- Bankroll management in betting strategy
- Understanding variance
- Different bankroll implications
Betting strategy - Betting bankroll management and awareness of variance are crucial skills for successful online casino enthusiasts. Understanding the intricate relationship between odds, edge, and variance is vital. How do these factors influence a gambler's bankroll over time? Continue reading to discover insights into these dynamics.
By grasping the expected outcomes across a range of bets, effective bankroll management aids a gambler in steering clear of cognitive biases such as Overconfidence Bias, Self-attribution Bias, and The Illusion of Skill. These biases can diminish expected profitability in the long run. This exploration delves into how odds, edge, and variance interplay, offering guidance on bankroll management for gamblers.
Bankroll management
For any gambler, from poker aficionados to sports betting experts, mastering bankroll management and understanding variance are indispensable. Traits shared among all adept gamblers include their knack for quantifying their advantage and attributing outcomes to luck, whether favorable or otherwise.
Imagine placing a wager at odds of 2.0, which suggests a 50% likelihood (without margin). If a gambler accurately deduces that the actual probability is 52% (a true price of 1.92), then the expected gain for each wager at 2.0 would be 4% (2.0/1.92 – 1). This is referred to as the gambler's ‘edge’.
Assuming a gambler begins with a bankroll of 100 units and stakes one unit per bet. After 100 such wagers, the bankroll could range from nothing to 200 units, but it's anticipated to be at 104 units—a 4% profit.
Simulating this scenario 10,000 times reveals how variance impacts the gambler's bankroll, as shown in the chart below.
Understanding variance
While the mean result was a near-four-unit boost in bankroll, the disparity between the best (+38 units) and worst (-30 units) results is significant. It's crucial for gamblers to comprehend variance and recognize that a 4% edge doesn't assure a 4% gain.
With 100 bets under this simulation, gamblers could anticipate a return between -12 and +20 units 90% of the time. A 10 unit decline from the initial bankroll is likely about 20% of the time, yet only 2% might experience a 20 unit decrease.
Interestingly, in 32% of cases, gamblers might find themselves at a loss after 100 wagers, despite having a 4% edge on every bet.
Enhancing the gambler's edge to 10% (true probability of 55% for a bet at 2.0) resulted in a loss 13% of the time after 100 bets.
The probability of a decline of 20 units or more was merely 0.4%. Clearly, as the edge increases, the chance of a losing streak diminishes, but what occurs when the number of wagers rises to, say, 5,000? The following chart illustrates the first scenario (52% true probability, betting at odds of 2.0) simulated 10,000 times.
While the most unfavorable outcome was a loss of 72 units, only 28 (0.28%) of the 10,000 simulations ended in a loss after 5,000 bets. In 90% of simulations, a return between +82 units and +314 units was produced. This illustrates a return on investment (ROI) ranging from 1.64% to 6.28%.
What changes if, instead of betting at 2.0, the odds are set at 4.0 (implied probability of 25%)? If the true probability is determined to be 26% (true price of 3.846), the expected return for each bet remains at +4% (4.0/3.846-1), but how does variance react?
How do the charts compare?
Comparison of the two charts shows a significant increase in variance despite identical bet sizes, number of bets, and expected returns. The standard deviation of returns rose from 1.4% to 2.4%. The range of simulated outcomes is 64% larger when betting at 4.0, with a 90% confidence interval 72% wider, showcasing an ROI between 0% and 8%.
In the first scenario, the gambler lost their entire 100-unit bankroll in only 2 of the 10,000 simulations (0.02%). In the latter scenario, the entire 100-unit bankroll was depleted in 6.3% of simulations. A 50-unit reduction was significantly more probable (25.7%) when betting on the 4.0 outsider compared to betting at 2.0 (2.0%).
In the direst scenario betting at 4.0, nearly three whole bankrolls (-276 units) were lost. This example illustrates that with a consistent bet size, number of bets, and expected return, variance escalates as the odds increase.
Therefore, a gambler primarily betting on underdogs should anticipate more significant and more frequent fluctuations in their bankroll than one betting on favorites, even if their edge remains constant.
Considering it might take years for a sports bettor to place 5,000 wagers, it's more pertinent to understand the bankroll implications when making a considerably lower number of bets.
Assuming a gambler finds a 4% edge at odds of 2.0 and stakes a fixed 1 unit, the chart below depicts the likelihood of experiencing a specific unit reduction from the starting bankroll over a range of 100 to 1,000 bets, based on 10,000 simulations.
Placing 1,000 bets at odds of 2.0 with a 4% edge, the probability of encountering a particular reduction seems to be nearing its maximum, particularly for smaller decreases. As the gambler's edge grows, the likelihood of a certain reduction diminishes. The following chart maps this probability across 1,000 wagers at odds of 2.0, based on 10,000 simulations.
For instance, with a 4% edge, the probability of a 20-unit reduction during 1,000 bets at 2.0 was 17.4%. Nonetheless, the chance of being down by 20 units or more after 1,000 such bets was only 2.8%. Understanding this distinction ensures a gambler can look beyond short-term variance towards their long-term advantage.
Different bankroll implications
What are the bankroll implications if the bet size and edge remain constant, but the odds vary? The chart below outlines the probability of various reductions (from the starting bankroll) when a gambler places 1,000 1-unit bets at different odds, with a 4% edge. Each series of 1,000 bets was simulated 10,000 times.
Remember, when betting at odds of 2.0, there was a 17.4% chance of being down by 20 units at some point through a series of 1,000 bets. At odds of 5.0, the likelihood of a 20-unit reduction increases to just under 60%. With the same stake, edge, and expected return from a series of bets, predominantly backing favorites or longshots leads to markedly different bankroll implications in terms of variance.
Therefore, recognizing what type of gambler you are is essential for managing the inevitable ups and downs you'll encounter.
To quantify this variance, consider once more a series of 1,000 bets. By altering the odds (implied probability from 10% to 90%) and edge, the chart below illustrates the standard deviation of returns.
It's evident that variance escalates as the odds increase (or as implied probability decreases), consistent with the analysis above. From the chart, making 1,000 1-unit bets with a 10% edge results in a standard deviation of 6.5% if all bets are placed at 5.0, compared to 2.5% when betting at 1.67. In both instances, the expected return is +100 units (+10%).
An intriguing finding is that for odds shorter than 2.0, as the edge (and thus expected return) increases, standard deviation actually diminishes. Finding an increasing edge with odds shorter than 2.0 is beneficial not only due to the higher expected return but also because it comes with reduced variance.
Summaries to be drawn from the data
This article has scrutinized the interrelations between odds, edge, and variance through simulating a series of bets with a positive edge.
While a larger edge and increased number of bets enhance the likelihood of overcoming a spell of bad luck, it's crucial for sports gamblers to recognize their gambling style and quantify their edge accurately.
This knowledge enables them to avoid discouragement during losing streaks or falling prey to overconfidence biases when on a winning streak.
While a gambler may not know their exact edge at the moment of placing each bet, previous discussions have highlighted the importance of using our closing price as an indicator of the true price.
Consistently outperforming the closing price indicates that a gambler is likely to achieve a positive long-term return.
However, if a gambler consistently profits at our closing prices, it might indicate an inefficiency in the market that has not been accounted for. 7x7Bets' policy of welcoming winners ensures that an edge remains accessible to any gambler as long as it exists.
Now you know what is the bankroll management and understand what is the odds, edge and variance. Sign Up Now or click HERE to play at 7x7Bets, the most reliable and trustworthy online casino in India. Don't forget to claim your withdrawable real money welcome deposit bonus, weekly cashback bonus and referral bonus!
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