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.

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'!

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.

Kelly's Multiple Personality Disorder

For the professional sports-trader, Kelly has three separate mathematical forms:

1. Single Event, Single Selection;
2. Single Event, Multiple Selections; and
3. Multiple Events, Multiple Selections.

Single Event, Single Selection

This is the basic case as outlined in cursory descriptions of the Kelly Criterion. We identify a selection, which gives us an edge over the market and calculate the optimal stake to maximize that advantage. For this purpose, the Excel Add-In offers the KellySingleStake sports-trading function that accepts Decimal Odds, Win Probability, and Multiplier parameters. The KellySingleStake function is the correct formula for AvB events such as moneyline markets in MLB, NBA, and the NFL.

Decimal Odds	Win Probability	Stake
2.00	53%	6.00%

Single Event, Multiple Selections

In AvB events, the general advice to only bet the overlay is technically correct. However, in an AvK event, such as horse-racing and golf with a number of mutually-exclusive outcomes this advice is not strictly correct. Kelly betting is predicated on maximizing the logarithm of the handicapper's bankroll over the long-term. But, in the short-term, that goal is translated into not losing specific events when the price is right! The key role played by overlays in mutually-exclusive events is that there must be at least one such betting option available in any event on which we wish to bet. Beyond that, the specific choices will only be governed by maximizing the logarithm of our bankroll! The sports-trading function, KellyMutExStakes (array formula), with Decimal Odds Range and Win Probability Range inputs will identify the optimal selections and stakes.

Entry	Decimal Odds	Win Probability	Trader Edge	Stake
Charlie	5.50	20.00%	10.00%	5.96%
Alpha	2.625	40.00%	5.00%	10.59%
Bravo	3.25	30.00%	-2.50%	6.25%
Delta	6.00	8.00%	-52.00%	0.00%
Echo	21.00	2.00%	-58.00%	0.00%

Multiple Events, Multiple Selections

For Nx(AvB) events, such as trading Ryder Cup golf singles matches or NFL games on Any Given Sunday, we need the sports-trading function, KellySimEvtStakes (array formula), with Decimal Odds Range and Win Probability Range parameters to identify the optimal stakes.

Entry	Decimal Odds	Win Probability	Trader Edge	Stake
SvWH	1.50	75.00%	12.50%	9.10%
BHAvB	1.40	80.00%	12.00%	11.70%
DvAV	1.60	70.00%	12.00%	6.825%
LvCP	1.30	85.00%	10.50%	14.70%

Note that Example #2 in the Pinnacle Guest article - The real Kelly Criterion - calculates the wrong stakes as can easily be confirmed by entering the decimal odds and win probabilities into the SBR Kelly Calculator for four independent events.

Kelly And Mutually-Exclusive Outcomes (AvK)

In AvB events, such as baseball, basketball, or football, the general advice to only bet the overlay is technically correct. However, in an AvK event, such as horse-racing, with a number of mutually-exclusive outcomes this advice is not strictly correct. For example, in the following racecard (sorted in decreasing e.v order), even though the handicapper has rated Bravo's win probability (π) at 29%, it is an underlay and not included in the list of bet selections:

h	p	π	e.v	Σ(π)	a.k	b.k	%
Echo	21.00	7.00%	1.470	0.070	0.048	0.977	2.49%
Charlie	5.50	20.00%	1.100	0.270	0.229	0.947	2.78%
Bravo	3.25	29.00%	0.943	0.560	0.537	0.951	0.00%
Delta	6.00	13.00%	0.780	0.690	0.704	1.047	0.00%
Alpha	2.50	31.00%	0.775	1.000	1.104	0.000	0.00%

But, if Bravo's price was to drift to 3.35, then this underlay is now added to the list!

h	p	π	e.v	Σ(π)	a.k	b.k	%
Echo	21.00	7.00%	1.470	0.070	0.048	0.977	2.56%
Charlie	5.50	20.00%	1.100	0.270	0.229	0.947	3.05%
Bravo	3.35	29.00%	0.972	0.560	0.528	0.932	1.18%
Delta	6.00	13.00%	0.780	0.690	0.695	1.015	0.00%
Alpha	2.50	31.00%	0.775	1.000	1.095	0.000	0.00%

Kelly betting is predicated on maximising the logarithm of the handicapper's bankroll over the long-term. But, in the short-term, that goal is translated into not losing specific events when the price is right! The key role played by overlays in mutually-exclusive events is that there must be at least one such betting option available in any event on which we wish to bet. Beyond that, the specific choices will only be governed by maximising the logarithm of our bankroll!

Note: Blindly backing high probability combinations such as Alpha, Bravo, and Charlie (total win probability = 80%) will eventually lead to ruin. In order to calculate the correct stakes, make sure table is sorted by column 'e.v.' in descending order!

Horse-Racing Overlays

Sports traders sometimes conflate AvB events with AvK (K = N-1) events (N = number of entrants). The prototypical AvB game is a football match and the equivalent AvK example is horse-racing. Some experts encourage traders to identify a single overlay in both events and bet accordingly. Whereas this advice is generally correct for AvB events, it is not correct for AvK events.
In horse-racing, you should only select those races with at least one overlay for further examination. But, as Ravi Phatarfod points out in his excellent 1996 paper Betting Strategies In Horse Races, a gambler is more likely to correctly assess that the winner will be one of three horses than being able to correctly assign win probabilities to each individual horse. And, as John Haigh illustrates in Taking Chances, Kelly betting on horse races also encourages us to spread our risk across multiple entrants including, on occasion, those from all three categories of bets, favorable (positive expected value), fair (zero expected value), and unfavorable (negative expected value) bets. This approach guarantees over time that you will minimize your risk of ruin (total loss of capital).

Note: You must enter horse details in descending e.v. order only.

Adaptive Boosting

Machine learning studies the design of automatic methods for making predictions about the future based on past experiences. In the context of classification problems, machine-learning methods attempt to learn to predict the correct grouping of unseen examples. In the mid-1990s, Freund and Schapire introduced the meta-heuristic, Adaptive Boosting (AdaBoost), “…an approach to machine learning based on the idea of creating a highly accurate prediction rule by combining many relatively weak and inaccurate rules… Indeed, at its core, boosting solves hard machine-learning problems by forming a very smart committee of grossly incompetent but carefully selected members…” (Boosting: Foundations And Algorithms). As context for how boosting might work, the authors introduce the following toy problem in the opening paragraph to A Short Introduction To Boosting: “A horse-racing gambler, hoping to maximize his winnings, decides to create a computer program that will accurately predict the winner of a horse race based on the usual information...” As discovered by the early pioneers of expert systems in the 70s and 80s and as acknowledged by the authors, the biggest stumbling block to using experts is that many of them are unable to detail their decision process or even to rank order the importance of key variables. In light of this issue, Freund and Schapire point out that the beauty of boosting is that it builds on what experts can do not on what they cannot do, namely, given a specific scenario they are usually able to make a judgment in favor or against a particular outcome. For instance, we can ask an expert handicapper if a specific scenario - course and distance winner in last outing a week ago - would lead him to believe that it was more likely to win again or to finish out of the money. Note the phrase “more likely to” – this is a key strength of boosting - it asks for the balance of probabilities and not for the highly probable. The boosting phase combines many such simple, scenario-based rules into an overall weighted decision for an upcoming event. In its original specification, the defining quality of boosting was that it aggregated an incomplete set of “if-then” rules (decision stumps) that recursively address unexplored regions (areas for which previously chosen rules would give incorrect predictions) of existing data sets. The inherent strength of this approach is that it automatically leverages the key dimensions of Wisdom Of Crowds, namely, diversity, independence, decentralization, and aggregation For a worked example applied to NFL prediction, see James McCaffrey’s Classification And Prediction Using Adaptive Boosting. For the underlying theory of why wisdom of crowds works, see Scott Page’s excellent The Difference. And finally, Malacaria And Smeraldi explore the relationship between the AdaBoost weight update procedure and Kelly’s theory of betting and also establish a connection between AdaBoost and Information theory in On AdaBoost And Optimal Betting Strategies.

Biased Coin (Haghani & Dewey, 2016)

A recent paper by Haghani & Dewey (2016) sheds an unflattering light on subjects formally trained in finance as to their lack of basic knowledge with respect to probability and uncertainty – “If a high fraction of quantitatively sophisticated, financially trained individuals have so much difficulty in playing a simple game with a biased coin, what should we expect when it comes to the more complex and long-term task of investing one’s savings?” Though an otherwise interesting study, there are a couple of key points which do not receive adequate attention in the paper:

• Financial: Though the median final bankroll of $10,504 is derived in the footnotes, there is not sufficient attention drawn to it in the paper itself. Time-Value automatically generates this value whereas Expected-Value generates the wholly unrealistic $3,220,637.
• Psychological: The fallacy of “Playing With House Money” – “…you are offered a stake of $25 to take out your laptop to bet on the flip of a coin for thirty minutes.” What would have happened if the subjects had to pay $25 to play instead of being given it for free? 

No less a luminary in both the financial and gambling worlds than Ed Thorp says: “This is a great experiment for many reasons. It ought to become part of the basic education of anyone interested in finance or gambling.”

Proebstings Paradox - Price Is Right If Marked-To-Market

Proebsting's Paradox refers to a situation in which a sports trader makes successively increasing bets on the same selection in a single market, ostensibly using the Kelly Criterion to calculate the stakes while taking advantage of better and better odds, only to ultimately face ruin.                 

 

The difficulty arises because the sequence of bets appears to cost more in total bankroll percentage than the Kelly Criterion would recommend as a standalone bet at the highest odds in the sequence.

In the Todd Proebsting example, the sports trader initially bets 25% of his bankroll on a 2/1 selection with a 50% win probability. Some time later, he is offered 5/1 on the same selection and calculates his Kelly stake at 22.5% leading to a total wager of (250 + 225) = 475. The problem with this result is that the Kelly stake for a single bet at 5/1 (assuming 50% win probability) is only 40% of bankroll (400) - leading to the theoretical possibility of ruin from betting at successively more attractive odds on the same selection in a single market.
To resolve this paradox, both Ed Thorp and Aaron Brown recommend that the sports trader should "mark to market" his bankroll after the initial 2/1 bet (reducing it by 12.5% from 1000 to 875) and use this updated position to calculate the 5/1 stake. In fact, to stay within the upper limit (400) defined by the standalone 5/1 bet, the trader also needs to reduce his estimated win probability to 42.5%! Using the above example, this would lead a sports trader to bet at most 146.25 = (16.71% * 875) at 5/1 giving a total wager of (250 + 146.25) = 396.25, which is roughly equivalent to the Kelly standalone stake (400) at 5/1 but with a lower upside!