Underdog Math

Analyzing Underdog Fantasy Pick'Ems and Peppers

Matt Moody
December 22, 2023

You may have seen, I’ve recently become more engaged on TikTok and am trying to learn the space. So far, I’ve learned about Girl Math, Boy Math, Dog Math, Cat Math (and a lot of cat videos.. those algos, they work!).. but what about UD math?

First: a few disclaimers.

I personally live in the state of Pennsylvania, and thus cannot use the “forced player parlay” sites such as Underdog Pick’Ems or Prize Picks. There are other entrants into this space (BETR, DraftKings just launched a peer-to-peer version of this game called Pick Six, etc.), but given the Morning Skate Podcast’s partnership with Underdog and UD’s awesome selection of Best Ball contests (including the premier NHL season-long contest in the world, in my opinion, Best Puck Classic), I am focusing on Underdog here, though this process should be applicable to any site with fixed payouts rather than dynamic payouts.
Given my locale, I have very limited personal investment in UD Pick’Ems, and have not tracked my ROI to any degree to flaunt here. It’s simply a very interesting math problem, and as we’ll get into, I wound up being completely wrong in my uninformed assumptions about the product before I put pencil to paper.
Please play responsibly. There will be swings not too dissimilar from Daily Fantasy. Even with “an edge”, this is not sports betting, in that you’re nearly guaranteed to exert your edge over the House if you have it over even the near-term in straight wagers. When you’re reliant on five events (as we’ll discuss the reasoning for focusing on five) to go your way for 20X, you’re going to have massive swings. This thread has a great visual representation of what a +EV player can expect over time, and that’s with an astronomical edge.
Recently UD is experimenting with applying correlation coefficients to correlated legs of a slip, rather than outright blocking users from combining them (in some cases) like they did earlier this year. This is new as of mid-December, and my math does not account for the slip-specific correlation factors that may be applied to your slip, so be careful when constructing your slips that new wrinkles aren’t being thrown in, or that you are at least accounting for them appropriately.

Today’s Primary Point: Let’s do some math about Underdog Fantasy Pick’Ems

If Underdog sounds like it might be for you, sign up using this link and help support me and the Morning Skate Podcast! You’ll get a 100% deposit match bonus worth up to $100 for your troubles.

⁠⁠⁠https://play.underdogfantasy.com/p-the-morning-skate-podcast

Underdog allows you to create slips featuring between two and five player-specific higher/lower selections, with the catch being you have to get everything right to be paid out. There’s a wonderful breakdown of the odds needed here and here if you are unconvinced that if you are building UD slips, you should be using 5-leg entries almost exclusively. I’ll let you peruse those articles on your own, however, as this math is table stakes at this point.

A clean summary of the fixed odds across sites from Unabated Sports

While Underdog doesn’t have a brick-and-mortar building with diamond-encrusted chandeliers and plush lounge chairs paid for entirely by their customers’ losses… they’re likely getting close to those big Vegas Casinos we all know and love. The reason why is simple:

It may seem tantalizing to get 20x when you correctly predict five things to occur.. but mathematically speaking, it really isn’t that amazing. As a matter of fact, if you were to simply flip a coin and be paid out fairly for the odds of guessing right five times in a row, that payout would be 32x!

you’re going to see a lot of these Excel-grabs here, so buckle up

In the above: 5 legs, listed across as 1 thru 5. To pay 32x, Root Odds indicate that the legs have to “average” 50% (which will be meaningful later). In this case, we can calculate (1/32)^(1/5) (which = ½, or 50%) to get these root odds that are uniformly applicable to each leg of the hypothetical slip. Critically, this 50% odds amount translates to a +100 American Odds sportsbook offering. If you bet $100 at +100 and win, you get your stake back ($100) plus your winnings ($100) for a $200 total return. The same would apply in this situation.

Since 32x payouts on 5-leg entries would result in UD trading profits back and forth with the average customer, and power users skew more skillful (and spend more money) than the average customer, this would result in a massive L for the company. So instead of using 32x as the payout, this number is reduced to 20x. Still a high-upside endeavor, but you can see now where that “expected value” shifts heavily in favor of the house (as it always does).

As a reminder, our edge is in the inability for the house to protect itself against every single bet listed. While we may be focusing on the NHL, or eSports, or whatever, the house has to monitor NFL, NBA, Tennis, Golf, so on and so on all around the clock and all around the globe. They are protected by paying 20x instead of 32x, but there will always be weaknesses. Our job is to determine whether the weaknesses are profitable enough to overcome the protection that the house grants itself.

-122 is the Magic Number

We can use the exact same approach we applied to the coinflips/32x to determine what our goal should be, then, when creating these five-leg slips at 20x.

Think of it the same as before, but I’ll spin it differently in case that last one didn’t quite catch: When you’re being paid out 20x, you need to win 5% of the time to break even. When you pick five legs all of which have a 54.928% probability of hitting, the independent probability of all five legs hitting (and thus you being paid out) is 5% on the nose. Translated to American Odds, this would be like picking out a sportsbook bet that was -122 every time.

What this means is that if you pick only legs that are -122, -125, -130, or better in your favor, you now hold the edge. 90%+ of the time, you’re still going to lose (as you can’t bend reality to your will, no matter how hard you try, see my variance disclaimer above…), but you’ll win often enough to overcome those losses.

If you are unsure of the process/math to arrive at the last two Excel-grabs I shared, please take another pass at them, or come bug me in the Morning Skate Podcast Discord server (or on Twitter) with any particulars you may not be grasping, as having this down pat is the #1 way to ensure you’re not bleeding expected value to Big Underdog.

Pepper Math

When Underdog introduced “Peppers” (Prize Picks is calling them goblins/ghosts/something like that, same principle but obviously different math would apply, I suggest you take the time to work through it yourself if you are heavily invested in Prize Picks!), I was initially highly skeptical. Any time you’re boosting payouts, my negativity is inclined to believe that there’s some “visual trickery” at play.

The first pepper I saw was a 1.5x, meaning that if you had a five-leg slip at 20x payout, you could include this leg as your fifth and receive a 30x payout instead, if it all hits. Without much thought put into it, I assumed the field would fall all over themselves to include these peppers in their slips (and using my friend and MSP co-host DJ Mitchell - a great follow if you like NHL UD Pick’Ems - as my reference.. they did), and that it was a mistake.

Ultimately, I moved on, as someone who isn’t playing pick’ems thanks to being in PA. Then one day, your triumphant hero had some time and had a particular nagging thought... “Only you can save the UD entrants from themselves”, Smokey Bear told me.

And this problem of quantifying the peppers is a bit more complex than it appears at first grasp. After all, to truly gauge whether a slip is worth it at 20x, 30x, or any multiplier, you need to know the true odds of the slip paying out, which you need precise probabilities of individual events to calculate.

The added wrinkle now is that you cannot assign the “risk” or the “odds” equally across all five legs. Since all five-leg entries consist of opportunity cost (no one is forcing you to use or not use a pepper as your fifth leg, not to mention multiple peppers can be included), the second you assume that the 1.5x leg is “equal” to a 1x leg, you should immediately be discarding all 1x legs.

My approach, then, was as follows:

Along the bottom, I calculated the “Root Odds” via the same formula as before, with a 1.5x pepper resulting in a 30x payout, so I did (1/30)^(1/5), which results in this 50.65% figure you see across the bottom. This number represents the odds, on average, I must beat with my five legs to have a profitable slip.

I then substituted in the now-accepted -122 odds on the first four legs of the slip, in order to more closely represent a “true decision” to be made with the fifth leg. The thinking is, I must have a profitable slip to begin with to determine whether a pepper is a +EV addition or not.

Once you fix the four legs, the fifth leg is a math problem again instead of a philosophical question, how do I make the top “FAIR MULTIPLIER” equal the bottom one (this time, at 30x):

If you must beat 3.33% odds (at 30x payout), and you have four legs of 1x that all hit at an X% rate in total, then we can determine the rate of Y%, or F28 in this screenshot, that the pepper must hit for the slip to be breakeven.

As seen in the first Excel-grab, this value is ~36.619%, which translates to a +173 sportsbook entry.

And this… this was a headscratcher. After all, if you are getting 1.5x, and the book is maintaining its edge, I would think that this value would hinge on +150, the “sportsbook odds” at which you would receive 1.5x back.

What we now know, however, is that any odds that are +173 or better, +160, +155, etc, would all be +EV additions to a 5-leg UD slip.

The math is the math, and this process requires assumptions to be made that stray further and further basic probability, but it was a positive experience to realize that perhaps the way that “normies” think about this math wasn’t being exploited to extract more money from the field than these -122 implied lines were already capitalizing on.

Beat these odds with your legs, and you’ll build profitable 5-leg UD entries!

Above is a simple table that can guide you through comparing sportsbook offerings to UD peppers to keep you on track with your slip creation process. For example, if an NHL player is listed at +200 to score a goal on the books (setting aside what their true odds are, of course), and that player is listed at a 1.75x pepper on UD, it would be a better use of your money to include that player in a UD slip than to bet it outright. On the other hand, if that same player was +200 on a book and 1.5x pepper on UD, it’s more profitable long-term to just bet the +200 and use different legs from other players on UD.

Combinations of Peppers

The other very cool thing before I get to the supporting math for all of the above odds, that I suppose I should have expected, is that these pepper probabilities are stackable. Meaning, that if I have a 1.5x and a 1.75x pepper in the same slip, these probabilities don’t change one iota. A good leg is a good leg, regardless of what surrounds it.

Keep in mind that UD caps all slips at 100x payout, so anything that goes over 100x does not stack.