Packs: Calculating Expected Value

Using simple probability to determine pack profitability

For as long as I can remember, I haven’t had to make any tough decisions about purchasing a pack in NBA Top Shot, and I’m certainly not alone. If we’re lucky enough to land a strong position in the line and have sufficient funds, we simply buy the pack. It’s not rocket science, but this is due to the simple fact that the value inside these packs outweighs their cost. In slightly more technical terms, this is due to the expected value (EV) of a pack, and more specifically, its positive expected value (+EV).

Expected value represents the average return of an action if the same action were to be repeated an infinite number of times. If a decision has +EV, it’s one we can make with confidence, knowing that in the long-run we’ll come out making a profit. While Top Shot packs have traditionally exhibited positive expected value, we’d be naive to assume this will always be the case. If we want to be best prepared for future pack drops, we ought to understand how we can estimate a pack’s EV and make a statistical decision about purchasing - let’s dive in.

Expected Value Basics

To get our probabilistic juices flowing, let’s walk through how we might calculate expected value for a simple coin-flipping game.

The rules are simple: we flip a coin, and if it lands:

• heads, you lose $100

• tails, you win $120

There are no tricks here, and we should be able to quickly see how this is a positive expected value (+EV) game for you to play - in the long run you will come out making a profit. To visually see this expected value, let’s simulate this game 20,000 times and calculate the average winnings over those simulations.

As we can see, the results after a few simulations are a bit blurry, but over time as the random chance cools off, we settle on an average value of $10. This is our expected value, and as we assumed earlier, it’s greater than 0 (+EV). I’d play this game as often as I could if I were you!

Running a computer simulation tens of thousands of times isn’t efficient or accessible, so let’s see how we can compute an expected value using simple probabilities. To calculate an expected value, we can use the following equation:

EV = (x1 * P1) + (x2 * P2) + ...

where x is the result of an even occurring, and P is the probability of that event occurring. Looking at our game, we know that flipping a tails nets you $120, and this happens 50% of the time. The other 50% of the time you lose $100. Plugging this into our equation we can calculate the EV:

EV = (120 * 0.5) + (-100 * 0.5)
EV = 60 - 50
EV = $10

And there we have the same value from our simulations, $10. Now let’s see how we can apply this to a given Top Shot pack drop.

2021 All Star Game Pack: Expected Value

The most recent Rare pack drop offered 7moments for $229, specifically:

• 1x Rare 2021 All Star Game moment

• 6x Base Set moments

Like most analysis, we can simplify our work by breaking it down into smaller, more approachable problems. For this analysis we can simplify things by calculating the average return for a single moment of each rarity in the pack (Rare and Base Set), and then combining these results. Let’s get started by analyzing the Rare moments.

Average Return: Rare 2021 ASG Moments

As the name implies, we’re guaranteed one Rare 2021 All Star Game moment in our pack, and the specific moment will significantly impact our value at the end of the day. While we’ll walk away with deeper Top Shot pockets if we pull a player like LeBron Steph, we’ll need to account for all possible scenarios. The first step in calculating the average return from our Rare moment is to find the value of each possible moment, and we’ll do this by taking the current lowest ask on the marketplace.

Determining these values with the moments on the marketplace is trivial, however we could make a reasonably accurate estimate for each moment before a pack drop by using some market capitalization approaches (more details on that here).

Next up, we need to determine the probability of pulling each Rare moment from our pack. In this pack drop there are an equal number of Rare moments minted for each player (2,021 to be exact). Knowing that there are 20 players with equal representation, the probability of getting a single moment is straightforward, and simply 1 / 20, or 5%.

Lastly, we can calculate the average return of our Rare moment by multiplying each moment’s value by its probability, and summing up the results. This results in an average return of $457.20. Based on the pack cost of $229, this Rare moment alone nets us a positive expected value of $228.20. Let’s factor in our Base Set moments to prepare ourselves for future packs where the margins may be slimmer.

Average Return: Base Set Moments

Things get a bit trickier when it come to calculating the average return of our Base Set moments. For starters, there are six in our pack, as opposed to just one. Secondly, calculating the probability of pulling an individual moment isn’t as straightforward as our Rare moments.

To determine the probability of receiving a given moment, we’ll have to take a closer look at the number of minted moments included in this pack drop for each individual moment. As a simple example, let’s look at two Base Set moments:

• James Wiseman’s Dunk (35k CC)

• LaMelo Ball’s Assist (4k LE)

With a higher circulation count, it’s no surprise that there are far more Wiseman dunks than LaMelo assists in this pack drop. Specifically, there are 3,147 Wiseman dunks, and only 13 LaMelo assists.

To calculate the probability of one of our Base Set moments being these moments, all we have to do is divide their number of minted moments by the total number of possible Base Set moments in the drop - 242,520. For Wiseman, this is 1.3%, and for Lamelo it’s a razor thin 0.005%. We can repeat this process for all of the Base Set moments in the drop.

Just as we did for our Rare moments, we can multiply each moment’s value by its probability and sum up the results, giving us an average return of $24.71. All that’s left to do is account for the six Base Set moments in our pack by multiplying this average return by six, and we’re left with an average return on all Base Set moments of $148.26.

Putting It All Together

With our average returns calculated for our Rare and Base Set Moments, all that’s left to do is sum up our average return:

total_avg_return = 457.20 + 148.26
total_avg_return = 605.46

That’s an average return of over $605. Subtracting the cost of our pack, we’re left with a positive expected value of $376! This Rare pack was a no-brainer at a price point of $229, but the data tell us that it would have been a statistically bad purchase at any price greater than $605.

Looking to the Future

While there haven’t yet been any tough decisions around purchasing Top Shot packs, the current trends don’t leave me confident that this will always be the case. As an example, consider the previous 3 Rare pack drops:

• Rare MGLE - $99, February 25th

• Rising Stars - $199, March 7th

• 2021 All Star Game - $220, March 19th

Using the same approach as before, we can calculate the expected value for each of these packs, and see them on a simple plot:

As we can see, the expected value from each of the previous 3 rare pack drops has decreased over time, in part due to the increase in pack prices. While they’ve remained +EV and profitable, they’re becoming less so over time. At this pace, I wouldn’t be surprised if we have some tougher decisions ahead of us. Fortunately, we can apply these strategies above to determine a pack’s EV, and make sound statistical decisions.

That’s all for this week! If you’ve made it this far, thank you for reading, and if you’re enjoying this content, the best way you can help out is by sharing with a friend.