The Small Bankroll, The Kelly Criterion, and You

I’ve read a decent amount of literature on the Kelly Criterion and was trying to relate it to the small bankroll player in DFS. As many of you know, the Kelly Criterion is a systematic form of wagering that can be used to determine how much of one’s total capital (ie bankroll) should be bet on a single outcome or game. Here is the formula for the Kelly Criterion copied from Wikipedia:

For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is:
Where:

  • f* is the fraction of the current bankroll to wager;
  • b is the net odds received on the wager (“b to 1”); that is, you could win $b (plus the $1 wagered) for a $1 bet
  • p is the probability of winning;
  • q is the probability of losing, which is 1 − p.

As an example, if a gamble has a 60% chance of winning (p = 0.60, q = 0.40), but the gambler receives 1-to-1 odds on a winning bet (b = 1), then the gambler should bet 20% of the bankroll at each opportunity (f* = 0.20), in order to maximize the long-run growth rate of the bankroll.

I suggest that you do some simple calculations with a pen and paper to get a grasp of how the formula works if you’re not familiar with it. Use simple numbers, then modify them as you practice more.

When we look at DFS, we have two possible outcomes, we either win money or lose our wager. Kelly Criterion is very easy to estimate for bets that are 1:1. Fortunately for us, there is a game in DFS that is very close to that 1:1 payout; the double up. Unfortunately for our odds, there is a little thing that keeps DFS sites afloat and allows us to play and enjoy these games: the rake. The rake forces us to win more than 50% of the time to show a profit.

What games should I be playing on a small starting bankroll?

For the purposes of this article, I am going to say that our starting bankroll is $100. I am also going to look at different types of games or tournaments that are possible to enter using that bankroll. I am analyzing these games using DraftDay’s structures.

20 Man Double-Up

With a 5% rake on all DraftDay double-up games, the odds on your wager are 9:11 in all double-up games. If you play a $1.10 entry and win, your payout is $2. If you win an $11 entry, your payout is $20. (Remember you have to calculate odds differently for DFS than a regular gambling wager. If you win $20 in an $11 DU, you are really only winning $9. In a typical 2:1 gambling wager you would receive your $1 you bet, plus your $2 you won.) DraftDay offers $1.10, $2.20, $5.50, $11, $22 and $55 entries that are able to be entered with our $100 bankroll. What level does the Kelly Criterion say we should play? To make things much easier, I am going to use the Kelly Strategy Calculator from Albion Research Ltd

Here is the calculation for the $1.10 Double Up:

Our bankroll is $100 and our odds are 9:11. We can make a minimum wager of $1.10 and the other buy-ins increase by a 1.1 multiple. Let’s assume that all players in the field are of equal ability. Our win % is therefore 50%. How much does Kelly Criterion say we should bet? According to Albion:

Results

  • The odds are against you – you should not bet.

Uh oh. Kelly Criterion is telling us we’re not at an advantage, so we should not wager. So when does the wager become advantageous? This is where you have to get honest with yourself. Can you win this game at least 55% of the time? If you can, then you are at a break-even proposition.

Now, let’s say you see AlSmizzle and DB730 in the $1.10 20-man Double-Up, and you know those two guys are fish and have no chance of winning. You now estimate that you have a 1/18 chance of hitting the top 10 spots, or a 55.5% chance. What do the results say we should do?

Results

  • The odds are in your favor, but read the following carefully:
  • According to the Kelly criterion your optimal bet is about 1.11% of your capital, or $1.10.
  • On 55.5% of similar occasions, you would expect to gain $0.90 in addition to your stake of $1.10 being returned.
  • But on those occasions when you lose, you will lose your stake of $1.10.
  • Your fortune will grow, on average, by about 0.01% on each bet.
  • Bets have been rounded down to the nearest multiple of $1.10.
  • If you do not bet exactly $1.10, you should bet less than $1.10.
  • The Kelly criterion is maximally aggressive — it seeks to increase capital at the maximum rate possible. Professional gamblers typically take a less aggressive approach, and generally won’t bet more than about 2.5% of their bankroll on any wager. In this case that would be $1.10.
  • A common strategy is to wager half the Kelly amount, which in this case would be $0.00.
  • If your estimated probability of 55.5% is too high, you will bet too much and lose over time. Make sure you are using a conservative (low) estimate.

The results now become very interesting. The optimum wager for our bankroll is now $1.10. But the key is the final statement! If you over-estimated your abilities (ie, Al and Dan really aren’t fish!), then you’ve wagered too much of your bankroll and are in an un-optimal situation with your money.

So basically there are no games I can play in that are optimal according to the Kelly Criterion? =(

Not so fast my friend. Some sites offer Double-Up games with a guaranteed prize pool! Let’s say that it’s 60 seconds to game time and you notice a $1.10 20 man double-up that only has 14 entrants. Should you hurry up and enter that 15th line-up in this game with your $100 bankroll? We will assume with only 15 entrants that we have a 66.6% chance of making the money.

Results

  • The odds are in your favor, but read the following carefully:
  • According to the Kelly criterion your optimal bet is about 25.78% of your capital, or $25.30.
  • On 66.6% of similar occasions, you would expect to gain $20.70 in addition to your stake of $25.30 being returned.
  • But on those occasions when you lose, you will lose your stake of $25.30.
  • Your fortune will grow, on average, by about 2.79% on each bet.
  • Bets have been rounded down to the nearest multiple of $1.10.
  • If you do not bet exactly $25.30, you should bet less than $25.30.
  • The Kelly criterion is maximally aggressive — it seeks to increase capital at the maximum rate possible. Professional gamblers typically take a less aggressive approach, and generally won’t bet more than about 2.5% of their bankroll on any wager. In this case that would be $2.20.
  • A common strategy is to wager half the Kelly amount, which in this case would be $12.10.
  • If your estimated probability of 66.6% is too high, you will bet too much and lose over time. Make sure you are using a conservative (low) estimate.

Whoa there! Maximum KC says that we should actually bet even more! So now you have 30 seconds left and you see a $22 20-man game with only 14 entrants as well! And they’re the same guys that are in the $1.10 game! It’s a dream come true! should we join this game instead? According to KC in order to maximize our profit, we should join the $22 game over the $1.10 game. As you can read in their analysis, a common strategy is to wager ½ the KC amount. If there was an $11 game meeting the same criteria, you could join it to follow this strategy. A very conservative strategy would be to stick to the $1.10 game.

Kelly Criterion and GPPS

What does KC tell us about GPPs? The $2.20 Wiz Walk-off features 250 available places and a $500 prize pool and pays out 70 places. When should we enter this tournament? Our bankroll is still $100. We will assume that each entry has a 70 in 250 chance of cashing or 28%.

First we have to throw in some Expected Value for a GPP since each spot pays out differently. If you have an equal chance of finishing in every spot from First to 250th and played the tournament 250 times, finishing once in each spot, you would make $2 on average every time you played. $500/250=$2. Our odds are 2:2.2 or 10:11.

Results

  • The odds are against you – you should not bet.

As I think we should expect at this point, on our small bankroll, we should not play this game. Let’s look at it though if only 200 people join. If we play it 200 times we now expect to win $2.50 each game or 2.5:2.2 or 12.5:11. Also, our chances of winning the $2.50 goes up to 35%.

Results

  • The odds are against you – you should not bet.

Originally trying to estimate the math, I thought this would be a profitable situation, to be in a GPP that is only 80% full. Turns out the KC is telling us we should not bet if we are equal in skill with all the other competitors in the game.

This is where you need to do the work

If you are on a small bankroll, be honest about your odds of winning. Do you think doing research gives you a 5% advantage over the field? Pop in 40% using the last example. Think the amount of weak players in the field gives you another 10% edge, making it a 50% chance of cashing? Look at that number.

My feelings are that using KC proves that players on a small bankroll should be sticking to double ups (especially DUs with overlay) until the bank roll and skill (Win %) is significant enough to begin joining GPPs. Should you enter the maximized amount that KC says you should, or use the ½ Kelly Method?

Is your bankroll bigger than $100? Are you Naapsterman and have $125k plus??? Use the calculator as a guide to figure out what games you should be playing and how much you should be wagering to maximize (or minimize your risk).

A great book on the Kelly Criterion, gambling and the stock market is Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone. I highly recommend checking it out.

Happy Grinding All!

About the Author

  • Curtis Brodzina (DillingerFour)

  • Curtis DillingerFour Brodzina considers himself the Dave Barry Rick Reilly of DFS. He tries to find the humorous side of the sports world. An avid Penn State, Broncos and Pens fan, he fades Kunitz way too often. RotoGrinder’s own Cal had these poignant words for him as he was lamenting his last place finish in the RotoGrinders FreeRoll Bowl: “Nobody expects you to win D4, they just expect you to be funny.”

Comments

  • tgowen

    • Blogger of the Month

    Quick Question:

    DraftDay double ups are $1 to win $1.80. Since I know nothing about the Kelly Criterion does that change things?

  • agendaman

    /

    fascinating dill.4 all the big names in the book shannon father of info. theory thorpe started card counting i think and wrote beat the dealer saw bk. for $3.86 online i took 3 statistics courses in college can even use some calculus on this stuff p.s. D4 got winner for canadian open jim furyk gl now

  • tgowen

    • Blogger of the Month

    Nevermind, I think I understand it all. On a site like DD it all relates back to what your probability of winning is when playing in a full 50/50 (as it likely is on all sites). Thanks for the easy to understand info.

  • DillingerFour

    FD Rep - DFBMeeMee Champ (x2)

    • Blogger of the Month

    You’re right tgowen. That does make it slightly different. Your odds are 8:10 instead of 9:11.

  • tgowen

    • Blogger of the Month

    I’m sure this has been discussed on here about a thousand times (I’m just too lazy to scroll the forums because I will become distracted by an outdated topic that I feel necessary to bring back to life), but out of curiosity what is the estimated win percentage in 50/50’s for the dominant players of the game?

  • marotw

    Simply fantastic article. This is the kind of stuff that I wish more people knew so that more educated discussion and learning could take place. This is also the stuff I wish less people knew so I could have more of an edge. Kudos!

  • DillingerFour

    FD Rep - DFBMeeMee Champ (x2)

    • Blogger of the Month

    There was an awesome small BR game on FTD that I think would fall into playable by KC. It was a .25 entry, paid out $10 to first place and $1 to 2nd through 71st. It ended up getting 196 entries. I need to run the math on it. They didn’t run it tonight, maybe because of short night schedule. I’m keeping an eye out for it tomorrow night.

  • madmanjayWV

    That’s a very interesting load of information. I’d never heard of that before.

    However, for the regular Joe Schmo who might not have but maybe 10% to possibly 50% of what you used as a low-ballers..i.e. small-time players bank roll of a C-NOTE…herein lies the problem(s)…

    My buddy explained it to me this way, so I too shall here.

    Now, we can go to the horse/dog track, look at the odds and show bet all day. Knowing that we have the skill to pick a dog/horse that’s going to get at least third.

    However, many folks don’t find much excitement in that. Rather, they’d just as soon play a $0.10 superfecta or attempt to hit a trifecta — that’s where the big money is at.

    The same holds true in DFS —— sure, say you got ‘X’ for your bankroll…Yes, you can stick to double-ups or 50/50s, but to me, personally, that’s like going for the show bet strategy, which little by little by little — if you pick well at all — you should find success.

    However, the allure of turning $1 into a couple of c-notes or $2 into perhaps $500 or $700 is the dream of any DFS player (let’s not even get into those “stacked” odds)

    Obviously, it might not be the best “strategy” but running the method you talk about takes a lot of patience and due diligence, for sure.

    Perhaps I’ll try this in the NFL season! Thanks for posting, JAY

  • DillingerFour

    FD Rep - DFBMeeMee Champ (x2)

    • Blogger of the Month

    Jay, I completely agree. I still play GPPs every day. It was more about digging into the system and seeing what results would come up. I think what I was trying to get at, is that if you’re playing on a $100 bankroll, and want to build it slowly, you should probably be sticking to 50/50s. If you want entertainment, then by all means go for it, but don’t expect to build a bankroll without some sort of system.

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