Best Sellers Fixed Matches

**Best Sellers Fixed Matches**

The Kelly Criterion has come to be accepted as one of the most useful staking methods * Best Sellers Fixed Matches* for sharp bettors. While most of us think we have an understanding of the Kelly Criterion. How it works, this is merely a simplified version of the formula. Our latest Guest Contributor has provided an in-depth explanation of the “real” Kelly Criterion. Read on to learn more.

**Examine the potential flaws**

Anyone who is unfamiliar with how the Kelly Criterion can be used to determine optimal bet sizes should read Dominic Cortis’ article on how to use the Kelly Criterion for betting. This approach works well in most cases, however, there are some situations where the * Best Sellers Fixed Matches* can give some head-scratching results.

Using the examples below, we can examine the potential flaws in using a simplified Kelly Criterion formula.

Example #1 – A * soccer fixed games* where both a visitor win and draw outcome provide the bettor with an edge:

The Kelly formula would suggest staking 2.5% of bankroll on both the visitor win and the draw, staking a total of 5% of bankroll. Looking at the * Handicap odds fixed matches *for the same

*changes how we might view the use of the Kelly Criterion.*

**soccer game fixed betting odds**Example #1A – The same soccer game in example 1 re-stated as a Handicap line:

A bet on the visitor +0.5 at odds of 2.50 is the equivalent to betting half the amount on both the visitor win and draw (both at odds of 5.00). So why does the Kelly formula give a different answer?

The answer is that the formula commonly known as the Kelly Criterion is not the real Kelly Criterion – it is a simplify form that works when there is only one bet at a time.

**How to use the COMBO FIXED MATCH**

Below is an explanation of how to apply the generalized Kelly Criterion to betting:

- Step – 1: List all possible outcomes for the entire set of
**COMBO FIXED MATCHES** - Step – 2: Calculate the probability of each outcome.
- Step – 3: For each possible outcome, calculate the ending bankroll for that outcome (starting bankroll plus all wins minus all losses). Leave the bet amounts as variables.
- Step – 4: Take the logarithm of each ending bankroll from step 3.
- Step – 5: Calculate the weighted average of the logarithms from step 4, weighted by the probabilities from step 2. Call this the “objective”.
- Step – 6: Find the set of bets that maximises the objective from step 5. These are the optimal bets according to the Kelly Criterion.

In order to find the set of bets that maximises the objective, simply use Microsoft Excel’s built-in “solver” module (see below) – this takes care of the complexities of advanced calculus and eliminates a tedious trial-and-error approach.

The result from using these six steps is as follows:

Note that this is identical to the result in Example #1A, where the simplified version of the Kelly Criterion does work. By making two mutually exclusive bets on the same game, the two bets act as a partial hedge for each other. Reducing the overall level of risk, which Kelly rewards by increasing the bet amount (compared to the calculation in Example #1).

**Additional uses from Manipulated fixed matches**

We have already seen how this generalized Kelly Criterion can produce completely different results than the simplified Kelly formula that most bettors will use when there are * multiple fixed matches *edges in the same game.

There are, of course, occasions when you might have multiple edges on different games, all taking place at the same time. The example below is one such situation:

Example #2 – Betting with an edge on four separate games that are all taking place at the same time.

Now while each of these bets make sense individually, using the simplified Kelly Criterion would result in staking 110% of bankroll – something that clearly doesn’t make sense. However, by applying the six steps stated above. We can see how the generalised Kelly Criterion produces a different set of results.

Because the four games are independent, the probability of each outcome can be calculated as the product of the probabilities of each game; for example, the first row probability would be calculated as:

75% x 85% x 80% x 70% = 35.7%

**Calculating the optimal staking amount**

In additional to calculating the optimal staking amount for a bet with multiple edges, the generalized Kelly Criterion can also be used when bettors have a viable hedging opportunity.

Example #3 – Hedging Garbine Muguruza to win Wimbledon in 2015.

Using the 2015 Wimbledon tournament example previously used in this hedging article, we can see how the generalized Kelly Criterion should be applied to a hedging opportunity.

If you had €1,000 starting bankroll, and you bet €10 on Muguruza at 41.00 to win Wimbledon 2015 outright, you would have to decide how much to hedge on Williams at 1.85 in the final. Let’s assume that the odds in the final are efficient, that is. They accurately reflect each outcome’s probability so that there is no edge on either side.

The more commonly used simplified Kelly formula

The more commonly used simplified Kelly formula would provide the following results in the scenario:

However, applying the generalised Kelly procedure as stated above yields the results below:

**Accurate sources fixed matches bets**

Using this method shows that optimal strategy would be to bet €183.41 on Williams at 1.85 to beat Murguruza in the final and win the tournament. This will hedge most, but not all, of your open position on Murguruza to win the tournament.

Taking the exponential of the objective gives an interesting number, called the “certainty equivalent”. In Example #3 above, the certainty equivalent is exp(7.072341) = 1,178.90. This means that, from a * Best Sellers Fixed Matches* perspective. The bettor would be indifferent between having the listed set of bets and having €1,178.90 in risk-free cash.

If we remove the hedge bet, we are left with the following:

So, the effect of the hedge bet is to raise the certainty equivalent from €1,166.87 to €1,178.90. So even though the hedge bet itself has a negative expected value. The resulting reduction in risk is so beneficial that from a Kelly perspective, it has created added value that’s equivalent to €12.03 in cash.

**Some other applications of Generalised Kelly**

Finding optimal bet sizes for a set of “round robin” combinations of parlays or teasers;

Finding optimal bet sizes for a set of futures bets on several different teams to win the same division or championship;

Deciding between different ways to hedge an existing bet (money line, spread, buying/selling points), especially if some options result in a “middle” opportunity;

Figuring out how much to add to, or exit from, an existing position after a line move.

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**A percentage staking bankroll calculator**

Provided all our bets have the same odds and stake percentage. We can build a calculator to put our bankroll growth equation to work, plotting. The distribution of possible bankroll growth figures for different win/loss rates. Using an Excel calculator that I have built for my own website. The charts below show outputs for various betting scenarios.

The first compares the performances of three different betting odds using a fully Kelly staking plan over 1,000 bets. With bets holding an EV of 5%, the percentages stakes for * Best Sellers Fixed Matches* 1.5, 2.0, and 5.0 respectively are 10%, 5% and 1.25%. Expected bankroll growth for these three odds scenarios are 147, 12.1 and 1.87. Whilst median bankroll growth figures are 12.7, 3.49. And 1.36. The green distribution is effectively a match for the Monte Carlo distribution above, given that the inputs for the model were the same. For

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**SURE FIXED MATCHES TOMORROW**The next chart shows how the bank growth distribution varies with EV. Three scenarios are shown: 1%, 3% and 5%, all with odds of 2.0. 1,000 bets and again with a full Kelly stake (1%, 3% and 5% respectively). Expected and median bankroll growth for these were 1.11, 2.46 and 12.1, and 1.05, 1.57 and 3.49.

**The size of the Rigged fixed matches 1×2**

The third chart illustrates how the bankroll growth distribution changes when we reduce the size of the Kelly fraction. With odds of 2.0 and EV of 5%, full, half and quarter Kelly stakes are 5%, 2.5% and 1.25% respectively. Expected and median bankroll growth for these were 12.1, 3.49 and 1.87 and 3.49, 2.55 and 1.73.

As mention earlier, fractional Kelly is often advocat to moderate risks. The distribution above illustrates why. Whilst the probability of poor performance is reduce significantly (compare the area to the left of bank growth = 1 for the blue and green distributions). The median * Best Sellers Fixed Matches* is only marginally smaller (2.55 compare to 3.49).

Granted, your expected (mean) bankroll growth is much bigger with full Kelly but most of the time you won’t see that. The median is arguably a better measure of what you should expect to happen in this context. With full Kelly you still have a 21.5% chance of making a loss. For half Kelly, that reduces to 11.8%.

**Expected and median COMBO FIXED MATCHES**

We can use the calculator to see how the expected bankroll growth will vary with the number of bets. We already know from our equations above that this will be logarithmic. The median bankroll growth also varies logarithmically. You can easily watch your revenue being maximized by using our * PREDICTION TIPS FIXED MATCHES TODAY*.

Finally, see how the median bankroll growth varies with your EV. Again, this scenario is for odds of 2.0, but you could create similar charts for other * betting odds fixed matches 1×2*.

**Halftime Fulltime fixed matches**

**Variable odds, variable stakes**

The equations and calculator described relying on all bets having the same odds and same stake percentage. How robust will they be under real world scenarios where both may vary? Testing against Monte Carlo simulations reveals that odds can vary considerably without having too much impact on the reliability of the calculator’s outputs. But only provided the stake percentages are all the same. Obviously, that will not be the case with Kelly staking.

The calculator is also robust for variable stake percentages. For example, those advised by the Kelly strategy, provided the odds don’t vary too much. A typical example would be * Asian Handicap fixed matches* or point spreads, where most odds are close to 1.95, with minimal deviation.

Reliability is weaker when EV for these bet types also varies. But again provided neither the odds nor the EV for those bets varies too much. The calculator offers a reasonable method of providing quick estimates of performance expectation.

We know that defining expectations from level staking is relatively straight forward. However, this article has shown that we can do the same for percentage staking too. Whilst level staking performances will be normally distribute, those from percentage staking are distribute log-normally.