Having delved into one of the classic betting questions in part one of his article, Joseph Buchdahl now takes his analysis of the price discovery model a step further. How many bettors are sharp? Read on to find out.
It’s commonly believed that bookmakers evolve their odds in a betting market via a form of price discovery. In part one of this resource, I attempted to build a rudimentary model to show how this might work. My first attempt captured the essence, but not the nuances of the odds evolution. Price movement was far too large; the model failed to adequately accommodate the action from large-staking bettors.
To deal with the influence of large bets, for my next set of model runs I increased the strength of the stake limitation, from 1 times existing traded volume to 1/5, 1/10 and finally 1/50 times existing traded volume. Here’s the chart for 1/50 stake limit, again with no skilled bettors. Initial market seeding was still 1 unit for A and B, implying the maximum initial stake permissible is 2/50 = 0.04 units. Evidently, this bookmaker would leave even the most cautious of bettors frustrated.
It certainly looks more like what we might expect from a real market but the fluctuations in prices still seem to be rather on the large side. We can measure how large by means of the standard deviation in odds over the course of their evolution. In this example the standard deviation was 0.116. This is considerably larger than observed market movements. Collecting a small sample of data for Premiership matches, I found the average price movement standard deviation to be about 0.04 in the total goals and Asian handicap markets.
We can reduce this figure closer to 0.04 if we introduce some skilled players into the model. In the model output below, 1 in 3 were skilled. This happens because there are fewer big-stake bets placed. Skilled bettors won’t bet when the price is below 2.00, so their extra volume that can influence price discovery is missing.
Whilst the odds evolution looks more promising (standard deviation = 0.041), we have a new problem. When I run this model 1,000 times as part of a Monte Carlo simulation, we find that the variation between opening and closing prices is just too small. Whilst typical opening to closing price ratios for total goals and Asian handicap from real Pinnacle markets show a standard deviation of about 0.055, this model produced an average of just 0.013 over the 1,000 model runs.
The table below show how the 1,000-model-run average standard deviation in the opening to closing price ratio varies for different model scenarios. I’ve applied a heat map to show where the model output is significantly higher (green), lower (red) and broadly similar (yellow) to the real market value of 0.055. We can thus use it to look for ‘Goldilocks’ scenarios that are ‘just right’. My choice of skilled bettor proportion might look a little odd; the scale is logarithmic and simply equivalent to 0, 10-4, 10-3.5, 10-3, 10-2.5, 10-2, 10-1.5, 10-1, 10-0.5 and 100 (or 1). 10-2, or 0.01, for example, is 1%.
Standard deviation in opening/closing price ratio for different stake limit and skilled bettor proportion scenarios (market seeding = 1 unit)
Similarly, the second table shows the average standard deviation in evolving prices over the 1,000 model runs. Remember, my real market sample had an average figure for this standard deviation of about 0.04.
Standard deviation in evolving prices for different stake limit and skilled bettor proportion scenarios (market seeding = 1 unit)
You might notice that all model scenarios show no variation at all where all bettors are skilled. For this model this is hardly surprising since if every bettor ‘knows’ the true odds to be 2.00, none of them are going to back 1.95, and hence the market will show no evolution.
For these scenarios there are none which fit the Goldilocks zone for both standard deviations together. Using an initial market seeding of just 1 unit for A and B is typically not enough to supress volatility in price evolution to realistic levels. We should try increasing the market seeding volume. With respect to the variation in opening to closing price ratios, you may observe a significant step change at around the 0.3 to 1% skilled bettor proportion. Fewer skilled bettors, too much variation; more skilled bettors, not enough variation. Look out for this again in later tables.
Changing the market seeding
Whilst a stake limitation factor of 50 does a pretty good job of reducing volatility in both price evolution and opening to closing price ratio, it’s probably too excessive. In reality, such draconian limitation would severely curtail a bookmaker’s turnover and annoy their customers. Instead of increasing this limitation, let’s change the size of the initial theoretical market seeding.
The next two tables show the two sets of standard deviations for the same skilled bettor proportions and a range of theoretical market seeding values. A market seeding of 100, for example, implies that the bookmaker has seeded the initial market with 100 theoretical units to both A and B. A stake limit ratio of 1 was applied to all scenario pairs.
Standard deviation in opening/closing price ratio for different market seeding and skilled bettor proportion scenarios (stake limit ratio = 1 unit)
Standard deviation in evolving prices for different market seeding and skilled bettor proportion scenarios (stake limit ratio = 1 unit)
As for the stake limit scenarios there isn’t really any Goldilocks pair where both standard deviations are close to observed values. The scenario pair 1% skilled bettors / market seeding 1,000 probably gets closest. But here’s an example evolution; it just doesn’t ‘look’ right, with sudden and occasionally large odds movements and reversals interspacing periods of very limited activity.
These arise because big stakes can still have an impact, even when the bookmaker has initially seeded their market with a large volume. Furthermore, would a bookmaker really want to seed their market with such a large volume in the first place? Doing so would severely limit the odds evolution for the smaller stakes that make up the vast majority of action, as can be seen in the chart.
What if we tried a combination of market seeding and staking limitation? Here’s one evolution for a combination of 250 market seeding and 1/25 stake limiting, for a 1% proportion of skilled bettors.
It looks a lot more realistic, doesn’t it? But again, we can’t get a Goldilocks match; the open/close standard deviation is too low (0.025) whilst the odds evolution standard deviation is too high (0.064). And we still see several unrealistic step changes. Is there way to deal with those?
There’s something wrong with the price discovery model
Thus far it’s been assumed that regardless of the model scenario and the type of bettor, whether skilled or unskilled, their action will influence the market, and the odds, in direct proportion to its size. But is this really the case? For skilled bettors, quite reasonably it might be. But for unskilled bettors?
Suppose unskilled bettors had a disproportionate preference for A over B, a bias reported to exist in both Over/Under and Handicap markets; why would a bookmaker, with a superior prediction model and the ability to avoid irrational judgments, pay any attention to them? Why not simply ignore their action for the purpose of recalculating the odds? Suppose a market of only unskilled bettors bet twice as often on A than B. Without ignoring some of the action, it might end up looking like this. Rich picking for skilled bettors.
In reality, of course, such large deviations away from ‘true’ outcome probabilities would soon be exploited away by the action from skilled bettors, and probably from unskilled bettors too once it becomes obvious the odds are ‘wrong’. There is only so much market inefficiency customers will be blind to, even relatively unsophisticated bettors. Nevertheless, such a view does bring into question the whole interpretation of balancing action through price discovery.
If bookmakers ignore, or at least partly ignore, the action of unskilled bettors, this will inevitably create situations where the bookmaker is forced to take a risk position, where one of the match outcomes might result in a substantial loss. Understandably, for creatures adverse to uncertainty, bookmakers would prefer not to find themselves there. However, if they have a means of understanding and managing those risk positions, and in the process create a bigger return, why wouldn’t they?
Ignoring unskilled action
In my final set of scenarios, I’ve applied a range of bookmaker reaction ratios to any unskilled action. A ratio of 1 implies the bookmaker reacts to all the action, as was the case for the model scenarios thus far, by including the whole stake for the purposes of evolving the odds. For a ratio of 2, only half is considered, whilst the other half is ignored. For a ratio of 64, just 1/64th of the action will be considered. For these scenarios I have applied a market seeding of 100 and a stake limit factor of 5 to guard against any excessive market reaction to a large stake from a skilled bettor. The standard deviation heat maps are shown below.
Standard deviation in opening/closing price ratio for different bookmaker reaction ratios to unskilled action and skilled bettor proportion scenarios (market seeding = 100 units, stake limitation factor = 5)
Standard deviation in evolving prices for different bookmaker reaction ratios to unskilled action and skilled bettor proportion scenarios (market seeding = 100 units, stake limitation factor = 5)
A few features stand out. Firstly, there is again a noticeable ‘wall of possibility’ in the proportion of skilled bettors beyond which there just simply isn’t enough variation in either the opening to closing price ratio or the stepwise market evolution. This appears at around 1%. For scenarios with more skilled bettors, there are now too many refusing to bet (no value). This restricts the odds movements to levels lower than those typically observed. Here’s a typical odds evolution with 5% of bettors skilled. It just looks far too ‘quiet’.
Secondly, for scenarios with a very low or zero proportion of skilled bettors, there remains too much variation in both measures unless a significant factor is applied by the bookmaker to the amount of action they will ignore from an unskilled bettor. Here’s a typical chart for all unskilled bettors where the bookmaker ignores only half their action. Far too ‘noisy’ and erratic.
Finally, we have a Goldilocks zone where the proportion of skilled bettors is relatively small (perhaps between 0.1 and 1%) and the bookmaker reaction ratio is relatively high (32). Much higher than this and again the odds fail to fluctuate. That’s hardly surprising. If the bookmakers ignored everything, then markets with no skilled bettors would not change at all. Here’s a typical odds evolution for this scenario with 0.3% of bettors skilled.
What have we learnt about price discovery?
We have built a model of price discovery that has attempted to replicate the real-world evolution of two-way betting markets like over/under, point spread and Asian handicap. We have found it necessary to have the bookmaker seed their market with some theoretical funds to kick-start the evolution of odds with action from bettors, and for the bookmaker to apply some limits to that action. Finally, it has been necessary to consider ignoring a lot of the action from unskilled bettors for the purposes of evolving those odds.
Bookmakers may evolve their betting markets via a process of price discovery, one where they arguably pay far more attention to a small proportion of skilled bettors than the vast majority of unskilled ones.
These model parameters seem to me to be representative of realistic behaviour on the part of a bookmaker. We know that they apply stake limits, with the limitation more significant during the earlier periods of market evolution where there has been less traded volume. It also makes sense to have some sort of theoretical market seeding to kick start a market. Without it, early prices would fluctuate all over the place far from the ‘true’ values.
Finally, it is commonly accepted wisdom that skilled bettors with meaningful information to bring to a market will influence it far more than those who just bet on a hunch. Pinnacle openly admit that their most skilled customers are used to refine the quality of the odds. Indeed, some of them may even be offered the opportunity to do so before the market opens publicly; a form of market seeding perhaps.
The question then remains: how many skilled bettors are there? Having tested a range of model scenarios, varying these three parameters, I believe I have found further support for the proposition that the likely proportion of bettors who are winners, as I defined them at the start of this two-part article, is low, and possibly between 0.1% and 1%.
To reiterate my warnings at the start of these articles, this model of price discovery and odds evolution is very much my own thought process. It is potentially full of inconsistencies and flaws. Furthermore, it hasn’t considered the possibility that a bookmaker might ignore some of the action of a skilled bettor as well, if they believe themselves to be even more skilled than some of those customers too.
I did, however, test it under conditions, typical of real markets, where biased bettors bet on A more often than on B. For a 2 to 1 scenario, and by further appropriate weighing of the amount of attention the bookmaker would pay to unskilled bettors of A versus B, I was able to reproduce very similar numbers and charts to the ones above.
Perhaps the most significant weakness is the restriction of the model to fixed number of 1,000 iterations (and typically about 500 bets). In reality, markets may have far fewer or far more than this, with implications for how this might influence the opening to closing price ratio variability.
In short, pretty much everything I have done is guesswork. Nevertheless, if nothing else it has shone a light on how bookmakers may evolve their betting markets via a process of price discovery, one where they arguably pay far more attention to a small proportion of skilled bettors than the vast majority of unskilled ones. If you want to be one of the former, you’ll have to work hard. One place to start is by reading the rest of Pinnacle’s educational resources.