The Kelly Method
The Kelly criterion is a beautiful and useful tool for determining optimal bet proportion of bankroll according to expectation in order to achieve maximum growth. That being said, its use as a hard and fast rule for DFS is questionable because of contest formatting and the volatility created by the variable in the Kelly equation: expected winning percentage.
If we multiply the expected winning percentage [W] by the payout percentage [P] and subtract from that the percentage chance of losing [L], we would then divide that result by the payout percentage [P]; this gives us the percentage of our bankroll that is advisable to bet [B].
(WP – L)/P = B
If we believe we have a 60% expectation of winning [W = .6] then we know the chance of losing is 40% [L = .4]. We also know that DK’s rake percentage is 10%, meaning that they pay out 90 cents of every dollar wagered.
So the payout percentage is 90% [P =.9].
(.6 • .9 – .4) ÷ .9 = B
(.6 • .9 – .4) = .14
.14 ÷ .9 = .156
B = .156
This formula is advising that the optimal bet is .156, which is 15.6% of our bankroll.
Kelly’s Figure Isn’t Perfect (Unless it’s Kelly Bundy)
The inherent problems that arise recurrently with the Kelly method in DFS as opposed to other types of wagering are:
- A) the fact that GPPs and leagues do not place equivalent probability on both winning and losing
- B) the variable nature of the expected winning percentage – this factor can be predicted based on the data from a person’s own win/loss history, but the whole purpose of diversification (as is discussed in “Successful Drafting According to Contest Type”) is to enable yourself to lose more contests than you win and still remain profitable.
While it is absolutely possible to lose more contests than you win yet continue to consistently grow your bankroll (unless you play only H2Hs and 50/50s), the Kelly method advises that no bet be made at all with an expected winning percentage of less than 50%.
A person who only plays in ten-person winner-take-all leagues, needs to win only 12% of the time in order to maintain an 8% return on investment and thus, sustain profitability long-term. Even when the factoring is adjusted to account for the lesser necessity of win percentage, fractional Kelly is still recommended over full-Kelly for bet sizing. Fractional Kelly, which is simply choosing a portion (e.g. half or a quarter) of the advised wager, decreases wild swings in return results by minifying the ambiguity of the variable in the algebraic statement.
The entire Kelly system must be adjusted to take into account the overall payout expectation of multiple wagers across different contest formats. A novice DFS participant will not yet have the data to analyze from hundreds of his own contest results to project an accurate expected winning percentage. But through mathematical assumptions, he can devise an appropriate strategy in order to be successful – this will be described in just a moment.
Expected Utility & Expected Value
I won’t go into too much detail about expected utility vs. expected value, but a summation will suffice to explain the concepts. Essentially, expected value forms a mathematical model that assigns a value to the probability of outcomes through the analysis of previous or predicted trials of an event; it basically describes how to arrive at an expected winning percentage by calculating a mean payout based on a certain number of hypothetical results.
Expected utility takes personal preference into account in order to satisfy mathematical anomalies which exist in decision making processes where reward does not justify risk to one extent or the other. The expected value of a two-dollar 50,000-person tournament is $1.80. The expected value of a two-dollar head-to-head is also $1.80. But because the top prize in the H2H is $3.60 and the top performer in the tourney may earn eight grand, the utility – or satisfaction of placing highly – may justify more or less risk from person to person. An individual’s own level of risk-aversion should be taken into account when managing bankroll.
Multiplying Your Winnings Through Division
The H2H features a 50% chance to win top prize, while the tournament – even with an advantage in knowledge and preparation – can never realistically be more than .1% per entry (the chance of cashing is of course much higher than the chance of winning top prize). Does that mean you shouldn’t throw your hat in the ring for two bucks? Of course not! Your wisely chosen picks are as good as (and can be better than) anyone else’s wisely chosen picks; it’s not the lottery – which offers no discernable strategy. And because many people don’t choose wisely, it makes for a lot of fish in the pond who are there to be gobbled up by you.
For those who do not yet have their own personal data from which to derive winning expectation, diversification meets bankroll management with the following precept. If you divide all of your wagers into groups of four contests, one contest from each group should have the capability of paying you back your entry fees of the other three contests. H2Hs and 50/50s with low-level buy-ins offer the highest probability of cashing. But because you may not be able to project whether or not you should expect to win above the break-even line (this will be explained shortly), it makes sense to hedge through diversification.
There is exactly a 50% chance of winning a 50/50 or a head-to-head (without previous data from which to extrapolate). There is a 33.3% chance of winning a three-person league and a 20% chance of winning a five-person league and so on. When starting out, winner-take-all leagues with more than five entrants should be avoided. If a person enters two $1 50/50s, one $1 three-person league, and one $1 five-person league, the probability doesn’t change by grouping these together but the possibility exists that the five-person league, which pays the winner $4.50, still puts you ahead even if you lose the other three. Or by winning the three-person league ($2.70) and one 50/50 ($1.80), the bettor still comes out on top if losing the other two. Whereas, if a participant plays four $1 H2Hs and wins two but loses two, he collects only $3.60 from $4 worth of entry fees. Large tournament entries should be factored into the bankroll management system with the expectation of not cashing, so that other contests with a higher probability of winning can absorb and cover the long shot losses. The purpose of a hedged wagering system is to keep a DFS player in the game long enough to calculate an expected winning percentage, beat the rake and realize a continual profit.
Beating The Rake (Without Stepping on it – Doh!)
Beating the rake involves calculating a break-even point and then consistently winning above that percentage. Because a DFS player is essentially investing in himself, it’s important to understand his cost of doing business. We know that DK pays out 90¢ of each dollar wagered. The somewhat confusing part of this is that 10¢ of each dollar wagered is kept by DK, meaning 10¢ of the losing dollar and also 10¢ of the winning dollar. So a $1 winning bet pays $1.80, hence the return is 80¢ on a dollar. This can be expressed as 10:8 or 10/8 which is 1.25 (10 ÷ 8 = 1.25). This is the price ratio (or cost), that is, for every ten units (dimes, dollars, etc.) coming in, eight units are paid back out. Divide the number of units you pay in, by the number of units you get if you win (not including the return of your own wager) to get the cost [c], which in this case is 10 ÷ 8 which equals 1.25. The break-even point [b] is figured by dividing the cost [c] by the sum of the cost [c] plus 1.
c/(c + 1) = b
1.25 ÷ (1.25 + 1) = b
1.25 ÷ 2.25 = b
1.25 ÷ 2.25 = .556
b = .556
The break-even point is 55.6%. So anyone who can consistently win 56% of his 50/50s and H2H matches will beat the competition and the rake frequently enough to be profitable over the long haul.
Without the benefit of prior contest information, a player can implement the Kelly strategy by substituting a conservative 56% expected winning percentage for 50% propositions (i.e. H2Hs and 50/50s) and then using the advised percentage to determine which portion of his total bankroll to risk during a given session. This Kelly-recommended dollar amount is then spread across his entire portfolio of entered contests. If this percentage of bankroll feels too high, it can be divided further and would then be fractional Kelly. Just don’t multiply it recklessly, that would be machine gun Kelly.
(.56 • .9 – .44) ÷ .9 = 7.1%
Putting It into Effect
Now that we know the total amount of bankroll we feel is optimal for a given day (or weekend) of wagering, let’s marry that with the diversification of contest types we discussed and explore an example.
If we have $500 in our bankroll, and Kelly advises us that 7.1% of that is the optimum, then we would figure to have $35 (or $18 in half-Kelly) to put in play this weekend. If we enter ten $2 50/50s [50s], four $2 five-person leagues [5s], two $2 three-person leagues [3s], and enter three different lineups in the same GPP tournament for $1 each [T’s], we have a total of $35 live – which looks like this:
50, 50, 50, 50, 50, 50, 50, 50, 50, 50
5, 5, 5, 5
T, T, T
And if we sort them thusly:
50, 50, 50, 5
50, 50, 50, 5
50, 50, 3, 5
T, 50, 50, 5
T, T, 3
We can see that if all but one contest is lost in any one of those groupings, there is still one contest in each grouping capable of absorbing the others’ losses by having a realistic chance of winning and paying back our entry fees for the whole group. If this were to occur, it would return $41.40 on $35 worth of wagering: a $6.40 profit.
It is far more likely for one of several other results to unfold, such as winning only one of the five-person leagues and one of the three-person leagues, but cashing in seven of the 50/50s, and hitting a safety net in a GPP for $2. This result would produce revenue of $41.60. Subtract from that the expenditure of $35 for the bets, and your profit is $6.60.
The ultimate profits realized from session to session are likely to be quite similar to one another (more so with fractional Kelly) despite the possibility of numerous different outcomes. This consistency is achieved through the implementation of appropriate drafting, diversification, and bankroll management strategies.
Results may vary 😉
*With an extensive set of personal results data, expected value and ROI can be used to calculate the optimal bet percentage among contests with more than two participants (return on investment is dissimilar to winning percentage in its derivation because ROI already has the rake factored in).
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