Total Less Than Sum Of Parts - Simultaneous Events

For Nx(AvB) ('multiple events, single selection') scenarios, such as simultaneous, Sunday, NFL games, we cannot just stake them as N separate events as this could potentially involve tying up a large portion of our bankroll.

Let us assume that we have lucked out and three 'home-dogs' are offered at unbelievable odds, as follows:

Treating them as simultaneous events gives us a total stake of 43% (approx.) of bankroll at Full-Kelly, but treating them as three independent events would require a total outlay of 80% (approx.).

Note that all stakes were calculated using the excellent SBR Kelly Calculator. Always keep in mind the specific advice given in Kelly's Multiple Personality Disorder, which outlines the differing incarnations of Kelly Staking depending on the context! Win percentages and odds in the above example are not necessarily realistic for these types of events. That said, last weekend, Betfair offered moneyline odds of 15.00 against the New York Jets winning away to the Los Angeles Rams for amounts any 'Weekend-Warrior' would have happily staked - assuming they estimated the Jets had better than a 7% (approx.) chance of winning!

In sum, if you are trading simultaneous events then the total stake should be less than the sum of the individual single stakes!

Some AvB Events Are AvK Events In Disguise

Many AvB contests (MLB, NBA, and NFL moneyline markets) are exactly what they appear to be - simple win-lose events. But, other AvB contests (soccer win markets) are actually AvK contests in disguise - there are three valid outcomes (win, lose, and draw). This turns a

'single event, single selection' contest into a possible 'single event, multiple selections' one - Kelly's Multiple Personality Disorder.

Treating this soccer match as a single event with three exclusive outcomes leads to an combined investment of 5.93% of bankroll on both the draw and away-win outcomes.

Alternatively, focusing on one or both draw and home-win outcomes as separate selections leads to an investment of 3.75% on the draw outcome and 1.00% on the away-win outcome for a total of 4.75%.

Given your assumed edge relative to the market, this amounts to 'leaving money on the table', in Kelly terms!

All calculations can be replicated using the excellent SBR Calculator.Win percentages are not necessarily realistic for this specific event.

Shannon-Fano Crowd Handicapping

Let us indulge ourselves in a thought experiment on how we might handicap the Crowd!

We have access to a betting-line (Betfair) for a graded-stakes race - with a low WCMI - as well as some simple, publicly-available data. Can we reverse-engineer what the Crowd is most likely factoring into its calculation?

Granted this is no Schrödinger's Cat, but nevertheless it might enlighten us as to whether or not the betting-line is vulnerable?

As outlined in Handicapping Twenty Questions Benford's Law And Shannon Entropy, taking our lead from Shannon-Fano Coding, we should iteratively divide the entrants into two approximately equal groups of win probabilities (i.e. 50%) and use Pairwise Comparison to eliminate the non-contenders using at most four questions.".

When the sub-divisions produced by the splits are approximately equal (50%) in terms of implied probability, then the one bit of information (question) used to distinguish them is maximally efficient. So, using the implied probability (I/P) of the betting-line odds (B/X) as our starting point, we can make an initial split into two groups:

• Alpha, Bravo; and
• Charlie, Delta, Echo, Foxtrot, Golf, Hotel.

Keeping our interpretation as simple as possible, it looks like the initial division is based on speed ratings. Then, in deciding between Alpha and Bravo, trainer rating appears to clinch it.

The next sub-division is:

• Charlie, Delta; and
• Echo, Foxtrot, Golf, Hotel.

Here, form ratings are the most likely rationale for the split with trainer rating again deciding the rank order within the group.

Next, we split the four remaining horses:

• Echo, Foxtrot; and
• Golf, Hotel.

Weight (proxy for fillies allowance) is the deciding factor here but it is not possible to easily account for the final, rank orders within these two groups. We have reached the limits of our simplistic approach. Obviously, the betting-line accounts for more factors than used by our naive approach. But, just because our model is wrong does not mean it is not useful!

In summary, speed ratings appear to be the primary driving factor for the betting-line with trainer rating as the qualifier. Given the likely high correlation between speed and form ratings, the fact that both are used suggests an element of double-counting by the Crowd and, consequently, may indicate a vulnerable betting-line.

Is Schrödinger's cat alive, dead or both?

Variable Weights - Entropy Method

In horse-racing, using fundamental handicapping, we try to derive predictor variables from mining past-performance data. As ever, our starting-point is to ask Bill Benter's fundamental question of handicapping:

What additional variables (if any) explain a significant proportion of the variance in results to date that is not already accounted for by the public odds (Wisdom of Crowds)?

Assuming that we have already identified a number of such variables that appear to influence the outcome of races, how do we weight those variables? Do we weight them separately for different codes (Flat, Jumps), different types (Maiden, Handicap), or different distances (Sprints, Routes) of races?

Obviously, we could use some form of Regression Analysis to derive the necessary weights but, perhaps, a simpler option presents itself! In the Multiple Criteria Decision Analysis process TOPSIS, the Entropy Weight method is used to objectively derive criteria (variables) weights based on the dispersion of scores across the alternatives being analysed. Translating into a handicapping scenario, the underlying assumption of this method is that the greater the difference in scores for contestants across multiple criteria, the greater the difference in predicted outcome for some future event! In other words, we are operationalizing the belief that it is the differences between horses on some key variables and not their similarities (or the differences between race codes, types, distances, and so on) that best determines the winner. Also, all races generate their own unique set of weights and there can be a mixture of positive (1,3,4,5,6) and negative (2,7) weights.

This method has some limitations (particularly relating to scores of zero and entropy values close to one). A number of solutions have been recommended to resolve these issues and the following approach shows promise - New Entropy Weight-Based TOPSIS for Evaluation of Multi-objective Job-Shop Scheduling Solutions.

WASP Trainers

The single most important influence on a horse's performance is the trainer. Current stable form can help explain some of the strange race results we observe daily. To that end, WASP (Winners Above Starting Price), can help us stay current with how in-form trainers are on a rolling weekly, fortnightly, or monthly basis.

• Using historical database of past performances, calculate wins per starting-price using Juvenile Finish Position Ratings algorithm.
• For each yard:
  • For each race in past performances during period
    • Calculate number of actual wins using Juvenile Finish Position Ratings algorithm.
    • Calculate number of expected wins using wins per starting-price.
  • Sum over all races and calculate both actual and expected win percentages.
  • Subtract expected win percentage from actual win percentage to calculate WASP.

Note: Numbers for illustrative purposes only.

The calculated difference tells us (if positive) that the stable is ahead of the market in terms of percentage of opponents beaten or conversely (if negative) that the stable is behind the market.

Thus, WASP informs us of both current stable form and stable value in a single number.

Longshot Stakes: Probability Or Edge

Notwithstanding the specifc advice outlined in Kelly's Multiple Personality Disorder and Kelly And Mutually-Exclusive Outcomes relating to AvK events, consider an idealized horse-racing scenario where you have identified two selections: High Expectations at 2/1 with a 40% win probability and In With A Chance at 20/1 and a 10% chance of winning. Assume further that you are planning to bet ¤50 (Bankroll: ¤500) on High Expectations. How much should you bet on In With A Chance?

Win Probability Stakes

Selection	S/P	Win%	Edge	Stake	Profit
High Expectations	2/1	40%	0.20	¤50.00	¤100.00
In With A Chance	20/1	10%	1.10	¤12.50	¤250.00

Edge Stakes

Selection	S/P	Win%	Edge	Stake	Profit
High Expectations	2/1	40%	0.20	¤50.00	¤100.00
In With A Chance	20/1	10%	1.10	¤27.50	¤550.00

If your answer is ¤12.50, then your handicapping is driven by win probability as High Expectations (40%) is four times more likely to win than In With A Chance (10%). Alternatively, if your answer is ¤27.50, then your handicapping is driven by edge as In With A Chance (1.10) has 5.5 times more edge than High Expectations (0.20).

The Kelly Criterion advises that you choose the stake so that the amount you win is proportional to your edge. Most punters choose stakes based on win probability and, as a result, they are not exploiting their advantage and are 'leaving money on the table'!

Session Handicapping: Stop Or Continue?

Intellectually speaking, we are value handicappers and if a market offers at least one value bet then we are prepared to trade it. However, psychologically speaking, we are session handicappers and we want to maximize our winnings at each session to minimize feelings of regret. These feelings can result from either getting ahead early and then losing later in the session or, alternatively, feeling that we are leaving money on the table because we quit the session too soon.

Bruss (2006) provides us with an approximation to an optimal stopping algorithm in these circumstances. For example, let us assume that we are trading multiple, sequential, sports markets on any given Sunday. We have completed seven markets of an eleven-market session and have accumulated two wins. Should we continue to the next market or stop for the day?

The decision formula is, as follows:

If (N – K) < ((K + 1 - G) / G)
Then "Stop"
Else "Continue"

In our example, with [N = 11, K = 7, and G = 2]  the decision is to continue. But, if after the next market, we have not secured another win [N = 11, K = 8, and G = 2] then the decision is to stop!

Kalman Filter Handicapping

Last Performance Or Median Performance

With respest to historical results, performance ratings and time ratings are routinely calculated, by various organizations, using proprietary formulae or algorithms that are unknown to the average handicapper. Given a track record of such past results, how should we evaluate a horse's latent ability:.

• last performance, or
• median performance.

Last performance captures current form but what if that last result was an unusually good or an unusually bad performance? In that context, we rely more heavily on median performance to better reflect latent ability!

As a guide to which is a better overall indicator (signal) of ability, we can adapt a Kalman Filter to track a dynamic model (changing ability of horse) using an error-prone, measurement process (time ratings: 67..115) to guide our intuitions. The filter predicts the next performance level (ratings: 90..116) beginning with an arbitrary starting-point (90), and proceeds through a series of predict-measure-update iterations using a Bayesian-like updating algorithm (predict=prior, measure=likelihood, update=posterior).

We are probably making unwarranted assumptions about the levels of process noise and measurement error in applying this filter to the current scenario. Plus, we are using a changing time interval between measurements that is non-standard. Nevertheless, we can see from the worked example that the last prediction (116) is a more accurate reflection of current ability than the median performance (108).

In order to experiment with the embedded worksheet, you may need to alter the parameters, as follows:

• X0 = Initial estimated rating,
• P = Average estimated rating error,
• Q = Average rating change per race, and
• R = Average calculated rating error.

Obviously, you will also have to provide the calculated ratings for all past performances of interest!

Practical Dominance (PD)

In terms of our ongoing efforts to improve the handicapping process, we can strongly assert that it is easier to evaluate a four horse race than a nine horse race (all other things being equal). Keeping in mind our strong preference for eliminating alternatives over confirming selections, we can look to the Even Swaps Method (ESM) for a useful concept called practical dominance.

• Select specific race using WCMI.
• For each horse in race (using past performances):
  

  * Evaluate each contestant's form on at most five to seven attributes. See Tsai et al, 2008, Slovic,  1973 and Do You Really Need More Information from the CIA on the positive impact of additional information on confidence (Figure 5).

  * Convert the absolute ratings on each attribute into rankings across contestants.

  * Eliminate those contestants that are either completely dominated (unlikely) or practically dominated (likely) by another entrant.
  

  • In the sample race below, Alpha practically dominates Charlie as his rankings are superior on all attributes except A6.
  • Foxtrot, Golf, Hotel, and Juliet are similarly dominated.

• Consider remaining contestants as potential trades using Kelly Criterion.

As ever, if we cannot find variables that account for sufficient variance in outcomes over and above that provided by market prices then we will not have an edge and we will lose our bankroll.