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Dependence in Kelly Bet Sizing
Dependence in Kelly Bet Sizing

Moving past one of the major limitations of the Kelly bet sizing strategy

Written by Steven
Updated over a week ago

Many bettors have heard of the Kelly bet sizing strategy, which is mathematically proven to be the fastest way to grow your bankroll. However, many bettors have also unfortunately found that, in practice, this strategy can be overly aggressive and result in wild bankroll swings.

One of the biggest drawbacks of Kelly's strategy is that, in general, it requires bets to be independent. This means that if you're finding value in a Best of 3 series, you could be wagering far too much if you blindly apply the standard Kelly bet sizing formula to every map and series bet available.

In order to account for dependence, you need to understand how your bets depend on one another. For example, in a best of 3 series, the Match Winner bet depends on the Map Winner bets. Due to the wide variety of ways that bets can depend on one another, it is impossible to create a general Kelly Bet Sizing tool that can handle arbitrary dependent bets.

However, for common scenarios, it is possible to create such a tool - closely following the idea behind derivation of the original Kelly criterion. The idea is to find the optimal bet sizes which maximize your expected bankroll growth. At Midnite, we've built a Dependent Kelly Bet Sizer that can handle standard map, series, and match handicap bets for best of two and three series - we plan on making this tool available publicly in the near future!

Let's use this dependent bet sizing tool to learn how using a standard Kelly bet sizer could mislead us from betting optimally. We'll assume our bankroll size is $2000. Keep in mind that following the Kelly strategy means maximizing expected bankroll growth, not expected value. Bankroll growth of course requires some positive expected value, but has a preference toward bets that are more likely to win.
Consider a Best of 2 series where we think our team has a 60% chance to win each map. Suppose we can find decimal odds of 2.00 for each map bet.

Our dependent bet sizer also needs to know the chance of our team winning 2-0 (ie winning the -0.5 handicap). To start, let's just use the product of both map odds, 36% = 60% * 60%.

The bet sizer calculates that we should wager $384 on each map bet, compared to $400 if we were to use the standard Kelly formula. The difference in amount wagered is less than 2% of our bankroll in total.

However, we haven't yet properly accounted for the dependency between maps 1 and 2 here. Looking at historical data, we generally  find that the odds of a team winning 2-0 is a bit higher than simply multiplying the map odds together. You'll often see betting markets reflect this after it becomes clear who has won the first map. A win on map 1 is a good sign that things are going well leading into map 2 - the outcome of map 2 is weakly dependent on map 1. We should add a couple percentage points to our view of our team winning 2-0 to account for this - let's try 39%.

Our dependent bet sizer now has us betting $345 per map - resulting in a total of $690 wagered instead of the $800 we would have bet using the standard Kelly formula.

The standard Kelly formulas would lead us to bet nearly 6% of our bankroll more than optimal - causing unwanted swings and lowered long run bankroll growth.

Let's now take a look at a best of 3 example to get a better understanding of how important it is to account for dependence when sizing our bets.

Let's say we have a Counter-Strike series, and map picks have just come out. We think our Team 1, who we see as an underdog in the series, secured some very good picks for maps 1 and 2, but doesn't have much of a chance if the game goes to a third map. We plug in our views and the bookmaker odds into the dependent Kelly bet sizer to determine our optimal bets - our bookmaker hasn't offered the -1.5/+1.5 handicap or a map 3 bet, so we leave these blank.

Let's take a close look at the bets suggested here. If you're used to using the standard independent Kelly bet sizer, the suggested bets here may look a bit surprising.

Notice that we have positive edges on Team 1 for Map 1 and 2 bets, as well as for the +1.5 handicap bet. The standard Kelly strategy places a bet on each of these - you may notice that the Team 1 +1.5 handicap bet with 4% edge has higher stake than the Map 1 bet with 5% edge, but this is because the Kelly strategy prefers the lower edge bet since it is significantly more likely to win (69% vs 47%). The expected value of the standard Kelly bets is +$11.52 which is more than what we see from our Dependent strategy, but the expected growth is significantly lower at +$1.64.

The Dependent Kelly Bet Sizer similarly places its largest bet on the +1.5 handicap, and a smaller bet on Map 1. However, it chooses not to place any stake on Map 2, as the edge is small and there is already exposure to Map 2 through the +1.5 handicap. Most interestingly, the strategy places a hedge bet - a negative edge bet - on Team 2 to win the series. The strategy understands that this hedge will increase the median bankroll in the long run, even though it has a negative expected value.

To test which set of bets will be best for our bankroll in the long run, we can simulate this scenario, over and over, to see which strategy will result in a higher median bankroll.

If you were to just look at the average final bankroll, the Standard Kelly Bet strategy appears to outperform the Dependent Kelly Bet sizer.

However, the Kelly strategy seeks to maximize the median bankroll - and the histogram here shows exactly why. The Standard strategy (red) has a small number of paths at the right end of the histogram with very large final bankrolls - these are what is pulling up the average final bankroll for this strategy. You can see much more red to the left of this histogram, indicating a much higher chance of losing if you do not follow the Dependent Kelly strategy.

This has shown that the Dependent Kelly Bet sizer is optimal in terms of bankroll growth when placing several dependent bets on the same series. Hopefully this new tool will make it easier to calculate your bet sizes easily!

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