 and 2 have very similar statistics, our confidence in the correctness of the simulation increases. However, the ratio of heads to tails should keep getting closer to 50—50 the more coins we flip.

The simulated gap between players 1 and 2 tells us something about the limitations on the accuracy of our simulation. I would be unwilling to state, for example, that player 7 has a better chance of finishing third than first because the observed differ- ence is so very small.

To help judge the correctness of the simulation, I compared its predic- tions to the outcomes of actual jai alai matches. A complicating factor is that many frontons have their matchmakers take player skill into consid- eration when assigning post positions. Better players are often assigned to less favorable post positions to create more exciting games. The actual variation in winning percentage by post position is less in practice than suggested by the simulation, and the data shows a little dip for player 4.

That matchmakers can influence the outcome of the matches is actually quite encouraging, because it suggests that we can further improve our prediction accuracy by factoring player skills into our model. Even more interesting phenomena reveal themselves in trifecta bet- ting, where the first three finishers must be picked in order.

However, as Tables 3. Trifecta 1—4—2 occurs roughly 10 times as often as 1—6—7, which occurs roughly ten times as often as 5—6—8, which occurs roughly 10 times as often as 5—8—7. Certain trifectas are 1, times more likely to occur than others! Each of these trifectas occurred over times in the course of the simulation, or more than 2. The advantages of these favorable trifectas show up in real jai alai results. These were the eight best trifectas listed above plus 1—2—5, 1—2—6, 4—2—1, 5—1—4, and the four symmetrical variants.

My simulation projected each of them should occur between 0. By contrast, the average trifecta occurs under 0. This is a significant bias that holds potential for exploitation, al- though the advantage of favorable trifectas is less pronounced than the disadvantages of unfavorable ones. That makes them at least a 10,to-1 shot, kiddies.

I would like to play poker against any- body who bets such numbers regularly. The paucity of these trifecta events is not an artifact of our simula- tion but is a phenomenon that really exists. Only 4 of the possible trifectas never occurred over this period: 5—7—8, 5—8—7, 6—7—8, and 6—8—7. These are exactly the four trifectas identified as least likely to occur by the simulation. The model expects them to happen only once every 25, games or so, meaning that our results are right on target.

For example, the 5—2—4 trifecta occurs almost three times as often as 5—3—4, and 6—3—5 occurs almost five times as often as 6—4—5. This is because neighboring players must play each other early in the game, and the loser is destined to return to the bottom of the queue with at most one point to his name. For both to do well, the point-winner has to go on to earn enough points to lock up second place and then lose to permit his neighbor to accumulate enough points to win.

This bias helps explains the Gang of Four rotten trifectas, because they all have the double whammy of neighboring high-post positions. Now we know the probability that each possible betting opportunity in jai alai will pay off. Are we now ready to start making money? Unfortunately not. A study of season statistics for players over several years reveals that their winning percentage stays relatively constant, like the batting averages of baseball player.

Thus, a good player is more likely to win than a bad one, regardless of post position. It is clear that a better model for predicting the outcome of jai alai matches will come from factoring relative skills into the queuing model.

Countless other people certainly noticed the impact of post position well before I did, including those who reported on similar simulations such as Goodfriend and Friedman, Grofman and Noviello, and Moser. Indeed, as we will see, the jai alai betting public has largely factored the effect of post position in the odds. Fortunately for us, however, largely does not mean completely. My simulation provides information on which outcomes are most likely.

It does not by itself identify which are the best bets. Payoffs are decided by the rest of the betting public. To find the best bets to make, we will have to work a lot harder. Generating Random Numbers Finding a reliable source of random numbers is essential to making any Monte Carlo simulation work. A good random-number generator pro- duces sequences of bits that are indistinguishable from random coin flips. This is much easier said than done. Generating random numbers is one of the most subtle and interesting problems in computer science because seemingly reasonable solutions can have disastrous consequences.

Bad random number generators can easily cause Monte Carlo simula- tions to give meaningless results. For example, suppose we tried a random- number generator that simply alternated heads and tails each time it was asked for another coin flip.

This generator would produce a sequence of coin flips having some of the properties of a truly random sequence. First, the real random sequence has an unbalanced number of heads and tails 27 heads versus 23 tails. This is not surprising. In fact, 50 coin flips should end up as exactly 25 heads and 25 tails only Likewise, in the real random sequence there is a run of four consecutive tails.

A sufficiently long sequence of flips should have substantial runs of consec- utive heads or tails if it is truly random. Such counterintuitive behavior helps explain why people are lousy at designing truly random-looking se- quences. No mat- ter how many games we simulated, only two different trifecta outcomes would ever be produced!

Suppose that whenever the first coin was heads, we assigned player 1 to be the winner of the first point against player 2. In either case, the outcome of the first coin flip always decides the winner of the match, and thus the results of our simulation are completely meaningless!

How, then, do we generate truly random numbers on a computer? Computers are deterministic machines that always do exactly what that they are programmed to do. In general, this is a good thing, for it explains why we trust computers to balance our checkbook correctly. But this characteristic eliminates the possibility of looking to computers as a source of true randomness. The best we can hope for are pseudorandom numbers, a stream of numbers that appear as if they had been generated randomly.

This is a dicey situation. In a roulette wheel, we start by rolling a ball around and around and around the outer edge of the wheel. Why do casinos and their patrons trust that roulette wheels generate random numbers? The reason is that the ball always travels a very long path around the edge of the wheel before falling, but the final slot depends upon the exact length of the entire path. Even a very slight difference in initial ball speed means the ball will land in a completely different slot.

So how can we exploit this idea to generate pseudorandom numbers? A big number corresponding to the circumference of the wheel times a big number the number of trips made around the wheel before the ball comes to rest yields a very big number the total distance that the ball travels.

Adding this distance to the starting point the release point of the ball determines exactly where the ball will end up. Taking the remainder of this total with respect to the wheel circumference determines the final position of the ball by subtracting all the loops made around the wheel by the ball. This is the idea behind the linear congruential generator.

It is fast and simple and if set with the right constants a, c, m, and R0 gives reasonable pseudorandom numbers. Finally, m stands for the circumference of the wheel. The mod function is just a fancy term for the remainder. Lurking within my simulation is a linear congruential generator with carefully chosen constants that have been shown to produce reasonable- looking pseudorandom numbers. Passing Paper I was a graduate student at the University of Illinois at the time I wrote this simulation.

I was living in Daniels Hall, a graduate student dorm whose layout consisted of pairs of extremely small single rooms that shared a one-seat bathroom between them. To say these rooms were small was no understatement; there was no place in my room where I could stand without being able to touch at least three walls. I had a little couch that rolled out to be a bed, after which I could not stand regardless of how many walls I was willing to touch.

The one saving grace in the tiny room was that my morning newspaper was delivered under my door, and I could pick it up and read it without ever leaving my bed. But it is the shared bathroom that is the relevant part of this story.

With two doors leading to a one-seat john, sooner or later you get to meet the other party. That year, my bathroom-mate was a guy named Jay French, a graduate student in the business school. As a future M. The Institute of Management Sciences publishes a semipopular journal, Interfaces. They have lots of articles analyzing optimal strategies related to sports. Daniels Hall rooms were so small you could only store a few back issues of anything.

But indeed, there were articles whose depth and topics were comparable to what I had done. But it is much less boring than the other journals they publish. The Spec- tacular Seven scoring system is unfair because equally skilled players have an unequal chance of winning. I tried varying the position where dou- ble points first start after the seventh point played in Spectacular Seven so as to discover the point that leads to the greatest equality.

After sim- ulating 50, games for each possible doubling point, it became clear that doubling near the beginning of a cycle is the worst time if you want to ensure fairness because the already favored first or second players are likely to be the first to emerge or reemerge from the queue.

Yet this is ex- actly what happens with the Spectacular Seven. It would be much better to double when the middle player is expected to leave the queue to play a point. You might be curious about how academic journals work. Identifying an appropriate set of referees for my article was probably somewhat difficult because there are few other academics with a clearly identifiable interest in jai alai. Instead, the editor probably sent it to experts in simulation or mathematical issues in sport.

Refereeing is one of the chores of being an active researcher. It takes time to read a technical paper carefully and write a report stating its merits and identifying its flaws. Thus, many people try to dodge the work. But peer review is the best way to ensure that journals publish only research articles that are correct and of high quality. These referee reports go back to the editor, who uses them to decide the question of acceptance or rejection. Anonymity ensures that referees are free to speak their mind without worrying that vengeance will be taken at a later date.

Referee reports contain ideas for improving the article, and thus even those papers recommended for ac- ceptance are usually revised before publication. For my efforts, I received a modest amount of glory but no money.

The authors of research papers receive no payment for their articles. A specialized academic journal might have a circulation of only or so, which is not enough to realize any significant revenue from advertising. To cover the cost of production, libraries get charged a fortune for subscriptions to academic journals, which can run from hundreds to thousands of dollars a year.

Nevertheless, most academic journals claim to lose money. This tight money issue lead to an amusing incident with this particular paper. My Interfaces article contained several graphs of statistical data related to fairness, which I had drawn and printed using typical late s computer equipment. The editor decided that the production quality of my graphs was too low for publication and that I had to hire a draftsman, at my expense, to redraw these graphs before the article would be accepted.

So I played the starving student routine. In my final letter responding to the journal I wrote as follows: Acting upon your suggestion, I found out that the university does in- deed employ a graphics artist. Since I had no grant to charge it to, they billed me at a special student rate. Thank you for your help and I look forward to seeing my paper in Interfaces. A few weeks later I received an envelope in the mail from the manag- ing editor of Interfaces.

The note said Maybe the enclosed will help you with your lunch problem. Instead of simulating 1,, jai alai games, why not simulate 10,, or more for greater accuracy? Even better, why not simulate all possible games?

In fact, there are only a finite number of possible ways that jai alai matches can unfold. Inherent in the Spectacular Seven scoring system is a tree of possibilities, the first few levels of which are shown here. The root of the tree represents the first point of any match, player 1 versus player 2. Each node in the tree represents a possible game situation.

The left branch of each node means the lower-numbered player wins the point, whereas the right branch of each node means that the higher-numbered player wins the point. The leaves of this tree correspond to the end of possible games, marking when the winning player has just accumulated his seventh point. The top of the tree representing all possible outcomes of the first four points. In such a tree, each path from the root to a leaf represents the sequence of events in a particular jai alai game.

For example, the leftmost path down the tree denotes the outcome in which player 1 wins the first seven points to claim the match. After appropriately weighting the probability of each path short paths are more likely to occur than long ones and adding these paths up, we can compute the exact probability for each of the possible outcomes. Compaq, the leading personal computer manufacturer, recently bought out DEC in a multibillion dollar deal — presumably to get hold of this proprietary jai alai technology.

Why is the program so fast to play all possible jai alai matches? Be- cause there are not so many possibilities. Moser built a program similar to that of the DEC guy and found that this game tree had only , leaves, or different possible sequences of events, even when factoring in the complicated tiebreaker rules. The number , is small in a world of computers that can process billions of instructions per second.

A typi- cal run of our Monte Carlo simulation played one million random games. This meant that we were playing many possible sequences more than once and presumably missing a few others. The net result is more work for the simulation. The primary reason was laziness. Getting the brute force program to work efficiently and correctly would have required more time and intellectual effort that the naive simula- tion. Further, we would have to be careful to do our probability computations correctly, which requires more intricacy than the simple accumulations of the Monte Carlo simulation.

The reason we could get away with this laziness is that the Monte Carlo simulation is accurate enough for our purposes. It measures the probability A portion of the tic-tac-toe game tree, establishing that x has a win from the position at the root of the tree.

Any betting system that required finer tolerance than this would not be in a position to make much money in the real world, and there is no reason to lull ourselves into a false sense of security by overoptimizing one aspect of our system. The idea of constructing a tree of all possible sequences of moves is the foundation of programs that play games of strategy such as chess. To evaluate which of the current moves is the best choice, the program builds a tree consisting of all possible sequences of play to a depth of several moves.

It then makes a guess of the value of each leaf position and percolates this information back up to the top of the tree to identify the best move. If such a program could build the complete tree of possibilities for a given game, as Moser did with jai alai, that program would always play as perfectly as possible. The game tree for tic-tac-toe is small enough to be easily constructed; indeed, a portion of this tree proving x has a win from a given position fits on a page of this book.

On the other hand, the game tree for chess is clearly too large ever to fit in any imaginable computer. Even though the computer Deep Blue recently beat the human champion Gary Kasparov in a match, it is by no means unbeatable. The game of Go has a game tree that is vastly bigger than that of chess — enough so that the best computer programs are no match for a competent human.

When playing games, it is always important to pick on somebody your own size. I received my doctorate in Computer Science from the University of Illinois with a thesis in computational geometry and found myself a faculty position in Computer Science at the State University of New York, Stony Brook. Jai alai would have to wait awhile.

As an Assistant Professor, your efforts revolve around getting tenure. Tenure gives you the freedom to work on whatever you want. You have to teach your classes, and you have to do your committee work, but otherwise what you do with your time is pretty much up to you. If I wanted to devote a little effort to an interesting mathematical modeling problem, well, nobody was going to stop me. By now my parents had retired to Florida, and each winter my brother Len and I would pay them a visit.

Each visit included an obligatory trip to watch jai alai, and so on January 17, , we spent the evening at the Dania fronton. Our second step was to convince ourselves of its infallibility. From bagel-heads like you who paid him good money for nothing.

Remember when he gave us that winning trifecta when we were kids. Pepe wins only once in a lifetime. After watching the early points play out, I looked at our ticket and thought ahead. Player 4 can then beat 8 to stop the run. Player 3 wins the next point to move into second. If 5 then wins his next three points, we are in business. Then 8 beat 1. Then 8 beat 2 to give him a total of five points.

My brother and I started giggling. Player 5 beat 6. We were laughing hysterically. Player 5 beat 7. Again we took the folks out to dinner. Each year, during our family fronton visit, I told my parents I could win at jai alai if only I spent the time to develop the method. I came back from Florida with a suntan and a determination to move forward with predicting jai alai matches by computer. The Coming of the Web To proceed with the project, I needed to provide my computer with an extensive database of jai alai statistics and results.

This meant obtaining the game schedules for a given fronton. This meant obtaining enough game results and player statistics to rank the pelotaris with some level of confidence. This meant accumulating the prices paid to bettors over enough matches to predict the payoff without making the bet.

Clearly, frontons use computers to compute payoffs and results; thus, I knew that all the in- formation I wanted must have been on some computer at some time. However, except for general season statistics, frontons had no interest in keeping detailed records on each and every game.

This means somebody at the newspaper must type them in in the first place. I figured that if I could get ahold of whoever did the actual typing, I could convince him or her to save the files for us to process. But I was wrong. Neither of these options was tenable. For jai-alai frontons, it was a no-brainer that they should provide sched- ules and results of play. Newspapers found the jai alai scores interesting enough to publish each day.

The more information frontons could provide to the betting public, the more money the betting public would provide to the frontons. Frontons on the net changed everything for this project. When several frontons started providing daily schedules and results in early , we started downloading it.

Retrieving a Web page by clicking a button on a browser such as Netscape or Internet Explorer is a simple task. It is not much more complicated to write a program to retrieve a given page without the mouse click. For more than 3 years now we have collected schedules and results each night from the Websites of several frontons, including Milford, Dania, and Miami. Dania and Milford proved the most diligent about posting this information; therefore, all of our subsequent work was performed using data from these frontons.

Our huge database of jai alai results promised to unlock the secrets of player skills — if only we could interpret them correctly. Parsing A Holy Grail of computer science is building computer systems that understand natural languages such as English. This is why there are so many computer programming languages out there such as Java, C, Pascal, and Basic. The first step in understanding any text is to break it into structural elements, which is a process known as parsing.

Parsing an English sen- tence is equivalent to constructing a sentence diagram for it the way you did in elementary school. Understanding means extracting meaning from a text, not just its structure. Understanding implies that you know what all the individual words mean in context, which is a difficult task because the same word can mean dif- ferent things in different sentences.

Understanding implies that you know what the context of the discussion is. Such complexities partially explain why it is so hard to build computers that understand what we mean instead of what we say. The field of artificial intelligence has attacked this problem for 50 years now with relatively little success. No computer has even come close to passing the Turing test. This difference between parsing and understanding language is part of the vernacular of computer science.

One of my primary missions as a professor is to teach computer science students how to program. Programming is best learned by doing, and the best students are always looking for interesting projects to hone their skills and learn new things. Properly harnessed, this youthful eagerness to build things is a terrific way for professors to get students to work for them without pay. The trick is presenting students with a project so interesting they are happy to work for free.

Building a parser for WWW files to support a gambling system was a sexy enough project to catch any undergraduate worth his or her salt. He was a soft-spoken 6-foot 8-inch Croatian with a small goatee and a tremendous eagerness to hack.

The language Dario had to parse was that which the frontons used to report the schedules and results on the WWW. For example, the start of a typical raw Milford schedule file looked like this: Milford Jai-alai Afternoon Performance - Sunday July 19, 1 Perhaps the most famous conversational program was Eliza, a s attempt to simu- late a Rogerian psychologist. Eliza briefly fooled surprisingly many people using very simple tricks that had nothing to do with intelligence. She met her match, however, when paired with another program designed to mimic a psychotic paranoid.

Eliza failed to cure him but did succeed in sending a bill. There is more subtlety to this task than immediately meets the eye. For example, how do we know that Aja—Richard represents a pair of teammates instead of the product of a modern, hyphenated marriage? The answer is that the heading above the column explains that this is a doubles match; consequently, we should be on the lookout for two players separated by a dash.

How do we know that Eggy—Fitz is the second team in Game 3 as opposed to Game 1 or 4? We know that this particular file format lists players for up to three games on each line and that the position of the names on the line implies which game is meant. The first task in parsing such a schedule file is associating which text goes with which game — a problem complicated by different numbers of games being played on different days. Each fronton posts its schedule and result information in different formats.

For example, it converted the calendar date July 19, to an absolute date 35, , which is precisely the number of days since January 1, Working with absolute dates can be much easier for computers than work- ing with calendar dates. The complexity of dealing with calendar dates was one of the pri- mary reasons for the infamous though ultimately innocuous millennium bug.

The parser also added the uniform number of each player. We main- tained a roster of all the players for each fronton, including their uniform number. After all, an unknown player named Doubles or Afternoon probably meant that the parser got confused in its interpretation of the file, meaning we had a bug to fix rather than a player to add. For example, to figure out what day of the week your birthday will fall on next year, simply add 1 to the day of the week it fell on this year, unless a leap day occurs between them in which case you must add 2.

The reason this works is that the days of a given year equals 52 7-day weeks plus one additional day to add to the current absolute date. They will kill enormous amounts of time customizing their personal computers to get everything to work just right and learn all the arcane details of the latest programming languages.

Students in this obsession phase are a joy to have around, largely because their professors have long since left it. These days, I am much more excited about finding interesting things to do with computers like predicting jai alai matches than I am in dealing with an upgrade from Windows 98 to Windows Fortunately, Dario did Windows, and a whole lot more.

Dario was particularly eager to learn the behind-the-scenes language that makes the Internet go, a programming language called Perl. Throughout most of the information age, computers spent the bulk of their time crunching numbers like predicting the weather or in business data processing doing things like payroll and accounting.

Most applications ran on expensive, mainframe computers that kept busy around the clock and charged users for every minute of computer time. Fast forward to today. We run increasingly elaborate screen-saving programs whose shimmering images decorate our desks as they protect the phosphors on our monitors. The truth is that the Internet is really about communication, not com- putation. An embarrassingly high percentage of the computing tasks associated with the Worldwide Web are basic bookkeeping and simple text refor- matting.

Perl is a language designed to make writing these conversion tasks as simple and painless as possible.

The partners have decided to continue the project after the expiration of the EU grant in April Chairing sessions on the gambling market and on senior sport at EU conference SportVision , arr. October Contributor to The National Danish Encyclopedia with several articles Various newspaper columns on sports politics. Chapters in anthologies, etc. Hans B. Awards: The Gerlev Award for personal efforts to promote freedom of expression in sports.

Received the Gerlev Award on behalf of Play the Game for its fight for democracy, transparency and freedom of expression in world sport. Understands Norwegian, Swedish and some Italian and Portuguese. Pelotas in play have been clocked at over miles per hour, which is twice the speed of a major league fast ball. The combination of hard mass and high velocity makes it a very bad idea to get in the way of a moving pelota.

Pelota is also used as the name for a sport with religious overtones played by the ancient Aztecs. Those guys took their games very seri- ously, for the losing team was often put up as a human sacrifice. Such policies presumably induced greater effort from the players than is seen today even at the best frontons, although modern jai alai players are able to accumulate more experience than their Aztec forebears.

At Milford Jai-Alai in Connecticut, this front wall is feet high and feet wide and is made of 8-inch-thick granite blocks. A wooden border the contra- cancha extends out 15 feet on the floor outside this box. The pelota makes an unsatisfying thwack sound whenever it hits the wood, signal- ing that the ball is out of bounds. A wire screen prevents pelotas from leaving the court and killing the spectators, thus significantly reducing the liability insurance frontons need to carry.

At Milford, the court is feet long, 50 feet wide, and 46 feet tall. Although courts come in different sizes, players stick to one fronton for an entire season, which gives them time to adjust to local conditions. The numbers from 1 to 15 are painted along the back walls of the court. The front court is the region near the small numbers, and the back court is near the big numbers. The lines marked 4, 7, and 11 designate the underserve, overserve, and serve lines, respectively.

The rest of the numbers function, like pin markers in bowling, that is, only as reference points to help the players find where they are on the court. In all of these sports, the goal is to accumulate points by making the other side misplay the ball. All games begin with a serve that must land between the 4 and 7 lines of the court. The receiving player must catch the pelota in the air or on the first bounce and then return it to the front wall in one continuous motion.

The players continue to volley until the pelota is missed or goes out of bounds. Three judges, or referees, enforce the rules of play. An aspect of strategy peculiar to jai alai is that the server gets to choose which ball is to be used. At each point, he may select either a lively ball, average ball, or a dead ball — all of which are available when he serves.

Once the server has chosen a ball, the receiving team may inspect his choice for rips or tears and has the right to refuse the ball should they find it to be damaged in any way. Jai alai matches are either singles or doubles matches. Doubles are more common and, in my opinion, far more interesting. The court is simply too long for any single person to chase down fast-moving balls. Doubles players specialize as either frontcourters or backcourters, depending upon where they are stationed. Frontcourters must be faster than the backcourters because they have more ground to cover and less time to react, whereas backcourtsmen require stronger arms to heave the pelota the full length of the court.

Understanding the court geometry is essential to appreciate the im- portance of shot placement. Although the ball does spin and curve, jai alai players rely more on raw power and placement than English1 to beat their opponents. If one is placed close enough to the crack in the wall, it becomes a. You will hear cries of chula every time it looks like the ball will get wedged into the crack between the back wall and floor.

The proper technique is to dive head first towards the wall, scoop up the ball, and then fling it forward from the prone position. This kill shot usually ends the point. This is the kind of shot that makes singles games boring, although it is trickier than it looks because of the spin of the ball.

As governed by the Spectacular Seven scoring system to be described later in this chapter in greater detail , the first two teams play, and the losing team goes to the end of the line as the winner keeps playing. Having eight teams in any given match greatly enlivens the space of betting possibilities.

The composition of the teams and post starting positions assigned to each player changes in each match. To help the fans and possibly the players keep everything straight, regulations require that the shirt colors for each post position be the same at all frontons. History of the Game Tracking down definitive information on the history of jai alai posed more difficulties than I might have imagined.

Unfortunately, if I were a pelotari, my nickname would be Monolingual. Therefore, most of the history reported below comes from less authorita- tive sources. Some cite legends that jai alai was invented by Saint Ignatius of Loyola, a Basque.

Others sources trace the origins of the game even ear- lier to Adam and Eve. These same legends assure us that they spoke to each other in Basque. The land of the Basques called Eskual Herria in the Basque language straddles the border of France and Spain, comprising three French and four Spanish provinces.

Legend states the Devil tried to learn Basque by listening behind the door of a Basque farmhouse. The Basques clearly are a people who did not mingle with outsiders. They defended themselves against the Phoenicians, the Greeks, the Romans, and the Visigoths. The Basque love of freedom continues today. More recently, the spectacular new Guggenheim Museum in Bilbao has put the Basque region on the map for something other than jai alai or terrorist activities. Indeed, the Basque region of Spain and France is a terrific place to spend a vacation.

A one-week trip could combine the unique architecture of Bilbao with the spectacular beaches of San Sebastian. You can drive winding cliff roads along an unspoiled rocky coast, stopping to eat fresh seafood and tapas, the little plates of savory appetizers that have spread throughout Spain but originated in the Basque country. You can stop in nearby Pamplona to see the running of the bulls made famous by Hem- ingway. And, of course, you can watch the finest jai alai in the world.

Players are not permitted to catch the ball but must hit it back immediately. The result is an even quicker game than cesta punta that is a lot of fun to watch. Played on a smaller court than cesta punta, it remains a fast-moving game with serves that can reach speeds of over 60 miles per hour.

Pala is more popular among amateur players because these clubs are considerably cheaper than baskets. Still, it amazes me that any- one succeeds in hitting a fast-moving ball with these foot-long clubs. The Spectacular Seven scoring system is in use primarily in the United States.

Much more common in France and Spain are partidos, in which two teams red and blue play to a designated number of points, usually 35 or The first player to get, say, 35 points wins the match. All championship matches are partidos.

Such matches can take hours to play, just like tennis matches. In partido betting, spectators are encouraged to bet even after the game has begun. This system is quite interesting. A bookmaker sits in the center of the room, updating the odds in a computer after each point is played.

The latest odds are immediately displayed on the scoreboard. The cashiers face the spectators with their own computer screens and a load of tennis balls. Any fan interested in placing a bet yells in Basque for the cashier to throw him or her a tennis ball, which contains a slot in which to deposit money. The fan touches his or her cheek to bet on red, or arm to bet on blue. The cashier processes this signal and the enclosed cash and returns a tennis ball with a ticket indicating the bet amount and current odds.

Setting the right odds at each point in the match presents a considerable challenge for the bookmaker. The first indoor fronton was built in in Markina, Spain. In France, the premier fronton is in Saint-Jean-de-Luz, a lovely village near the sea. Basque players dominate world jai alai. Jai alai has been played whereever Basques have lived. The Havana fronton was one of the best in the world before Castro outlawed the sport in the late s.

At least until recently, jai alai was played professionally in Italy, the Philippines, Macao, and Indonesia. Jai alai achieved international recognition when it was played in the Barcelona Olympic Games as a demonstration sport. It was almost immediately destroyed in a hurricane but then quickly rebuilt. Ten years later, in , wagering on jai alai was legalized in Florida. After the Basques, Americans constitute the largest population of pro- fessional jai alai players. Many of these players learned the sport at a long-standing amateur facility in North Miami or the more recent am- ateur fronton at Milford, Connecticut.

Jai alai underwent a big boom in the mids. At its peak in , there were 10 frontons in Florida, 3 in Connecticut, 2 in Nevada, and 1 in Rhode Island. Referenda to expand the sport to New Jersey and California failed by narrow margins, but further growth seemed inevitable. The first problem was the long and nasty players strike, which lasted 3 years start- ing in and left serious wounds behind. The strike poisoned relationships between the players and the frontons, significantly lowered the quality of play through the use of underskilled scab players, and greatly disenchanted the fans.

It was a lose—lose situa- tion for all concerned. Since , when the Florida Lottery started, the number of operating fron- tons there has dwindled to five Miami, Dania, Orlando, Fort Pierce, and Ocala , the last two of which are open only part of the year. Several prominent frontons skate on thin financial ice and are in danger of suffering the fate of Tampa Jai-Alai, which closed down on July 4, The primary hopes of the industry now rest on embracing casino gam- bling, and owners have been lobbying the governments of Connecticut and Florida to permit frontons to operate slot machines on the side.

Be- sides competition, fronton owners complain about the amount of taxation they must pay. Each fronton is owned and operated by private businessmen but licensed by the state. The fronton seats people and claims an annual attendance of over , Phone: —— Their Worldwide Web site is updated daily, which will prove crucial for the system described in this book. Open since , the Orlando—Seminole fronton seats Its new Worldwide Web site is quite slick.

Address: South U. Highway 17—92, Casselberry, Florida They aggressively promote amateur jai alai, through several schools in Spain and France and one in Miami. Address: N. Ocala serves as somewhat of a farm team for American players, and thus it is a good place to see up-and-coming domestic talent.

Highway , Orange Lake, Florida More recently, frontons have opened and closed in Acapulco and Cancun. The jai alai palace is the classiest structure on Revolucion Avenue in the tourist part of Tijuana. In front of the fronton, a statue of a pelotari with his cesta aloft strides the world. Alas, no gambling is allowed at the matches played Friday and Saturday nights in the Jai-Alai Palace, al- though there is a betting parlor next door that simulcasts games from Miami.

There are much easier ways to lose your money in Tijuana — eas- ier but ultimately less satisfying than jai alai. Address: Revolucion Ave. All told, there are about active players in the United States. The rest of this book will consistently ignore the fact that players are people. Player Pos. Height Wgt. Age Nat. Jean De Luz, Fr. This section is the only portion of this book in which we will ignore the numbers and look at the people who have stories to tell.

Just as with soccer players, it is traditional for jai alai players or pelotaris to adopt a one-word player name such as Pele. Many players use their actual first or last name. The Basques often use shortened versions of their last names, which can approach 20 letters in their full glory. Naturally, players prefer the fans to call them by name rather than uniform color or number. There are stars in jai alai as there are in every sport. Many old timers consider Erdorza Menor to be the best player of all time.

Perhaps the best American player was Joey Cornblit, known as Joey, who was a star for many years beginning in the early s. Heaving the pelota the full length of the court. Can you find it? Hint: Look in a corner of the photo. He honed his game playing summers in Spain after turning professional at age As in baseball, many of the best players throw hard.

This is almost twice the speed of a top-notch fast ball. Fortunately, the playing court is long enough to enable players and fans to follow the action. Since the s at least four players have been killed by a jai alai ball. The only U. He died a few days later. In , a champion player named Orbea was hit in the head, and he lay in a coma for weeks.

Ultimately he recovered, eventually becoming the player-manager at Dania and Milford jai alai. Fortunately, there have been few instances of serious head injuries ever since. Legend recounts at least one instance of the pelota being used for self- defense. Perkain was a champion player who fled to Spain to escape the guillotine during the French Revolution.

Still, he could not resist returning to France to defend his title against a French challenger. When threat- ened with arrest, he succeeded in making his escape by beaning the law enforcement official with the ball. Jai alai players come in all shapes and sizes. The players on the Milford roster ranged in height from 5 feet 6 inches to 6 feet 3 inches and in weight from to pounds. Frontcourt players are typically shorter and quicker, for they must react to balls coming at them directly off the frontwall.

Backcourt players must be stronger and more acrobatic to en- able them to dive for odd bounces yet recover to toss the ball the length of a football field in one smooth motion. It is not that unusual to see play- ers sporting substantial bellies, but appearances can be deceiving.

These are highly skilled, conditioned athletes. According to pedometer studies, each player runs about one mile per game, and each player typically ap- pears in four to six games per night, five nights per week. As in tennis, the players must be versatile enough to play both offense and defense. The sport is not as easy as it looks.

Three of the members of the Milford roster including Alfonso, shown in the figure on the page 22 have played at Milford since at least As in baseball, professional players range in age from less than 20 to over Both youth and experience have their advantages on the court. Not all players have such long careers, of course. The granddaughter of Claire Barry, one of the sisters, recently mar- ried a professional jai alai player named Bryan Robbins.

The open wall of the court results in an asymmetry that makes it very undesirable to have the cesta on the left hand. Therefore, all pro- fessionals today are right-handed, or at least use that hand for playing jai alai. There have been exceptions, however.

Marco de Villabona man- aged to be a competitive player after losing his right arm. A nineteenth- century player named Chiquito de Eibar was such a dominant player that he was sometimes required to play with the basket on his left hand as a handicap. Jai alai is a male sport, although a few women have played the game on an amateur level. Perhaps the best-known amateur player was Katherine Herrington back in the s, who went on to write a book on the sport after playing her last exhibition at Saint-Jean-de-Luz, France, in The legendary Tita of Cambo, a French Basque, was reputedly so strong that her serves damaged stone walls.

These terms should be familiar to anyone ac- quainted with horse racing, and we will use them throughout the rest of the book. There are eight possible win bets at a standard fronton. You will receive the same payoff regardless of whether your team is first or second. This is a less risky bet than picking a team to win, but the payoff is usually less as well. There are eight possible place bets at a standard fronton. There are eight possible show bets at a standard fronton. The order in which your two teams finish is irrelevant — so long as they finish 1 and 2 you receive the quiniela price.

Personally, I find the quiniela bet to be the single most exciting choice for the spectator because it seems one always has a chance to win at some point in the match. If you pick a 2—6 exacta, it means that 2 must win and 6 must come in second. If you play a 2—5—3 trifecta, then 2 must win, 5 must finish second place , and 3 must come in third show.

Trifectas are the riskiest conventional bet, but the one that typically pays the highest returns. Different frontons operate under slightly different betting rules. One aspect that varies is the size of the minimum bet allowed. Frontons tend not to have maximum bet limits because those are imposed by common sense.

As will be discussed in Chapter 6, jai alai is a pari-mutuel sport, and thus you are trying to win money from other people, not from the house. Any bettor is free to make any combination of these types of bets on any given match. Indeed, frontons provide certain types of aggregate bets as a convenience to their customers.

Ordering a box can simply be a convenience, but certain frontons allow one to bet a trifecta box at a cost that works out to less than the minimum bet per combination. Indeed, we will exploit this freedom with our own betting strategy. For example, a 1—2 trifecta wheel de- fines bets on the following six trifectas: 1—2—3, 1—2—4, 1—2—5, 1—2—6, 1—2—7, and 1—2—8. Certain venues presumably allow one to bet a tri- fecta wheel at a cost that works out to less than the minimum bet per combination.

The Spectacular Seven Scoring System This book reports our attempt to model the outcome of jai alai matches, not horse racing or football or any other sport. The critical aspect of jai alai that makes it suitable for our kind 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. Every point is a battle between only two players, with the active by pair determine by their positions in a first-in, first-out FIFO queue. Initially, player 1 goes up against player 2. The loser of the point goes to the end of the queue, and the winner stays on to play the fellow at the front of the line.

The first player to total typically seven points is declared the winner of the match. Because seven is exactly one point less than the number of 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. 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 their turn to compete. The two players not on this queue play the next point. As always, player 1 starts against player 2, and everybody be- gins with 0 points. Suppose player 1 beats player 2 the event reported on the center of the first line. Player 1 collects his first point and continues playing against the next player in line, player 3.

The loser, player 2, sulks his way back to the bench and to the end of the player queue. Continuing on with this example, player 1 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 player sitting at the top of the queue, player 2, the loser of the opening point.

Here, the scoring system gets slightly more complicated. In order to reduce the disadvantage of late post positions, the Spectacular Seven scoring system increases the reward for each winning volley after the seventh physical point from one to two points. Our illustrative game is now at the midgame division line. Player 8 goes against player 2 and scores on a well-placed chic-chac. After winning 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 for player 8 goes from 1 to 3. Player 1, the winner 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, player 1 needs only the next point for victory remember it counts for 2. But number 5 is alive and knocks player 1 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.

Player 1 with five points and player 3 with four points stand alone for place and show, resulting in a 5—1—3 trifecta. We have now seen two aspects of the Spectacular Seven 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 never get another chance 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 their first seven opponents to win without ever going back to the queue, whereas player 8 only has to win his first four volleys the first of which counts for 1 and the last three of which count for two points each. Because it is rare for any player to win seven in a row, the early players are penalized, and the system is supposed to even out.

The Spectacular Seven scoring system was introduced in the United States in the s 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 wager in between matches.

The ratio of action- to-waiting is much better in jai alai than horse racing because each race lasts only 2 or 3 minutes. The Spectacular Seven 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. This is not always the case. Our second example shows a match in which two players are tied for second at the moment player 1 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 Spectacular Seven describe how to resolve such complicated sce- narios. Sometimes higher point totals are required in Spectacular Seven matches; for example, 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. Bookies and bettors alike are not interested in wagering on professional wrestling, which is a situation unique among nationally televised sports. Professional wrestling has no chance to succeed as a gambling venue because the betting public understands that the results of wrestling matches are choreographed in advance.

Hence, to someone in the know, there is no uncertainty at all about who will win. Many people are afraid to bet on jai alai because they are betting on players who happen to be people. Professional players want to win.

In horse racing, you can be pretty sure that the horse did not bet on the race, but such confidence seems misplaced in jai alai. Anything that scares away potential bettors is a funda- mental threat to their business. Every fronton pays players both a fixed salary and a bonus for each game they win, and thus they have incentive to play hard and win.

The frontons have strong rules against match fixing, and any player not on the up-and-up will become persona non grata at every fronton in the world. In the course of my research for this book, I have uncovered only lim- ited discussions of crooked jai alai betting. Nasty things apparently oc- curred in the United States in the late s, which no one likes to talk about today, but several Florida and California state documents from the s and s I studied stress that the sport had no whiff of scandal up to that point.

They credited this to a strong players union and the close-knit structure of the Basque community, which polices its own. It is hard for an outsider to fix a game with a player who speaks only Basque. The one game-fixing scandal I have seen documented occurred in Florida, apparently during the strike years, when underskilled and unded- icated scab players roamed the court. Groups of three or four players per match were bribed by the fixer to play dead, who then bought multiple quiniela boxes covering all pairs of honest players.

The betting volume required to turn a profit on the deal was also high enough to catch the attention of the fronton. Eventually, it was used to help convict the head fixer in criminal court. The nature of jai alai particularly lends itself to suspicions of fixing.

Players have to catch a rock-hard ball hurled at miles per hour using an outsized basket strapped to their arm. The width of the cesta in the area where the ball enters is only 3 to 3. This leaves only an inch or so as the margin of error, which is not much — especially when the ball is curving or wobbling. That our sys- tem predicts the outcome of jai alai matches much better than chance tells us that most games are not fixed.

Even the most cynical bettor will admit that performing a successful fix requires a certain amount of en- ergy, investment, and risk. These considerations dictate that only a small fraction of games will be fixed. For a system like ours, which relies on making lots of small bets instead of a few big ones, fixes can be written off as a cost of doing business.

Spend too much time worrying about fixes and you turn into a conspiracy theorist and then a nut case. I have the serenity to accept the things I cannot change. Other Betting Systems Betting systems have existed for as long as gambling has. A betting system is either bogus or clever, depending upon whether it is based on a sufficiently deep understanding of the given game so that there is some method to the madness. Gambling systems, even bogus ones, are always interesting to hear about because they say something about how people perceive or misper- ceive probability.

Walk into the casino and bet a dollar on black. If it wins, boldly pocket your earnings. If it wins, you are back to where you started. After each loss, keep doubling up. Inevitably, you are going to win sometime, and at that point you are all caught up. Now you can start again from the beginning.

Nothing really, so long as you have an infinitely deep pocket and are playing on a table without a betting limit. If your table has a betting limit or you are not able to print money, you will eventually reach a point at which the house will not let you bet as much as you need in order to play by this system.

At this point you will have been completely wiped out. This doubling or Martingale system offers you a high probabil- ity of small returns in exchange for a small possibility of becoming homeless. Casinos are more than happy to let you take this chance. After all, Donald Trump has a much deeper pocket than either you or I have.

Set aside enough money for a ticket on the next plane to South America. Bet the rest on one spin of the roulette wheel at even money. If you win, return the principal and retire on the rest. Otherwise, use the plane ticket. Mathematically, the key to making this work is being bold enough to wager all the money on a single bet rather than making multiple smaller bets.

You pay more tax each time you re- bet the winnings, thus lowering your chances of a big killing. Lottery numbers are selected by draw- ing numbered balls from a jar, or some equivalent method. As we will see, poor random number generators certainly exist; I will talk more about this in Chapter 3.

During the Vietnam War, the U. A total of balls, each bearing one possible birthdate, were tossed into a jar, and unlucky year-olds were mustered into the army if their birthdate was selected. It was fixed for the next year, which was presumably small consolation to those left marching in the rice paddies.

Although each lottery combination is just as likely to come in as any other, there is one formally justifiable criterion you can use in picking lottery numbers. For this reason, playing any ticket with a simple pattern of numbers is likely to be a mistake, for someone else might stumble across the same simple pattern. I would avoid such patterns as 2—4— 6—8—10—12 and even such numerical sequences as the primes 2—3—5— 7—11—13 or the Fibonacci numbers 1—2—3—5—8—13 because there are just too many mathematicians out there for you to keep the prize to yourself.

There are probably too many of whatever-you-are-interested-in as well; thus, stick to truly random sequences of numbers unless you like to share. Indeed, my favorite idea for a movie would be to have one of the very simple and popular patterns of lottery numbers come up a winner; say, the numbers resulting from filling in the entire top row on the ticket form.

This will not be enough to get members of the star-studded ensemble cast out of the trouble they got into the instant they thought they became millionaires. The key decision for any player is whether to accept an additional unknown card from the house. This card will increase your point total, which is good, unless it takes you over 21, which is bad. However, a sufficiently clever player does know something about the hand he or she will be dealt. Suppose in the previous hand the player saw that all four aces had been dealt out.

If the cards had not been reshuf- fled, all of those aces would have been sitting in the discard pile. If it is assumed that only one deck of cards is being dealt from, there is no possibility of seeing an ace in the next hand, and a clever player can bet accordingly. By keeping track of what cards he or she has seen card counting and properly interpreting the results, the player knows the true odds that each card will show up and thus can adjust strategy ac- cordingly.

Card counters theoretically have an inherent advantage of up to 1. Equipped with computer-generated counting charts and a fair amount of chutzpah, Thorp took on the casinos. Most states permit casinos to expel any player they want, and it is usually fairly easy for a casino to detect and ex- pel a successful card counter. Even without expulsion, casinos have made things more difficult for card counters by increasing the num- ber of decks in play at one time.

If 10 decks are in play, seeing 4 aces means that there are still 36 aces to go, greatly decreasing the potential advantage of counting. For these reasons, the most successful card counters are the ones who write books that less successful players buy. Thorp himself was driven out of casino gambling in Wall Street, where he was reduced to running a hedge fund worth hundreds of millions of dollars.

Still, almost every mathematically oriented gambler has been intrigued by card counting at one point or another. A few years ago, the American Physical Society had its annual convention in Las Vegas, during which the combination conference hotel and casino took a serious financial hit.

The hotel rented out rooms to the confer- ence at below cost, planning to make the difference back and more from the gambling losses of conference goers. However, the physicists just would not gamble. They knew that the only way to win was not to play the game.

But another group of physicists did once develop a sound way to beat the game of roulette. A roulette wheel consists of two parts, a mov- ing inner wheel and a stationary outer wheel. Things rattle around for several seconds before the ball drops down into its slot, and peo- ple are allowed to bet over this interval.

However, in theory, the win- ning number is preordained from the speed of the ball, the speed of the wheel, and the starting position of each. All you have to do is measure these quantities to sufficient accuracy and work through the physics. Finger or toe presses at reference points on the wheel were used to enter the observed speed of the ball. It was necessary to conceal this computer carefully; otherwise, casinos would have been certain to ban the players the moment they started winning.

Did it work? Yes, although they never quite made the big score in roulette. Like Thorp, the principals behind this scheme were even- tually driven to Wall Street, building systems to bet on stocks and commodities instead of following the bouncing ball. Their later ad- ventures are reported in the sequel, The Predictors. The bigger a jackpot, the more that people want to play. The pool grows very large whenever a few weeks go by without a winner.

The interesting aspect of large pools is that any wager, no matter how small the probability of success, can yield positive expected re- turns given a sufficiently high payoff. If nobody guesses right for a sufficiently long time, the potential payoff for a winning ticket can overcome the vanishingly small odds of winning.

For any lottery, there exists a pool size sufficient to en- sure a positive expected return if only a given number of tickets are sold. But once it pays to buy one lottery ticket, then it pays to buy all of them. This has not escaped the attention of large syndicates that place bets totaling millions of dollars on all possible combinations, thus ensuring themselves a winning ticket. State lottery agents frown on such betting syndicates, not because they lose money the cost of the large pool has been paid by the lack of winners over the previous few weeks but because printing millions of tickets ties up agents throughout the state and discourages the rest of the betting public.

Still, these syndicates like a discouraged public. The only danger they face is other bettors who also pick the winning num- bers, for the pool must be shared with these other parties. Given an estimate of how many tickets will be bought by the public, this risk can be accurately measured by the syndicate to determine whether to go for it.

Syndicate betting has also occurred in jai alai in a big way. Palm Beach Jai-Alai ran an accumulated Pick-6 pool that paid off only if a bettor correctly picked the winners of six designated matches. This amount was more than it would have cost to buy one of every possible ticket.

I found myself in high school taking a course in computer programming and got myself hooked. It was very empowering to be able tell a machine what to do and have it do exactly what I asked. All I had to do was figure out what to ask it. I Was a High School Bookie During my sophomore year of high school, I got the idea of writing a program that would predict the outcome of professional football games.

It seemed clear to me that writing a program that accurately predicted the outcome of foot- ball games could have significant value and would be a very cool thing to do besides. In retrospect, the program I came up with now seems hopelessly crude. It first read in the statistics for teams x and y; stats such as the total number of points scored this year, the total number of points allowed, and the number of games played so far.

The cham- pion Cowboys had scored points and given up , whereas the peren- nial doormat Saints had scored and given up points, each team having played 10 games. I would then adjust these numbers up or down in response to 15 other factors, such as yards for and against and home field advantage, round the numbers appropriately, and call what was left my predicted score for the game.

This computer program, Clyde, was my first attempt to build a mathe- matical model of some aspect of the real world. This model had a certain amount of logic going for it. Good teams score more points than they allow, whereas bad teams allow more points than they score. If team x plays a team y that has given up a lot of points, then x should score more points against y than it does against teams with better defenses.

Similarly, the more points team x has scored against the rest of the league, the more points it is likely to score against y. Suppose team x has been playing all stiffs thus far in the season, whereas team y has been playing the best teams in the league. Team y might be a much better team than x even though its record so far is poor.

This model also ignores any injuries a team is suffering from, whether the weather is hot or cold, and whether the team is hot or cold. It disregards all the factors that make sports inherently unpredictable. And yet, even such a simple model can do a reasonable job of pre- dicting the outcome of football games.

As an audacious year-old, I wrote to our local newspaper, The New Brunswick Home News, explaining that I had a computer program to pre- dict football games and offering them the exclusive opportunity to publish my predictions each week. Remember that this was back in , well be- fore personal computers had registered on the public consciousness.

In those days, the idea of a high school kid actually using a computer had considerable gee-whiz novelty value. To appreciate how much times have changed, check out the article the paper published about Clyde and me. I got the job. Clyde predicted the outcome of each game in the Na- tional Football League. It was very cool seeing my name in print each week and monitoring the football scores each Sunday to see how we were doing.

As I recall, Clyde and I finished the season with the seemingly impressive record of — As I recall, we all finished within a few games of each other, although most of the sportswriters finished with better records than the computer. Instead, the Inquirer included me among 10 amateur and professional prognosticators. Each week we had to predict the outcomes of four games against the point spread.

The point spread in football is a way of handicapping stronger teams for betting purposes. Think back to the Cowboys and Saints football game described earlier. It would be impossible to find a bookie who would let you bet on the Cowboys to win at even-money odds because any Saints victory required a miracle substantial enough to get canonized in the first place.

Instead, the bookies would publish a point spread like Cowboys by 14 points. If you bet on the Cowboys, they had to win by at least 14 points for you to win the bet. The Saints could lose the game by 10 points and still leave their betting fans cheering. The point spread is designed to make each game a 50—50 proposition and hence makes predicting the outcome of games much harder.

We did somewhat better on the game we selected as our best bet of the week, finishing 12—8 and in second place among the touts. Clyde finished his career with 4 years of picking the results of University of Virginia football games for our student newspaper, The Cavalier Daily. Our results were about the same as with the pros. We went 35—19—1, correct on the easy games and wrong on the hard ones.

Randy was a linebacker, 6 feet 6 inches and pounds. One day I asked him what he thought of Clyde in the newspaper, not letting on that I was the man behind the program. Back to Jai Alai Every other winter or so our family migrated down to Florida for fun in the sun and a night at the fronton.

This mixed record impressed upon me the benefits of finding winners for ourselves. The more jai alai I watched, the more it became apparent to me that the Spectacular Seven scoring system exerted a profound effect on the outcome of jai alai matches. Even a cursory look at the statistics revealed that certain positions were far easier to win from than others.

It was simply not the case that good teams would usually beat bad ones, because the arbitrarily chosen position from which you started in the queue made a big difference in how many chances you had to score the required points.

If a good team got a bad starting position, its chances of winning might be less than that of a bad team in a good starting position. A good team in a good starting position had a real advantage; them that has, gets. Modern business ethics teach us that, whenever you see an unfair situation, you should exploit it for as much personal gain as possible.

How could I exploit the biases of the Spectacular Seven scoring system? The simple ideas underlying my football program were simply not sufficient for such a complex reality. However, I could get a handle on the situation using the powerful technique of Monte Carlo simulation.

Monte Carlo Simulations Simulations provide insight into complex problems. Simulation is used in economics, engineering, and the physical sciences because it is often im- possible to experiment on the real thing. Economists cannot play with the U. Rather, they will make a computer model and study the effects of such spending on it. The significance of the simulation results depends on the accuracy of the model as well as how correctly the model has been turned into a computer program.

There are a wide variety of computer simulation techniques, but we will employ a curious method known as Monte Carlo simulation. However, this connection is even deeper because the whole idea of Monte Carlo simulation is to mimic random games of chance.

Suppose we want to compute the odds of winning a particularly exotic bet in roulette, such as having the ball land in a prime-numbered slot either 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, or 31 three out of the next four times we spin the wheel. The most naive approach would be to watch a roulette wheel in action for a spell, keeping track of how often we win. If we watched for trials and won 91 times in this interval, the odds should be about 1 in To get a more accurate estimate, we could simply watch the game for a longer period.

Now suppose instead of watching a real roulette wheel in action we sim- ulate the random behavior of the wheel with a computer. We can conduct the same experiments in a computer program instead of a casino, and the fraction of simulated wins to simulated chances gives us the approximate odds — provided our roulette wheel simulation is accurate.

Monte Carlo simulations are used for far more important applications than just modeling gambling. A classical application of the technique is in mathematical integration, which if you ever took calculus you may recall is a fancy term for computing the geometric area of a region. Calculus-based methods for integration require fancy mathematical tech- niques to do the job. Monte Carlo techniques enable you to estimate areas using nothing more than a dart board.

Suppose someone gave you this Picasso-like painting consisting of oddly shaped red, yellow, and blue swirls, and asked you to compute what fraction of the area of the painting is red. Computing the area of the entire picture is easy because it is rectangular and the area of a rect- angle is simply its height times its width. But what can we do to figure out the area of the weirdly shaped blobs?

Now suppose that I start blindly throwing darts in the general direction of the painting. Some are going to clang off the wall, but others will occasionally hit the painting and stick there. If I throw the darts randomly at the painting, there will be no particular bias towards hitting a particular part of the painting — red is no more a dart-attractor than yellow is a dart-repeller.

Thus, if more darts hit the red region than the yellow region, what can we conclude? There had to be more space for darts to hit the red region that the yellow one, and hence the red area has to be bigger than the yellow area. Suppose after we have hit the painting with darts, 46 hit red, 12 hit blue, 13 hit yellow, and 29 hit the white area within the frame.

We can conclude that roughly half of the painting is red. If we need a more accurate measure, we can keep throwing more darts at the painting until we have sufficient confidence in our results. Picasso Measuring area using Monte Carlo integration. Random points are distributed among the colored regions of the painting roughly according to the fraction of area beneath.

It would be much better to perform the entire experiment with a computer. Suppose we scan in an image of the painting and ran- domly select points from this image. Each random point represents the tip of a virtual dart. By counting the number of selected points of each color, we can estimate the area of each region to as high a degree of accuracy as we are willing to wait for. I hope these examples have made the idea of Monte Carlo simulation clear. Basically, we must perform enough random trials on a computer to get a good estimate of what is likely to happen in real life.

I will explain where the random numbers come from later. Building the Simulation A simulation is a special type of mathematical model — one that aims to replicate some form of real-world phenomenon instead of just predicting it. The simulated football game would start with a simulated kickoff and advance through a simu- lated set of downs, during which the simulated team would either score or give up the simulated ball. After predicting what might happen on every single play in the simulated game, the program could produce a predicted final score as the outcome of the simulated game.

By contrast, my football-picking program Clyde was a statistical model, not a simulation. Clyde really knew almost nothing that was specific to football. Indeed, the basic technique of averaging the points for-and- against to get a score should work just as well to predict baseball and basketball scores. The key to building an accurate statistical model is pick- ing the right statistical factors and weighing them appropriately.

Building an accurate football-game simulator would be an immensely challenging task. Jai alai is, by contrast, a much simpler game to simulate. What events in a jai alai match must we simulate? All players start the game with scores of zero points each.

Each point in the match involves two players: one who will win and one who will lose. The loser will go to the end of the line, whereas the winner will add to his point total and await the next point unless he has accumulated enough points to claim the match.

This sequence of events can be described by the following program flow structure, or algorithm: Initialize the current players to 1 and 2. Initialize the point total for each player to zero. So long as the current winner has less than 7 points: Play a simulated point between the two current players. Add one or if beyond the seventh point, two to the total of the simulated point winner. Put the simulated point loser at the end of the queue. Get the next player off the front of the queue.

End So long as. Identify the current point winner as the winner of the match. The only step in this algorithm that needs more elaboration is that of simulating a point between two players. If the purpose of our simulation is to see how biases in the scoring system affect the outcome of the match, it makes the most sense to consider the case in which all players are equally skillful.

To give every player a 50—50 chance of winning each point he is involved in, we can flip a simulated coin to determine who wins and who loses. Any reasonable computer programmer would be able to take a flow description such as this and turn it into a program in the language of his or her choice. This algorithm describes how to simulate one particular game, but by running it 1,, times and keeping track of how often each possible outcome occurs, we would get a pretty good idea of what happens during a typical jai alai match.

Breaking Ties The Spectacular Seven scoring system, as thus far described, uniquely determines a winner for every game. However, because of place and show betting, a second- and third-place finisher must also be established. It is very important to break ties under the same scoring system used by the frontons themselves. To show how complicated these rules are, we present for your amusement the official state of Florida rules governing the elimination games in jai alai: 1.

After a winner has been declared, play-off rules to decide place, show and fourth positions vary according to the number of points scored by the participating players or teams, and shall be played according to the players or teams rotation position not post posi- tion , i. When there still remain five or seven players or teams, all of which are tied without a point to their credit, the play-off shall be for a goal of one point less than the number of post positions represented in the play-off.

When there still remain five or seven players or teams, all of which are tied without a point to their credit, the play-off shall be contin- ued until the player or team reaches the number of points desig- nated for the game. In case of two ties, after a Winner has been declared official, and there are still two players or teams tied with the same number of points, the place position shall be awarded to the player or team making the next point, and show position goes to the loser of said point.

In games where a fourth position is required, if: a Two ties remain after win and place have been determined, the show position shall be awarded to the player or team making the next point, and fourth position shall go to the loser of said point. In case of three ties, after a Winner has been declared official, place, show and fourth positions shall be decided among the three players or teams, with the same number of points, through elimination, according to rotating position.

In case of four or six ties, after a Winner has been declared official and there remain four or six players or teams, tied for place, show or fourth, or all three, play-off shall be through elimination according to their rotating position. The first two players or teams will play the first point. The next two players or teams will play for one point, and the remaining in case of 6 players or teams will also play for one point. Winners of the above points will play additional points to decide place, show and fourth position, as the case may require.

In games where a fourth position is required and no place, show or fourth position has been determined and there remain four ties, the losing players or teams of the elimination play-off return and play a final point to determine fourth. If at any time during a play-off a player or team reaches the des- ignated number of points the game calls for, said player, or team, shall immediately be awarded the place, show or fourth position, as the case may be and the remaining players or teams shall forfeit the right to play for said position.

As you can see, the complete tiebreaking rules are insanely compli- cated.

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Cs go wild betting 2021 military | This model had a certain amount of logic going for it. It is illegal to copy, distribute, or create derivative works from this book in whole. The degree to which one can do this in my tiny toy domain tells us something about our potential to foresee larger and more interesting futures. This scoring system has interesting mathematical properties that just beg the techno-geek to try to exploit it. It is hard for an outsider to fix a game with a player who speaks only Basque. If you wanted to view all these WWW pages, you had better buy the Microsoft product because it is the only company supporting this new standard. |

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Daniel gibbs csgo betting | We can conclude that roughly half of the painting is red. The hotel rented out rooms to the confer- ence at below cost, planning to make the difference back and more from the gambling losses of conference goers. Soccer betting professor pdf editor more profitable direction nowhere rewards How to change direction and find the road to reward A more profitable direction Your reward program could be harming your profits, and your relationship with. When there still remain five or seven players or teams, all of which are tied without a point to their credit, the play-off shall be for a goal of one point less than the number of post positions represented in the play-off. Jai alai underwent a big boom in the mids. |

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This advantage was evident regardless of whether reward opportunities were relative i. Notably, experts did outperform novices. Nonetheless, even experts were reliably better in predicting winners when making general bets than when making specific bets. It seems that even in cases where greater knowledge may offer an advantage, the act of focusing on that knowledge can disrupt decision-making.

Thus, while a lifelong baseball fan is more likely to pick the winning team than someone who has never watched a game, for either person a quick prediction about the winner is likely to be more accurate than one that follows deep reflection. Yoon's team confirmed this notion by assessing the kinds of information participants were using to make their predictions.

In addition, reliance on global information significantly predicted success for all participants. Even for those in the specific score group, use of detailed knowledge e. These data align with lessons learned from research on basic personal decisions. Whether choosing a jelly bean flavor , rating the attractiveness of a face , or selecting a poster to hang in a room , people are more satisfied with their selection and less likely to change their minds when they make their decisions quickly, without systematically analyzing their options or mulling over the reasons for their choice.

The advice is thus the same whether considering complex scenarios or simple situations: Don't overthink it. Today, more than ever before, we have access to extensive data that we can consider when making complicated decisions like selecting a mutual fund or betting on a baseball series.

While reviewing that information may prove useful in developing an accurate overall view of the options, the results from Yoon and colleagues suggest that focusing on the details during the decision process will prove detrimental. It is best to trust your instincts and make up your mind already. Are you a scientist who specializes in neuroscience, cognitive science, or psychology? And have you read a recent peer-reviewed paper that you would like to write about?

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Check several bookmakers to get the best odds before making a bet. Understand moneylines. Moneyline odds are a given number in the hundreds or thousands, accompanied by a plus or a minus. Calculate potential profits with positive moneyline odds by dividing the odds by , then multiplying the result by your initial stake. Calculate potential profits with negative moneyline odds by dividing the odds by , then dividing your stake by the result.

Understand the outcome probabilities. Probabilities indicate how likely something is to happen or not happen. A high value indicates the team is not favored, and likely to lose. Probabilities do not guarantee an outcome. Part 2 Quiz Why should you research soccer before betting?

To make knowledgeable bets Right! To calculate the odds Try again! To show your bookmaker you're a serious bettor Not exactly! Part 3 of Bet on a two-way moneyline. In this betting arrangement, bettors will bet on one team winning and another team losing. If the team you bet on wins, you will receive a payout.

If the team you bet on loses, you will have lost your bet. Bet on a three-way moneyline. Compared to the two-way moneyline, three-way bets usually have lower odds. Place a bet on totals. This means that you are betting that the average goals per team in the game will be over or under 2. For instance, assume you made a bet that totals would be under 2. If both teams scored three goals, the total will be six. Divided by two the number of teams in the game , you end up with the number three.

In that case, you would lose the bet, since the total was above 2. Bet on goal lines. When betting on goal lines, you usually need to call the bet as well by indicating which team you think will win. The higher the goal line bet you make, the riskier it is to win it. This type of bet is also known as betting on point spreads. Make a split bet. Sometimes you will get refunded and win the other half of the bet.

Part 3 Quiz You bet that Spain will win its game against Germany by at least 5 goals. A split bet Close! A totals bet Definitely not! A two-way moneyline bet Not quite! A goal lines bet Correct! Part 4 of Decide on your maximum budget. Decide how much you want to spend on betting. This amount -- known as your bankroll -- will vary widely depending on your income. Be honest with yourself about how much money you have to invest in betting.

On the one hand, a budget that is too high may result in personal bankruptcy. A bankroll that is too low, on the other hand, will limit your ability to reap big rewards from betting. Do not bet on soccer unless you have sufficient funds. Decide on your maximum bet. This amount is known as your betting unit size. You risk losing it all at once.

Place your bet. Choose the game or team you want to bet on. Do not put all your money on one outcome or one game. Distribute your risk by betting on multiple games. Limiting your maximum bet will increase your chances to win. If you do lose, be prepared to walk away, no matter how tempted you are to continue. You don't want to lose even more. Know when to quit. If you win, don't get carried away. Enjoy what you've won by cashing out and coming back later. Set specific guidelines for success when betting on soccer.

Your goals should be realistic and achievable. Make rules for yourself to prevent excessive loss. Gambling can be addictive. Go to source If you are experiencing a problem with gambling, tell someone or call your local helpline. Part 4 Quiz If you're new to gambling, how much of your bankroll should you bet at a time? Not Helpful 5 Helpful Not Helpful 1 Helpful It's normal to lose sometimes. Don't worry about it and just continue to make the best bet you can.

However, if you lose a lot, you should not continue betting on soccer. Check odds regularly. Helpful 0 Not Helpful 0. Monitor your success and failure over the long term. Make a spreadsheet with information about each math you bet on, including the teams, the date of the game, how much you bet, and how much you won or lost.

This will give you a big picture view of your success or failure as a bettor. Be patient when betting on soccer. Start with small wagers and work your way up to larger bets. It takes at least 10, hours of practice at something to become really good at it.

There are no quick and easy tricks to win when betting on soccer. Consider the odds in each match and make your bet only after careful consideration. Submit a Tip All tip submissions are carefully reviewed before being published. If you gamble with a large betting unit, you could win a lot of money, but you can also lose a lot.

Helpful 6 Not Helpful 1. Related wikiHows. More References About This Article. Co-authored by:. Co-authors: Updated: October 8, Categories: Soccer. Article Summary X The easiest way to bet on soccer is to bet on something simple, like which team will win. Italiano: Scommettere sul Calcio. Nederlands: Gokken op voetbal. Thanks to all authors for creating a page that has been read , times. In fact I've only won once. Thanks to your tips I think I'm heading for the jackpot. Thanks wikiHow. Sipho Phiri Sep 8, Having lost thousands in the past months, I'm finally ready to rake in a few consolation dollars.

Moses Ntee Jun 17, Sometimes it's good to research first. I did that, and I'm glad, thanks to the team. Keep up good work. Anonymous Mar 28, Betting in actual fact is risk taking, otherwise you end up being bankrupt. Sunday Simeon Jun 20, Satrust Oct 4, Anonymous Apr 18, Anonymous Dec 13, Anonymous Jan 5, Gibs Quao Dec 4, Melinda Szab-Jrgensen Feb 6, Rated this article:.

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