# Daily Fantasy Football: Optimizing Your Wagers

The most frequent question I get asked by daily fantasy enthusiasts is how much money should be wagered on lineups each week. It’s an important question—one that’s vital to your overall success—and perhaps the most crucial factor in your overall profitability.

In my initial daily fantasy post here at 4for4, I briefly discussed money management and its relation to balancing your risk and reward. I’m going to expand on that article today, more thoroughly explaining an issue everyone wants to know—how your money (and how much of it) should be dispersed among lineups in any given week.

This is a tougher topic to tackle than you might think. Unlike in sports betting, most of your daily fantasy “wagers” aren’t going to yield a clear black-and-white result. There are three reasons for that:

1) You can use the same players in multiple lineups.

2) You can use the same lineups in multiple leagues.

3) The same lineup can both win and lose.

The third point is the most crucial. If you’re playing in multiple heads-up leagues with the same lineup—as I recommend—you’re going to see mixed results. As I explained in last week’s article on heads-up and 50/50 strategy, you’ll basically be paid as you finish if you participate in enough heads-up leagues. By that, I mean that if you could theoretically compete in one million leagues and you finished in the 70^{th} percentile, you’d win 70 percent of the time and lose 30 percent. The more heads-up leagues in which you compete, the greater your odds of “getting what you deserve.”

That limits the inherent volatility of heads-up leagues in such a way that you can put more cash on a lineup than you might place on a single sporting event at a sportsbook. Whereas there’s no gray area at a sportsbook—you (almost always) either win or lose—your heads-up results are entirely a gray area. You’ll win *and* lose every week, so you’re just looking to tilt the scales in your favor.

On the other hand, 50/50 leagues are far more volatile over large samples. If you participate in one million 50/50 leagues and finish in the bottom 10 percent, you’ll still fail to make a dime (well, you might make *one *dime). Because of that risk, we need to adjust the total cash we place into 50/50s accordingly. Enter the Kelly Criterion.

## Kelly Criterion

The Kelly Criterion is a formula that helps determine optimal bet sizes. Sports bettors typically use it to maximize profit, although most implement a more conservative approach since the Kelly Criterion is usually considered high-variance. Nonetheless, the Kelly Criterion is useful primarily for two factors it utilizes: the rake and expected winning percentage. As the rake declines or your expected winning percentage increases, your bet sizes—or the total money you have in play in daily fantasy—should increase. The Kelly Criterion captures both effects.

The most basic form of the Kelly Criterion is P = (o*w – l)/o, where P = percentage of bankroll wagered, o = odds received (i.e. the rake), w = expected winning percentage, and l = probability of losing. Let’s go through a few examples.

**Example 1: 9-to-11 payouts, 60 percent expectation**

First, assume you’re in a heads-up match on a site that gives 9-to-11 payouts (meaning for every $1.10 wagered, 90 cents will be paid out). Based on your game history, you see that you can expect to win 60 percent of heads-up matches over the long run. The formula would be P = (0.82 * 0.6 – 0.4)/0.82. The first number is generated by dividing 9 by 11—the payout—and the second number, 0.6, represents the 60 percent expectation. The result is 0.91/0.82 = .111, or 11.1 percent. Thus, the optimal long-term percentage to wager with a 60 percent expectation and 9:11 odds is 11.1 percent of your total bankroll.

**Example 2: 9-to-11 payouts, 50 percent expectation**

Let’s see what happens if we keep the payout steady but decrease the expectation. Our new formula would be (0.82 * 0.5 – 0.5)/0.82. That’s equal to -0.09/0.82 = (-0.110), or negative 11 percent! But how is that possible? Well, since your expected winning percentage of 50 percent wouldn’t be enough to overcome the rake, the Kelly Criterion suggests that you don’t bet at all. And if you know you’re a long-term loser, that’s the right call; don’t waste your money.But what if you’re new to daily fantasy and just don’t know your expected winning percentage? Or what if it’s been hovering around 55 percent—right around what you need to maintain to break even on most sites—and you want to continue to play so that you can improve to become profitable? In those cases, it makes sense to still play as if you have a small advantage; otherwise you wouldn’t be able to play at all. Play like you have a 57 percent win expectation—a realistic number that will yield a small advantage on daily sites. At that expectation in a league that pays out 9-to-11, the optimal bet (or money in play) would be 6.7 percent of your bankroll.

**Example 3: 10-to-11 payouts, 60 percent expectation**

Lastly, take a look at what decreasing the rake can do to our optimal wagers. At 10-to-11 (meaning the site pays out 90.9 percent), the Kelly Criterion formula would be (0.91 * 0.6 – 0.4)/0.92 = 0.160, or 16.0 percent. Just by increasing the money paid out, the optimal money in play jumps from 11.1 percent to 16.0 percent.

## Total Money in Play

It’s important to note that when we’re discussing daily fantasy money management, we shouldn’t be too concerned with the buy-in amount, but rather the total money placed on each lineup. If you enter five leagues with a $5 buy-in, the result is the same as one league at $25 (actually, the lower buy-ins are better, assuming the rake is the same, because you can diversify your results and play inferior competition). So the Kelly Criterion has absolutely no effect on your individual buy-in amounts, but rather the total amount you put into each lineup.

By the way, I took you through the formulas just so you could get a sense of how changes in rake or winning percentage alter your optimal plays, but you can and should use a Kelly Criterion calculator.

## Kelly Criterion’s Relation to League Type

You have to remember that the basic Kelly Criterion formula is meant to optimize bets for black-and-white, independent outcomes. As discussed, that’s pretty much the case with 50/50 leagues, but not at all the case in heads-up leagues. So if you’re placing your funds on a lineup in 50/50 leagues, expecting a 60 percent win expectation over the long-run on a site that pays out 9-to-11, you should probably put down around *half* (half!) of what the Kelly Criterion suggests—or about 5.6 percent. That’s typically what professional sports bettors do because it minimizes some of the risk associated with the high-variance Kelly betting strategy.

So if you have a $1,000 bankroll, your optimal total cash in play for a single lineup with a 60 percent win expectation would be $56 (so you could place between $50 and $60 on that lineup in 50/50s, total, over a few different leagues).

When entering heads-up leagues—the backbone of our strategy this year—we can be far more aggressive. Remember, if you finish in the 30^{th} percentile among heads-up lineups in a given week, you’ll still get paid in 30 percent of leagues. So if you put down $200 on a single lineup in heads-up leagues and win 30 percent of the time, you’ll still lose less than $100 (the exact amount depends on the rake). In 50/50 leagues, you’d likely lose the entire $200.

Because of those numbers, you can effectively double the Kelly Criterion for your optimal heads-up lineup. So while you’d spend 5.6 percent of your bankroll on your top 50/50 lineup, you could put down 22.2 percent of your bankroll on your best heads-up lineup. That might seem excessive—and it would be if the expected outcome were black-and-white—but since we’re working with a range of outcomes through which you’ll still get paid even with a below-average lineup, you can be more aggressive in order to maximize profit.

## A Sample Week

The nature of how you disperse your funds in a given week is really subject to a variety of factors—your expected winning percentage, the rake, the number of players you like, and so on—but let’s go through a general example. We’ll assume 9-to-11 payouts.

Optimal Heads-Up Lineup – 60 percent expectation (22.2 percent)

Optimal 50/50 Lineup – 60 percent expectation (5.6 percent)

No. 2 Heads-Up Lineup – 57 percent expectation (13.4 percent)

No. 2 50/50 Lineup – 57 percent expectation (3.4 percent)

In this hypothetical, we’d have 44.6 percent of our bankroll allocated as “money in play”—35.6 percent in heads-up leagues and 9.0 percent in double-ups. Remember, this is for someone with an established 60 percent winning percentage. If you’re unsure of your long-term winning percentage, these numbers would change quite a bit. And if you’re unsure, lower is probably better.

Also note that this example doesn’t account for tournaments or other league sizes, which I’ll discuss at a later date. Typically, we’ll want to set aside between five and 10 percent of our bankroll for high-return tournaments, and that can affect the money in play on each of the lineups above. On the average week, though, we’ll probably have between 30 to 45 percent of our total bankroll in play, with the majority of that money in low-risk heads-up leagues.

## Using Kelly Criterion for Buy-In Decisions

Generally, I advocate getting as much money into leagues with low buy-ins as possible; it’s suitable to play 10 leagues at $5 to two leagues at $25 because the latter will likely have stiffer competition. But it’s not always best to go lower, particularly if the higher-entry leagues have a reduced rake. To determine the optimal buy-in amount, we can use the Kelly Criterion.

Assume you have a 60 percent win expectation in low-entry leagues, and you receive 8-to-10 payouts. The Kelly Criterion suggests the optimal amount to wager on that lineup (total money in play) would be (0.8 * 0.6 – 0.4)/0.8 = 0.1, or 10 percent.

But what about if you can enter a league with a greater entry fee that charges less to play? What if you can get 9-to-11 odds, but your win expectation drops to 57.5 percent? Let’s check the calculator: (0.82 * 0.575 – 0.425)/0.82 = 5.7 percent. Since the calculator is telling you to bet less, it means your long-term expectation is lower, i.e. stick with the smaller leagues.

Let’s change some of the parameters. Now we can get 10-to-11 odds and we have an expectation of 58 percent. That calculation would be (0.91 * 0.58 – 0.42)/0.91 = 11.8 percent. In that situation, we’d be justified in moving to the higher stakes, even with a two-point drop in expecting winning percentage, because the rake is so much lower. Over the long-run, we’d be maximizing profit, despite winning less.

And regardless of your daily fantasy goals, we don’t want to maximize winning percentage at all costs just because it feels good to win; we want to generate as much profit as possible. With any luck, that’s what we’ll do in 2013.