Notes on Probability and Horse Racing

By djones@mtest.teradyne.com (Dave Jones)

A few years ago a group of mathematicians from Santa Cruz tackled a problem: Casinos allow bets to be placed at the roulette table after the mechanism is set in motion. The mathematicians wanted to beat the game by tracking the spinning ball and wheel with relays hidden on their persons and wired to miniature computers. The computers would use this physical data to predict the final resting spot of the ball while bets were still allowed. What they were doing was probably unethical then, and is certainly illegal now, but it was fun.

They didn't need to be right every time. They just needed to be right often enough.

As thoroughbred handicappers, we might say they were attempting to use physical data to produce a new "true" odds line. But how "true" was it? If you allow the collection of physical data, the game no longer has one set of "true odds". The odds depend on the data-gathering technique and on the quality of the calculations. Ultimately the result of the spin is determined by physics. In their new game, (one the house does not want you to play), the odds no longer reflected inherent, planned properties of the game, but instead they measured the limitations of the player's data and calculations in doing the physics. If both data and calculation were perfect, the player would be able to assign a probability of 1 to the slot the ball was destined to land in, and a probability of 0 to all others. But the Santa Cruz mathematicians didn't need to get it that fine. All they needed to do was to produce what we would call an "overlay".

Unlike us, they were in the very enviable situation of being able to know when they had an overlay, because in their game, as opposed to ours, the competition was not allowed to use physical data and calculations to set the payoff odds. The house was forced by the rules to keep the wheel balanced and to pay 35:1 regardless of how the dealer spun the wheel. The house was not allowed to have their own (possibly superior) computers grinding away, posting ever-changing payoff odds on a toteboard as the ball spun around the wheel. Even with all that going for the mathematicians, they were not home free. They discovered that small, unavoidable errors could produce enormous variance of results. They developed a theory for the phenomenon, and they aptly named it "chaos".

Like the "Santa Crustaceans", we seek a method for reducing our uncertainty below that of the opponent, but in our game it is both ethical and legal to do so. However, it is also legal for our opponents to gather physical data and use it to set the odds that we will receive if we win. That complicates our problem far beyond what the chaos guys faced at the wheel. We must estimate not only our own uncertainty, but also we must learn to "give respect" to the intelligence and intelligence gathering abilities of the pari mutuel crowd.

How do we do this?

Obviously, very precise Newtonian calculations of the type that a computer can do to calculate the trajectory of a roulette ball are far beyond our current handicapping abilities, and beyond our data gathering capacity as well. Nevertheless, we attempt to obtain evidence about the physical abilities of the horses and information about other physical attributes of the race such as track condition, post position, even the trainer's mental state and intentions. We attempt to use that evidence to predict the outcome of the race.

We have a wide range of very imposing obstacles, which we can lump together under the name "uncertainty". I once bet on a horse named Zignew, who was going off at double digit odds. He was moving well in the stretch and looked like he was going to win, but a swan flew across the track, right into his face, causing him to lose his stride, and the race.

I think we can agree that there is no way anyone could have calculated before the race that Zignew would lose because of a swan attack. There is also no way anyone can determine precisely the physical condition of any of the horses. The calculations we perform on the physical data we gather are inherently prone to error. We must somehow get a handle on all this uncertainty, both ours and that of our pari mutuel opponents.

Probability theory looks like a good place to start looking for a way to reconcile the limits of our handicapping with the odds posted on the tote board. But if it is, it's only a start.

Probability divides everything into the "known" and the "random". But we have a much more complicated situation. We have not only the "known-by-us", but also the (unknown to us) "known-to-them", in addition to the unknowable or "chaotic". Unlike roulette and card games, our game has only a few rules that prohibit intelligence gathering. There is therefore no theoretical reason that the "true odds" on the eventual winner should not be 1 and the "true odds" on the others should not be 0. But in practice we can not do nearly that well. Not unless we know an awful lot about the migratory habits of swans.

Dave

)  Date: Wed, 16 Nov 94 19:53:42 -0500

)  From: andrew ivan <ivan@alpha.ces.cwru.edu>
)  Subject: (long) Neural Net/probability missive
)
)  ...
)  I think you could say that Old Blue's chances are one in thirteen. What does
)  that mean? The same thing as saying that the chances of drawing a king are
)  one in thirteen. It will happen, on average, one time in every thirteen
)  identical trials.

But what is an "identical trial"? If the trials were absolutely identical, right down to the last molecule, the result would be the same every time. But even a misplaced butterfly in an adjacent field might cause a swan to fly into the path of one of the horses. Does moving a butterfly in the next field make the trial not identical? How about running the 8" blade over the track before the race? Where do you draw the line? In games of chance, the line is drawn for us: The rules of the game separate the constant elements from the random or chaotic. There is information we are simply not allowed to have, calculations we are not allowed to make.

If we attempt to extend the concept of probability to horseracing, we get lots of "true odds" lines -- one for every knowledge-set. Probability is determined by what is known and unknown, and the rules of handicapping leave that division tantalizingly with the player.

) Alas, no two horse races are identical. Furthermore, there ) are so many factors governing the outcome that the computation to arrive at ) this probability is very difficult, possibly impossible, unlike the method of ) determining the probability of drawing a king which is rather obvious.

I think it goes deeper than that. It is not only a matter of the difficulty of computing a number. We don't even know how to begin to compute the number, because we don't know how to define it. Indeed, what does the number represent that we seek to compute?

When that which can be known is not constrained by rules, there is no single number that qualifies as *the* probability of a horse winning.

Probabilty is a quantification of our ignorance, not a measure of a horse's ability.

)
) Well, if you make an odds line for a race, aren't you (at least implicitly) ) assigning probabilities to each horse?

I am attempting to quantify *my* uncertaintly. An odds line is an admission of ignorance. But I hope that my odds lines will out-perform the toteboard. Specifically, I am hoping that I have assigned a higher probability to the eventual winner than the tote board has. In effect, I am saying to the toteboard, "Nyah, Nyah, Nyah! My uncertainty is better than *your* uncertainty!"

I am NOT claiming to approximate some golden, exhaulted state of uncertainty called "true odds", which somehow qualifies as the "best uncertainty" against which all other states of uncertainty are measured.

)
) Why not? Why does the concept of probability not apply to the outcome of ) horse races? This is the main thing you're saying that I don't understand.

I hope that this posting and the previous one about swans help to explain why I think the concept of probability does not translate to the province of horseracing.

-- Dave