Ensemble vs Time: Betting Strategy Simulation

Ensemble vs Time: Betting Strategy Simulation

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Imagine a lively casino brimming with daydreamers hoping to spin meagre coins into great fortunes one bet at a time. Others position themselves on the sidelines, cheerfully tallying the incoming bets. The daydreamers are the players, and the sideliners are the house.

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Volatility Drag

In the bustling arena of sports trading, a hidden force exists called volatility drag: the mathematical difference between geometric and arithmetic averages. This difference resembles a tax due to the mathematics, which imposes a lower compound return when returns vary over time. This difference between the average outcome that a bookmaker tallies (from countless eager bettors) and a single bettor's actual lived experience can become very wide.

To illustrate, we have created a Betting Strategy Simulator that reveals how luck may bolster or batter your bankroll.

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Two Sides of the Same Bet

1. Time Perspective (Median)
   Regard this perspective as the lone trader, placing multiple bets in sequence. Each triumph or tumble weighs heavily on their well-being. Over many tries, their median final bankroll can suffer from volatility drag, meaning a few unlucky tumbles may gouge deeper than occasional victories can heal.
2. Ensemble Perspective (Mean)
   Consider the bookmaker presiding over a torrent of simultaneous wagers. In that swirling chaos, all results average out. While heartbreak and jubilation strike individuals, the house sees a calm, aggregated mean that, if well-calculated, coasts along in a far more stable fashion than a single bankroll can hope for.

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Key Parameters

• Probability of Winning: A Percentage.
• Odds: For example, odds of 2.30 yield the stake plus 130% more if we win.
• Stake: The size of each bet (percentage).
• Number of Bets: Length of the trader's journey or the repeated steps over which the bookmaker aggregates.
• Number of Simulations: Number of scenario replications.

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Running Simulation

1. Inputs
   • Enter probability, odds, stake, and so on.
2. Process
   • Click on Run Simulation to launch the simulation.
3. Outputs
   • Monte Carlo (Time) Results: The Median Final Bankroll for the solitary trader forging through a sequence of wagers. The geometric rise (or descent!) becomes evident here.
   • Monte Carlo (Ensemble) Results: The Mean Final Bankroll from the vantage of the house. With each bet in parallel, the chaos yields a predictable average - an expected value.

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Under the Hood

• Time: We line up $(N)$ traders, each living out $(M)$ consecutive bets. Their final bankrolls vary widely, but the median is the honest sentinel of their fortunes.
• Ensemble: We replicate $(N)$ parallel bets for each of $(M)$ rounds, calculate the mean outcome each time, and watch the bankroll grow in that aggregated manner.
• Volatility Drag: If there is one lesson to learn, it is that a 50% dip requires a 100% surge to climb out of the pit. The bigger the stake, the more each stumble stings and volatility seldom shows mercy.

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Reading Results

1. Single Bettor's Plight
   The median bankroll can unravel if luck sends you through a rough patch. Even with a favourable win probability, sustained drawdowns hurt more than fleeting upticks help.

2. Parallel Paradise
   The bookmaker's many concurrent bets form a serene environment where the mean bankroll (averaged across countless outcomes) marches forward in lockstep with the basic mathematics of expected value.

3. Practical Wisdom

   • As a punter, carefully consider the violent power of sequential losses. Bankroll management becomes your shield, lest a string of flops knock you out.
   • As the house or aggregator of bets, relax behind a wide net of players, diluting the wilder swings of misfortune.

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Looking Forward

Volatility drag is a subtle and cunning opponent that shrinks big dreams. The simulator reminds us that mean vs. median can diverge drastically. The sports trader sees the ephemeral illusions of large short-term gains, recognising the risk that a run of losses can carve away capital faster than big wins can restore it. Meanwhile, the bookmaker basks in the calm assurance of ensembles: a stable accumulation of profits gleaned from the grand churn of wagers.

If nothing else, remember that early wins are crucial in the time dimension but not so in the Ensemble dimension!

Enjoy!

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Note: An LLM generated the first draft of this post based on our simulator code listing.

Bellman Bets Meets Shannon's Demon

Bellman Bets Meets Shannon's Demon

As 'Royal Ascot' week is upon us with a splendid array of graded stakes races, it would be nice to enjoy some convex betting opportunities without risking too much capital and ending the week with either a small loss or a small gain.

To that end and with a certain amount of tongue-in-cheek attitude, we present our 'Bellman Bets meets Shannon's Demon' (or Session Handicapping meets Market Rebalancing) strategy. The ideal conditions for using this approach include:

• Markets should be volatile (i.e. range of short-, medium-, and long-priced winners),
• Markets should be negatively correlated (i.e. Local Track and Royal Ascot) or uncorrelated (e.g. cash), and
• Rebalancing costs should be very low or zero.

Let us assume that we have the following parameters for both markets:

• Number of Races = 7
• Current Bankroll = 117
• Target Bankroll = 525
• Win Probability (Avg) = 0.15
• Decimal Odds (Avg) = 7.00
• Current Race = 1

In order to put the protocol through its paces, we simulate the first four Bellman bets (assuming three losses and one win):

[Local Track; 13:45]
java.exe GamblersRuin 7 117 525 0.15 7.00 1
Success = 0.23789
Stake   = 13.0

[Local Track; 14:20]
java.exe GamblersRuin 7 112 525 0.15 7.00 2
Success = 0.22352
Stake   = 29.0

[Royal Ascot; 14:30]
java.exe GamblersRuin 7 103 525 0.15 7.00 1
Success = 0.21001
Stake   = 36.0

[Local Track; 14:55]
java.exe GamblersRuin 7 131 525 0.15 7.00 3
Success = 0.24309
Stake   = 19.0

leaving us with the a small profit. Note that the winning bet at Royal Ascot assumes it was on one of multiple selections in that particular market.

Ideally, we should consider quitting for the day if we either win or lose 50% of the initial bankroll. If the former outcome occurs, then we should also revisit our estimates for average price and average win percentage. Either way, our 'portfolio' is automatically rebalanced for the next day.

Mathematically, Bellman maximizes our probability of reaching a specific target using a limited number of events and Shannon reduces the impact of volatility drag on our portfolio by rebalancing after every event.

Tread carefully, do your own research, and enjoy!

Ensemble Averages Vs Time Averages

Evaluating Gambles Using Dynamics outlines his idea of time averaging in contrast to ensemble averaging in a discussion of the St Petersburg Paradox. Note the parallels with volatility drag, median outcomes, and expected values!

Analytical (Kelly) And Numerical (Solver) Solutions

Many sports trading problems yield to both an analytical and a numerical solution.



In the above example, the numerical solution (using Solver in Excel) to minimizing the difference between expected value and volatility drag over a sequence of similar bets equals the analytical solution (using Kelly) for the same sequence!

Volatility Drag

Aaron Brown, author of The Poker Face Of Wall Street, makes a strong case for the negative impact of volatility drag on expected value with respect to the Kelly Criterion in the following posts:
  * Short-Term Variance
  * Risk Of Ruin And Kelly Betting
  * Bankroll Performance Simulator
  * Betting Strategy.

The above before and after illustrations show a worked example of setting stakes to match a zero difference between expected value and volatility drag.