Bankroll Management

Cracking DraftKings: Bankroll Management (From Chapter 5)

Bankroll Management

By Maurile Tremblay

 

Bankroll management will mean very different things to different types of daily fantasy players.

Many players play fantasy sports, including DFS, only as a hobby. Ideally, they bet only what they can afford to spend on entertainment — just as someone who skis for a hobby will spend only what he or she can afford on lift tickets — and any winnings will just be a happy bonus.

Some players play DFS for a living. For them, bankroll management is of utmost importance. They must bet an amount each week that takes into account their expected return (the more of their bankroll they bet each week, the more money they can make), but is appropriately balanced against their risk of ruin (the more of their bankroll they bet each week, the more likely they are to go broke). For full-time professionals, winnings are not merely a bonus. They are rent, car insurance, and food. A professional will need to take a (generally somewhat fixed) amount out of his bankroll each month in order to meet living expenses.

A player‘s bankroll is not simply what he has currently on deposit in his DFS accounts. For either the hobbyist or the professional, we'll define his bankroll as the amount of money he has set aside to wager in DFS contests, such that if he lost that amount, he'd be unable to place any more bets until he finds an outside source of additional cash — a paycheck from another job, a loan or funding arrangement with a backer, etc.

Now that we've defined bankroll, we need to define risk of ruin. It‘s what it sounds like — it‘s the chance that a player will go broke over a given number of bets.

Sound bankroll management means increasing our bankroll by the greatest amount possible over the long run. This necessarily means taking our risk of ruin into account, and here‘s why.

Suppose we have $100 to wager on a coin flip. Heads, we double our amount wagered; tails, we lose our amount wagered. How do we maximize our expected return if the coin is fair? We can't. Whether we bet 0%, 32%, 71%, 100%, or any other percentage of our bankroll, our expected return will always be zero dollars, because the amount we win when victorious is exactly equal to the amount we lose when defeated, and we'll win half the time.

But now suppose that it‘s a weighted coin that comes up heads 60% of the time. Now how do we maximize our expected return, measured in dollars, on a given bet? It should be obvious that the answer is by betting 100% of our bankroll. When we bet $100, we expect to win $20 on average. (Suppose we play five times, winning $100 three times and losing $100 twice, for a net gain of $100. A gain of $100 over five flips is $20 per flip.)  Any other amount will produce a lesser return: if we bet $50, for example, we will win only $10 on average.  If we are allowed to play any game only a single time, and if we have a positive expectation in that game, we maximize our expected return by betting 100% of our bankroll.

But now suppose that we are allowed to play the game more than once. Suppose we are allowed to play as many times as we want unless and until we go broke.  Do we still maximize our expected return by betting 100% of our bankroll?

No, we don't. If we bet 100% of our bankroll on the first trial, there‘s a 40% chance that we'll go broke. Then we'll have to sit on the sidelines for the rest of our lives winning $0 while we watch our friends continue to make money in thispositive-expectationgame.

Going broke is terrible because it deprives us of the opportunity to keep wagering, and to keep making money (on average).

That‘s why we have to balance two competing interests: we want to maximize our expected return in a given trial, but we also want to minimize our risk of ruin. We can't do both at once, it turns out, so we need to find an appropriate compromise.

This is where the Kelly Criterion comes in.

According to the Kelly Criterion, the percentage of our bankroll that we should bet in a given contest is equal to (bp - q)/b, where is the net odds we are being offered (e.g., 1-1 in an even-money contest, or 2-1 in a fair-odds game that we will win only 33% of the time), is the probability of winning, and is the probability of losing. (Since we ignore pushes, q = 1 - p.)

The Kelly Criterion originally comes from the world of finance, but it is just as useful in any type of wagering situation. Betting using the Kelly Criterion maximizes our median bankroll over the long run.

Let‘s return to the example above, where we have $100 to start with and can bet as many times as we want (until we go broke) on a coin flip that will land on heads 60% of the time. It turns out that we will maximize ourlong-runrate of return by always betting 20% of our current bankroll. (See Kelly Criterion formula above.) So our first bet will be just $20, well short of the $100 we'd bet if we were trying to maximize our return on only a single wager.

This is a special case of the Kelly Criterion: in any game that pays even odds, the percentage of your bankroll that you wager should be equal to your advantage in the game. When we win 60% of the time and lose 40% of the time, we have an advantage of 20%, and should therefore bet 20% of our bankroll on each trial ((1 * 0.6 - 0.4) / 1 = 0.2). If the game changes such that heads occurs only 55% of the time, we should bet 10% of our bankroll on each trial ((1 * 0.55 - 0.45) / 1 = 0.1). If heads occurs 51.5% of the time, we should bet 3% of our bankroll on each trial ((1 * 0.515 - 0.485) / 1 = 0.03).

And here we have our first application toDFS—specificallyto Double Up contests. In a Double Up, the game pays even odds. Whatever your entry fee is, that‘s how much you win when victorious, and it‘s also how much you lose when defeated. So if you know what percentage of the time you expect to finish in the money in a Double Up, you also know what percentage of your bankroll you should bet on each independent contest. Just double the amount by which it‘s over 50 percent.

Notice that I said in each independent contest. Different DFS contests are not always independent of each other. To take an extreme example, suppose you enter the same lineup in 20 different large Double Ups in some particular week. Suppose you believe, based on your track record, that you have a 54% chance of finishing in the money in each contest. If the contests were independent of each other, you'd expect to win around 11 of them in a typical week. But in fact, in this example, you are usually going to win either 20 of them or 0 of them. You will hardly ever win anything like 11 of them. If your lineup is awesome in the first contest, it will be awesome in the others as well — because it‘s the same lineup. Your results from entering the lineup in 20 contests at $1 apiece will be pretty much the same as your results from entering the lineup in a single $20 contest. Therefore, you do not want to enter 8% of your bankroll on each individual contest; rather, you want to enter 8% of your bankroll total in all such contests.

If multiple even-money contests with 54% success rates are truly independent of each other (say, you are entering two contests — one for 1pm games only and another for 4pm games only) then you can spend 8% of your bankroll on each of them, for 16% total. But the more that the two rosters overlap with each other, the more you'll have to drop down from 16% total toward 8% total. By the same token, the less the two rosters overlap, the more you'll be able to move up from 8% total toward 16% total.  

Moving on to other contests besides Double Ups...

Suppose we play a 50/50, such as a head-to-head contest. It‘s easier to finish in the money in a 50/50 than in a Double Up, but we don't get paid as much when we do. In a standard 50/50 where we wager $10 to win $8, suppose we can expect to win 60% of the time (compared to 54% in a Double Up — giving us a 20% advantage over the average player in each case). Plugging the numbers into our Kelly formula, we'd be justified in wagering 10% of our bankroll in such a contest. So with the same 20% advantage in a 50/50 as in a Double Up, we're justified in wagering more money in the 50/50 (10% of our bankroll as opposed to 8%).

Let‘s go to the other extreme and consider some GPPs with top-heavy payouts.

Let‘s first consider a contest with 100 entries that pay the top 30 spots. The average fantasy player has a 30% chance of finishing in the money, but in keeping with our 20% advantage, let‘s assume we have a 36% chance of finishing in the money. If we pay a $10 entry fee, and the top 30 spots get paid, the average winner will net $20. (The $1,000 in entry fees, after the commission, will constitute a $900 prize pool. That $900 will be spread over 30 winners, so the mean win will be $30, which is a net win of $20 over the entry fee.) That means the average winner is getting 2-1 odds (($30 - $10) / $10 = 2). Plugging all of that into the Kelly formula, we should wager 4% of our bankroll in this sort of contest ((2 * 0.36 - 0.64) / 2 = 0.04). That‘s about half as much as we'd enter in a Double Up with a similar advantage.

Let‘s go all the way to the extreme and consider a $10 winner-take-all contest with 100 entries. The average DFS player has a 1% chance of winning. In keeping with our 20% advantage, that gives us a 1.2% chance of winning. If we win, we get paid $900 (after the $100 commission) for a net of $890. So we're getting odds of 89-1. Plugging those numbers into the Kelly formula, we should wager only 0.0009% of our bankroll in this contest.

So let‘s recap what we've learned.

1. Going broke is bad, so don't wager your whole bankroll, or anything close to it, on a single contest.

2. The bigger advantage you have in a contest, the more of your bankroll you can wager on it. If you have a 10% edge, you can wager twice as much as if you have a 5% edge.

3. If you are entering multiple contests, you can increase the total amount you wager only to the extent that the different contests are independent of each other. If your roster overlap is zero, go ahead and bet twice as much (total) on two contests as you'd bet on one contest. But if your roster overlap is total, you should bet the same amount (total) on two contests as you'd bet on one — so half as much per contest. If your roster overlap is partial, bet somewhere in between. (This is a bit of an oversimplification. Contests are not completely independent just because there is no roster overlap if the rosters were made using the same set of projections. But it‘s probably close enough for our purposes.)

4. Holding your advantage over the field constant, you can wager more in contests that pay out a larger percentage of the field. If you have a 20% edge over the average DFS player, for example, you'd be justified in betting 10% of your bankroll in a 50/50 or head-to-head contest that pays 50% of the entrants, 8% of your bankroll in a Double Up that pays 45% of the entrants, 4% of your bankroll in a league or tournament that pays 30% of the entrants, and just 0.0009% of your bankroll in a winner-take-all that pays 1% of the entrants.

5. As a rough approximation, for any specified amount you intend to wager in a given week, a decent rule of thumb is to put about 70%-80% of it into cash games and about 20%-30% into tournaments.

The following charts show the percentage of bankroll you should put into play in each independent contest according to the Kelly Criterion, given the specified advantage you have over the field.

With a 10% edge over the field or less:

contest type

players

entry

prize pool

winners

% of bankroll

H2H

2

10

18

1

0.00%

50/50

100

10

900

50

0.00%

Double Up

100

10

900

45

0.00%

Tournament

100

10

900

30

0.00%

Winner-Take-All

100

10

900

1

0.00%

With a 15% edge over the field:

contest type

players

Entry

prize pool

winners

% of bankroll

H2H

2

10

18

1

4.37%

50/50

100

10

900

50

4.37%

Double Up

100

10

900

45

3.50%

Tournament

100

10

900

30

1.75%

Winner-Take-All

100

10

900

1

0.04%

With a 17.5% edge over the field:

contest type

players

entry

prize pool

winners

% of bankroll

H2H

2

10

18

1

7.19%

50/50

100

10

900

50

7.19%

Double Up

100

10

900

45

5.75%

Tournament

100

10

900

30

2.88%

Winner-Take-All

100

10

900

1

0.06%

 

With a 20% edge over the field:

contest type

players

entry

prize pool

winners

% of bankroll

H2H

2

10

18

1

10.00%

50/50

100

10

900

50

10.00%

Double Up

100

10

900

45

8.00%

Tournament

100

10

900

30

4.00%

Winner-Take-All

100

10

900

1

0.09%

With a 22.5% edge over the field:

contest type

players

entry

prize pool

winners

% of bankroll

H2H

2

10

18

1

12.81%

50/50

100

10

900

50

12.81%

Double Up

100

10

900

45

10.25%

Tournament

100

10

900

30

5.13%

Winner-Take-All

100

10

900

1

0.12%

"I believe records and stat-keeping is the most underrated aspect of bankroll management, especially for beginners.

A large part of record-keeping is being proactive. This means planning out your week ahead of time, and in many cases having at least a bi-weekly plan. Keep track of the type of contest, the buy-in, the number of entrants, and (+/-) profit. This also goes hand in hand with game selection, as you should be browsing the contests ahead of time and then coming up with a plan for that week.

Record-keeping is also the best way to be impartial on when you should be moving up and down stakes." - BJ VanderWoude