Pokémon turns 30 this year. Beginning life as a video game which spawned a wildly successful trading card game and an anime series, the franchise - based around collecting an assortment of fanciful "pocket monsters" - has become an inescapable piece of the world's pop-culture furniture.
Author
- Nathan Garland
Lecturer in Applied Mathematics and Physics, Griffith University
A dedicated 30th-anniversary card set is due to be released to celebrate the occasion. A wave of nostalgia has created high demand for trading cards, with resale prices rising and sellers adopting anti-scalping measures and guarding against increasingly audacious thefts .
Behind the hype is a simple question familiar to generations of players: if you try to "catch 'em all" by opening booster packs - sets of randomly allotted cards - how long would it actually take? And would you get bored before then?
Opening packs, virtually
To tackle these questions, we can turn to mathematics. I built and ran a computer simulation of six recent Pokémon Trading Card Game sets from the current Mega Evolution series: the base Mega Evolution set plus the expansion sets Phantasmal Flames, Ascended Heroes, Perfect Order, Chaos Rising, and Pitch Black.
The simulation model uses published "pull rates" - how often particular cards turn up - estimated from thousands of real booster packs opened by collectors.
I then created 10,000 virtual collectors, and simulated each of them opening packs one at a time.
Every simulated pack contained ten cards. The simulation recorded which cards were new, which were duplicates, and how many packs each collector needed to complete the numbered base set.
This base set includes the ordinary numbered cards, up to the regular rare Pokémon EX cards - often around 100 cards per set. This excludes special illustration and ultra-rare "chase" cards that appear beyond the main set.
The distinction between the "base" and "full" sets matters, as acquiring every secret and special card is vastly harder as they are designed to be a rare, lottery-like feature. The more relatable question may be how long it takes to assemble the main base set.
The answer? More than 100 packs, usually
For five of the six sets I examined, the typical simulated collector needed about 110-130 packs to complete the base set.
Pitch Black, the newest set in the model, had a median of about 129 packs. Mega Evolution required about 130, Phantasmal Flames 131, Chaos Rising 133 and Perfect Order was the kindest at about 111.
That is a big pile of wrappers! The nature of random chance will also give different collectors very different outcomes, even when they open the same number of packs.
In the above chart, you might notice one line looks very different to the others. The Ascended Heroes set required a median of about 771 packs, because it contains 39 "double rare" cards, compared with only nine or ten in the other sets.
This is the cruel feature of random collecting games: finding the first few cards is easy. Finding the final missing cards can take far longer than intuition suggests.
When does opening packs stop feeling fun?
Completing a set is only half the story. Opening packs is also entertainment and is meant to be fun.
We can define a "freshness" measure to try and analyse this. It is not a psychological test of fun, but a simple number to try and capture the novelty: the percentage of cards in a new pack that you have not seen before.
At the beginning, almost every card is new. But freshness falls quickly!
For most of the simulated sets, after roughly 15 packs, fewer than half the cards in the next booster are expected to be new. By 30 packs, a typical pack contributes only one or two unfamiliar cards in most sets, and around three in Ascended Heroes.
Ascended Heroes stays fresher for longer because its base set is so large. But that variety comes with a trade-off: it also takes far longer to complete.
Our simulation shows how maths can help explain a familiar collector experience. Early packs feel full of discovery. Later packs become a stack of repeated "common" and "uncommon" cards, punctuated by the small chance of something genuinely new.
This is an old mathematical problem
Mathematicians call this the " coupon collector's problem ".
Imagine cereal boxes contain one random coupon and you want the full set. At first, almost every coupon helps. As your album fills, duplicates dominate. The last missing coupon can take an unexpectedly long time.
Pokémon packs present a more complicated version of the problem, because they contain multiple cards and different rarities have different probabilities. Computer simulation lets us reproduce those details while keeping track of the variation between collectors.
The same mathematics has serious applications. Ecologists use related models to estimate how much sampling is needed to observe all species in a population. Similar ideas appear in epidemiology, animal-behaviour research, and industrial quality control.
The results of my simulation don't say people should stop opening packs. Opening them can be fun, social and nostalgic. But it does reveal what you are buying: an experience whose novelty fades much faster than a collection approaches completion.
Thirty years after Pokémon began, the slogan remains "gotta catch 'em all!" But as we now know, that might mean catching many of them again and again and again.
Nathan Garland does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.
