Mass multi-entry guaranteed prize pool tournaments (MME GPPs) form the backbone of present-day daily fantasy sports. They are the focus of much of the marketing. And they also represent a major percentage of all play on a daily basis.
As central as these contests are to today’s daily fantasy, I feel like they are poorly understood by most people. They’re also a flashpoint for controversy, and people regularly debate the merits of entry limits and single-entry contests in forums and on Twitter. During these debates, however, I frankly hear more fuzzy logic and downright myths (on both sides) than I do sound arguments.
I write this article to explain how mass multi-entry GPPs work from a mathematical perspective. My goal is to dispel the myths and give everyone a clear understanding of how these things actually play.
My impetus for writing this article is that it’s clear this debate will soon spill out of players’ spaces and into the public sphere. Politicians and regulators will soon be grappling with these issues. For example, Massachusetts recently published a proposed set of DFS regulations for the state, and those proposals included entry limits.
I am personally neither for nor against entry limits, but the level of misunderstanding surrounding mass multi-entry GPPs is at present so severe that without a significant elevation of the community’s understanding of these core contests, any regulations cemented in law regarding entry limits are bound to be both arbitrary and rooted in ignorance.
The basics
Mass multi-entry GPPs allow a single player to enter multiple times—up to the maximum entry limit. For the purpose of this article, let’s assume a $10 contest with 10,000 total entries, a $90,000 prize pool, and an entry limit of 100.
Let’s start by assuming that you enter such a contest a single time. You do your research and build the best lineup you can. If you are exactly average at building lineups, you have a one in 10,000 chance to win. Your lineup will have a –10 percent return on investment (ROI) where 10 percent is what the site takes as rake. In the event the GPP had overlaid (i.e., the site sold fewer than 9,000 entries), you (as an average player) would have had a positive ROI.
Now say you are more skilled than the average player. You still enter the contest a single time. You build the best lineup you can. But your chance to win is better than average. Your skill might improve your chance to win to one in 5,000.
Your lineup will now have a positive ROI. For simplicity (though this assumption doesn’t quite hold for real play), let’s assume that your skill makes you twice as likely to hit every prize-winning place in the GPP. So you’re twice as likely as the average lineup to win first, twice as likely to win second, and twice as likely to win 2000th, the last minimum-cash place. And, naturally, twice as likely to hit any place in between.
Your ROI in this contest is +80 percent. You can determine this quickly with our assumptions. The average player wins on average $9 in prizes (0.0001 of the prize pool). With double the chance to win, you win $18 in prizes (0.0002 of the prize pool). Subtract the $10 entry fee, and you’re up $8 per contest, or 80 percent of your entry fee.
If you’re twice as good as the average player, your ROI is sky-high. What if you’re just a bit better than average? Let’s say you are only 11.1 percent better than the average player. Then you will win in prizes $9.99 on average for your $10 entry, and you’ll be roughly break-even.
This is the first point I want to make about mass multi-entry GPPs. If—using your skill at the game—you can build lineups that are roughly 12 percent better than the average lineup, you will have a positive ROI playing DFS. This is true even if you only submit a single entry into tournaments that other players are entering 100 times each.
Ultimately, what determines whether your play will win or lose is your skill at building lineups compared to the average lineup in the field. Period. That’s it. Build superior lineups and you will have a positive ROI.
Multi-entry basics
Now let’s talk about multi-entering lineups into the same contest. The key concept is that your lineups are independent and self-competitive.
Here’s what that means. Let’s say you enter two lineups—lineup A and lineup B. And, for now, let’s assume that the lineups are completely different. They don’t share any of the same players. Let’s also assume that they are roughly equally good—and both average.
The ROI of lineup A is -10 percent. It’s an average lineup entered into a GPP, just like in the previous example. The ROI of lineup B is also -10 percent. It too is an average lineup entered into a GPP. The total ROI of entering both lineups together is -10 percent—though the average dollar loss is twice as much because you are spending twice the entry fees.
The lineups perform independently of one another. Lineup A can win money and B can lose. Or lineup B can win and A can lose. Or both A and B can win money. Or they can both lose. The presence of lineup B doesn’t help A, nor does the presence of A help B. They don’t care much about each other at all.
Here’s the key point to independence—if lineup A is your best possible lineup, then you cannot enter any other lineup B that will improve the net ROI of both lineups combined. At best, A and B can ignore each other. They each stand alone on their own merits.
Lineups are also self-competitive. This is a very simple concept with complex implications. Here’s how they’re self-competitive—both lineups can’t win first place. Either one wins, or the other wins, or they tie (and split the prize pool).
Let’s assume that first place is $10,000 in our example GPP, and second place is $4,000. We enter lineups A and B into the GPP, and they finish first and second. We win $14,000. Not bad.
Now consider an alternative universe where you can clone the GPP—that is, there are now two simultaneous and separate GPPs with roughly the same fields and prize pools. You enter lineup A into contest 1 and lineup B into contest 2. In other words, you are no longer multientering.
In this case, since lineup A is better than all the other lineups in the field, it wins first place. And since lineup A has been removed from the field, lineup B is now also better than all the other lineups in the field, and it wins first place. We win $20,000.
This difference between the $20,000 win in the alternate universe and the $14,000 win in the present universe is the cost of self-competition. You can think of it as the tax you pay for entering the same GPP multiple times rather than multiple GPPs a single time each.
The frequency with which you happen to finish a GPP winning both first and second determines how high this self-competitive tax is.
The way we designed lineups A and B, without any overlapping players, makes it extremely unlikely for the two lineups to finish first and second. It’s virtually a lock that if lineup A is good enough to win, someone else will be playing a lineup that has some key players from A, and that lineup will end up second. Lineup B likely won’t crack the top 10, since we know it doesn’t benefit from any of the great players in lineup A.
Because we’ve built these lineups this way, if they are both average, the combined ROI stays at the average lineup ROI of -10 percent.
Now let’s do something different. Let’s make lineup A and lineup B identical. They have the exact same players. These two lineups are perfectly self-competitive. If one lineup places, the other is guaranteed to tie it. Self-competitiveness hurts the combined ROI of both lineups.
We can estimate what happens to the combined ROI with a little trick. Let’s say the lineups win first place (which is the most important case for self-competitiveness). Since the lineups tie, they split the $14,000 of first and second place. This happens roughly one time in 10,000. The rest of the time, the lineups perform as an average lineup would if the first-place prize were set to zero. So 99.99 percent of the time, the combined lineups will have a net win of about $16 (an average result in an $80,000 prize pool—removing the first place prize). The other 0.01 percent of the time, the combined lineups will have a net win of $14,000. This gives an expected prize of:
EP = 0.9999 x $16 + 0.0001 x $14,000 = $17.40
The entry fee for these two lineups is $20, so the ROI is -13 percent. The actual number is slightly worse than this, because the self-competitive effects cascade down the prize structure as well, but this analysis is sufficient for our purposes.
By entering the same average lineup twice, your ROI dropped from -10 percent to -13 percent. The 3 percent difference is the cost of self-competitiveness. That is, it’s the cost you suffer from the fact that your lineups are trying to beat each other.
Here is the bottom line with independence and self-competitiveness. With the first lineup you enter, you establish an ROI. Assuming this first lineup is the best one you can possibly make, each additional lineup you enter cannot possibly increase your ROI, since the lineups are independent. They can, however, decrease your ROI through the self-competitiveness effect. The more similar your lineups are to one another, the higher a self-competitiveness tax you pay.
Why multi-entry?
“Okay Ed,” you might ask, “If this is true that adding entries can only cut your ROI, why does anyone do it?”
Well, it’s because if you have a sufficient skill advantage over the field, you make more money by entering many times with a lower ROI than fewer times with a higher ROI.
Let’s take the example of the highly skilled player who wins with a single lineup twice as often as the average player. With a single lineup, his ROI is +80 percent.
Now let’s say that he can construct and enter 100 lineups. In constructing these lineups, he uses many combinations of players selected from a relatively small pool—identifying the pool of good players to use is the skill he leverages to generate that 80 percent ROI advantage in the first place. Because he’s using a lot of players from the same pool, he has to suffer the self-competitive tax, because if his players go off, there’s a fairly decent chance he’s going to end up winning both first and second place. And he’s also going to pay down the line as the lesser lineups also compete with each other.
Furthermore, each additional lineup isn’t quite as good as the first to begin with, so by the time he gets to lineup 100, he’s both entering a lineup that’s clearly not as good as the first one, and one that likely self-competes with the first one.
Say this effect brings his total ROI down to +20 percent. That’s a big drop in ROI, but it’s easy to see that winning 20 percent of $1,000 is a lot better than winning 80 percent of $10. So that’s why the skilled player submits multiple entries.
The bottom line
The bottom line is that a single player multi-entering a contest cuts his ROI in the contest essentially without exception. This is true even in a perfect world where the multi-entering player doesn’t make mistakes. The reality is that it’s a lot harder to build and manage 100 lineups without making errors than to build just one lineup, and multi-entering players make bad lineups all the time—lineups with players who aren’t starting or for whom late negative news broke.
The skilled player eats these disadvantages hoping to make up for it (and more) with volume. And in today’s DFS, it’s very easy to make up for these disadvantages with volume.
Myth-busting
Now I’d like to do a little myth-busting of some of the most common errors in logic that I hear surrounding the multi-entry debate.
Myth No. 1. Multi-entry takes the skill out of DFS
The skill in DFS comes from constructing lineups that win more often than the average lineup. There are a few different ways to go about creating this advantage which are beyond the scope of this article. But absolutely nothing about multi-entry affects the source of this skill in any way whatsoever. As I demonstrated above, a player of average skill multi-entering the example contest will have at most an ROI of -10 percent—and that’s if the player perfectly manages the self-competitiveness. In reality, with how most players build lineups, the player of average skill will over-compete with his own lineups, and his ROI will creep a few points below -10 percent.
Simply put, an average player entering 100 times the lineups loses (at least) 100 times as fast. It is absolutely impossible to improve your performance (measured in ROI) simply by adding lineups.
Myth No. 2. Multi-entry makes DFS a contest of who has the biggest bankroll.
The size of your opponents’ bankrolls is essentially irrelevant to you. Say I told you that one of your opponents in the above contest is going to be CSURAM88. And he’s going to enter 100 times.
But he’s going to give you a choice. He’ll agree not to enter any lineups if you can convince Donald Trump to enter the same contest 100 times.
I’m not an expert on CSURAM88’s finances, but I assume he has a smaller bankroll than Donald Trump by a good bit. Would you want Trump to play? Or CSURAM88?
Hopefully you picked Trump. Why? Because I bet Donald Trump sucks at multi-entry GPPs—certainly compared to CSURAM88. The size of the bankroll is irrelevant. The only important variable is the skill with which the player makes the lineups that you are competing against.
Indeed, a bad player with an enormous bankroll is the ideal opponent. A sharp player with a shaky bankroll is a bad opponent. Multi-entry GPPs are in no way a contest of bankrolls.
Myth No. 3. Multi-entry lets the same players win almost every time.
If you have one lineup in a 10,000 entry field, your chance to win if you are an average player is one in 10,000. If you have 100 lineups in a 10,000 field, your chance to win is one in 100—exactly 100 times greater—if you can completely eliminate the effects of self-competition. Since no real player eliminates these effects, your chance as an average player to win in a 10,000 entry field is less than one in 100.
If you are a skilled player with one lineup in a 10,000 entry field, your chance to win if you are an extremely skilled player might be one in 5,000. If you have 100 lineups in a 10,000 field, your chance to win is worse than one in 50—since being skilled makes it essentially impossible to perfectly manage self-competition. Obviously, a somewhat worse than one in 50 chance to win is much better than a 1 in 10,000 chance—but the skilled player is paying one hundred times for the privilege.
More to the point, what does it matter to you if one skilled player enters 100 times and wins one in 50 contests, or 100 skilled players enter one time each and win one in 50 contests? Neither scenario affects your chance to win.
More generally, the screen names attached to the lineups are irrelevant. What if DFS were played anonymously? What if you didn’t see any screen names attached to the lineups—a contest page just showed the lineups themselves, the points, the usage percentages, and so forth. Would it be a different game?
Maybe it would feel different to you, but from a mathematical perspective—how often you were to win or lose and what ROI you could maintain—it would be unchanged.
Arguments against multi-entry
Okay, so I’ve spent the majority of the article making an argument that could be characterized as “mass multi-entry is no big deal.” Now I will argue against multi-entry.
The thing mass multi-entry does—really the only thing it does that tangibly hurts the recreational player—is it makes fields tougher. It raises the sharpness level of the average lineup, because 10 to 30 percent of the entries are submitted by mass multi-entering players who as a group tend to build sharper lineups than the average person.
If you think it’s harder to win as a recreational player in a GPP that allows mass multi-entry than it is in a single entry GPP, you are correct. It is harder. But it’s not because mass multi-entry represents some intrinsic advantage for players with big bankrolls. It’s because mass multi-entry allows players who are already skilled to enter more lineups into the field and thereby bring up the strength of the average lineup. Far from eliminating the skill of DFS, it allows those who possess skill to leverage it more thoroughly.
That’s it. That’s what it does.
Here’s the thing. Just because it’s harder to win in a multi-entry GPP than it is in a single-entry format, doesn’t mean it’s impossible. Far from it. All you have to do is build lineups—in any way that you choose—that are roughly 12 percent better than the average lineup. Yes, multi-entry brings up the average a bit, but the 12 percent bar is still not that hard to clear. Maybe one day it will be, as sharp players get more sophisticated. But today, right now, you can—using only your brain and the tools publicly available—consistently build lineups that win.
Is mass multi-entry good policy?
I’ll finish with my opinion. I think GPPs are a game that permit skilled players to create large ROIs (because of the GPP prize structure), and allowing these skilled players to enter many times ensures that many recreational players will lose considerably more than the -10 percent average number. I don’t think there’s anything inherently unfair about this. Daily fantasy is a game, and many games allow skilled players to win consistently at the expense of less-skilled players. Indeed, without the skill component to the game, few of us would find DFS so engaging.
If you lose consistently at DFS, it’s not because other players are out-bankrolling you or out-scripting you or building magic multi-entry lineup sets that somehow guarantee a victory. It’s because the lineups you build are average or worse than average. That’s the reason. I don’t think there’s a thing that’s unfair about that.
But from the industry perspective—and from a game design perspective—I think mass multi-entry GPPs are problematic. They permit skilled players to apply their advantages many times over, and this will in time—as the skilled players get better and more numerous—create an increasingly weaker experience for recreational players who are the lifeblood of the game. Put in the terms of this article, the average lineups in GPPs will get increasingly sharper, so someone who is average today might be significantly worse than average in the near future. If this causes too many recreational players to quit, the fields will dry up.
I think the steps the major sites are taking to limit multi-entry are probably good for the longevity of DFS as a pastime that many people can enjoy. I question whether multi-entry limits belong in state regulations of DFS. The best solution is probably to allow the sites to react to changes in the game ecosystem to strike the right balance between a fun game with big prize pools now and the long-term health of the game we love.
Ed Miller is an MIT-trained engineer and the author (or co-author) of nine best-selling poker strategy books. including “The Course: Serious Hold ’Em Strategy For Smart Players.”