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Solving the 50/50 Multi-Entry Dilemma

You and your buddies are at a bar watching the game. Your team is down by four points with one minute left and they’re lining up for an onside kick. One of your friends asks you if you think your team is going to recover it,…

You and your buddies are at a bar watching the game. Your team is down by four points with one minute left and they’re lining up for an onside kick. One of your friends asks you if you think your team is going to recover it, and knowing that teams recover onside kicks around 20 percent of the time, you respond with a simple ‘no.’

But then you have an idea. You know your friend has no clue how often onside kicks are recovered, so you offer him a proposition: “I’ll give you $1 if the onside kick fails, but you need to give me $5 if we recover this thing.” He agrees.

Was this a smart move on your part? The answer, which deals with a concept known as expected value, is yes. You’re getting the right odds to make this friendly wager, assuming the true probability of a recovery is near 20 percent. In that case, there will be four failed attempts for every recovery, but you’re getting better odds (5-to-1) that the kicking team will recover.

We can calculate your expected value in this situation by playing out each scenario and doing a little math. Over the long run, the kicking team will recover one out of every five kicks. So let’s say we play out this same situation 1,000 times; you’ll win around 200 times, and your buddy 800 times. If you’re putting down a buck each time to win $5, the outcome will be you giving away $800 bucks but receiving one grand (200 multiplied by $5). With $1,000 on the line after 1,000 onside kicks and $200 in returns, your expected value on each individual kick is 20 cents. This is where the terms positive expected value (+EV) and negative expected value (-EV) come from. There are all types of nuances involved daily fantasy sports strategy, but all of them are geared toward becoming +EV. In the onside kick example, you’re maintaining a healthy positive expected value. If you were to receive only $3 in return for the $1 you post, you’d be -EV.

50/50 Expected Value

One dilemma in the world of expected value is if you should enter the same lineup more than once into a 50/50, and if so, how many times. I actually think this is a pretty complex issue with a lot of variables (and to be honest, I don’t even know “the answer,” if there is one, as I type this), but I’m going to do my best to walk through the problem and see what I can figure out.

I’ve seen and heard discussion on this topic that claims that each entry into a 50/50 has the same expected value. Whenever I’m thinking about conundrums in daily fantasy sports or any area of life, really, I like to break things down to one extreme or the other and see if a particular premise holds water.

In this case, let’s imagine you’re in a two-man 50/50. These exist in reality, but they’re called ‘head-to-head’ games. In effect, a heads-up match is a two-man 50/50; the top half of the field gets paid, but in this case, that’s just one person.

So what would happen if you were to multi-enter a two-man 50/50? You’d obviously lose money (whatever the rake is for that league). If you’re in a $50 head-to-head game and theoretically entered both sides, you’d profit maybe $40 for first place and lose $50 for second place; your expected value would be losing $10.

Let’s extend this example out just a bit to a four-man 50/50, again for $50. Now the numbers become more challenging. We’re going to assume that you’re playing just a single lineup and deciding whether or not it makes mathematical sense to enter it twice. There are only three possible combinations where you can finish (if we ignore ties): 1st and 2nd, 2nd and 3rd, 3rd and 4th.

In the first scenario, you’d win all the money, which would be $180 on your $100 investment for an $80 profit. In the second scenario, you’d profit $40 for second place, but lose your entry fee for third place. Ultimately, you’d bring in $90 on your $100 investment. And finally, finishing 3rd and 4th nets you nothing, so you’d lose $100.

So the potential outcomes for you if you enter the same lineup twice into a four-man 50/50 would be profiting $80, losing $10, and losing $100. That’s a total $30 loss, or 10 percent of the total money in play when you add up each scenario.

Let’s extend out the league size one more time to see what happens. This time we’re going to enter the same lineup twice into a six-man 50/50. The possible outcomes are 1st and 2nd, 2nd and 3rd, 3rd and 4th, 4th and 5th, 5th and 6th. With a $50 entry, this is how things would shake out in terms of your return: $80 profit, $80 profit, $10 loss, $100 loss, $100 loss. That’s a total of -$50 (again 10 percent of the total money in play after adding up each potential outcome).

Now let’s try entering even more lineups.

What if we want to enter a $50 six-man 50/50, except this time enter three lineups? The potential finishes (and returns) would be as follows:

1st, 2nd, 3rd (Profit $120)

2nd, 3rd, 4th (Profit $30)

3rd, 4th, 5th (Lose $60)

4th, 5th, 6th (Lose $150)

The end result is a loss of $60, which is yet again 10 percent of the total money in play from each outcome ($150 spread out over four scenarios).

The conclusion is obvious: if you are an average player, it doesn’t matter if you enter the same lineup into a 50/50 multiple times; your ROI will be the same (equivalent to the rake).

Note that all of this assumes we’re dealing with an average player. Obviously you need to be better than average to make money long-term, but the point is that your ROI won’t change by entering 50/50s more than once. But that doesn’t mean it’s always a good (or bad) idea.

The Value of Multi-Entering 50/50s Depends on Your Win Probability

Let’s go back to the hypothetical two-man 50/50. If you are an average player and you win 50 percent of your cash games, you’ll likely have a -10 percent ROI (because of rake), and that won’t change whether or not you enter the two-man 50/50 once or twice.

If you are a profitable player and you win 60 percent of your cash games, however, your ROI would clearly decrease if you had both entries into a two-man 50/50. And the opposite is true, too; horrible players would actually benefit from entering 50/50s multiple times. A 30 percent long-term winner would cash half of his entries by eating up all of the spots in a 50/50, which would be more profitable for him than playing users who are superior. It follows that entering a 50/50 multiple times actually decreases the expected value for a profitable player because it just takes him or her closer and closer to losing his money via the rake, it doesn’t matter at all for average players, and it improves the ROI for below-average players.

Size Matters (for 50/50s, Anyway)

The ROI for each additional entry into a 50/50 improves for below-average players and decreases for profitable players, but that doesn’t mean that profitable players shouldn’t ever multi-enter 50/50s. Even if there’s a declining ROI, all we really care about is being as +EV as possible. As long as a slight decline in ROI isn’t accompanied by a drop to a negative expected value, an additional 50/50 entry is fine.

Further, the ROI and expected value decrease become less and less significant as the size of the 50/50s grows. If you enter a lineup twice into a four-man 50/50, that’s 50 percent of the field (which takes you very close to losing your money via rake). If you enter the same lineup twice into a 1,000-man 50/50, that’s 0.2 percent of the field; in that case, the drop in ROI and expected value would be very inconsequential.

Everyone’s goal should be to become profitable with a long-term positive expected value, and as such I think it doesn’t make sense to multi-enter small-field 50/50s. The exact math on what your optimal entry total should be is much more complicated with a lot of variables, expected win probability being the most vital.

However, I think there’s definitely a point at which entering a 50/50 more than once (perhaps even quite a few times with the same lineup) can make sense. DraftKings’ huge “World’s Biggest 50/50s” are an example of that; with 20,000-plus entrants in some of them, even 100 entries equates to less than 0.05 percent of the field, which is the same as entering a lineup only five times in a 1,000-man 50/50.

So while profitable players probably shouldn’t enter a lineup twice into a 10-man 50/50, for example, a multi-entry strategy isn’t a poor idea in bigger 50/50s. Yes, each additional entry decreases the ROI of a profitable player (even if slightly), but the ultimate goal isn’t per-entry ROI, but rather increasing the overall return, i.e. make as much money as possible.

As long as an extra entry doesn’t take you into -EV territory (which isn’t likely in a large-field 50/50, assuming you’re long-term +EV), go ahead and fire another.