# Rare stickers

In an earlier post we looked at the number of packets of Panini stickers you’d be likely to need in order to complete an album. In that post I made the throwaway and obvious remark that if some stickers were rarer than others, you’d be likely to need even more packs than the standard calculations suggest. But is there any evidence that some stickers are rarer than others?

The official Panini response is given here. In reply to the question “Does Panini deliberately print limited edition or rare stickers?”, the company answer is:

No. Every collection consists of a number of stickers printed on one or two print sheets, which in turn contain one of each sticker and are printed in the same quantity as the albums.

So that seems clear enough. But the experience of collectors is rather different, with many suggesting that some stickers are much harder to find than others. So, what’s the data and what type of statistical methods can be used to judge the evidence?

The first thing to say is that, as we saw in the previous post, the nature of collecting this way can be a frustrating experience. So that even if the average number of packs needed was less than 1000, we saw that some collectors are likely to need more than 2000, even assuming all stickers are equally frequent. To put this into perspective, suppose you’ve already collected 662 of the 682 stickers, and need just another 20 stickers to complete the album. By the same type of calculation that we made before, the expected number of further packs needed to complete the album is 491, which is well over half the expected total number of packs needed (which was 969). You might like to try that calculation yourself, using the same method as in the previous post.

In other words, once you reach the point of needing just 20 more stickers, you’re not even half-way through your collection task in terms of the number of packs you’re likely to need. This can make it seem like certain stickers – the 20 you are missing – are rarer than others, even if they’re not.

But that’s not to say that rare stickers don’t exist – it’s just that the fact you’re like to have to wait so long to get the remaining few stickers might make you feel like they are rare. So again: do rare stickers really exist?

Well, here’s a site that tried to conduct the experiment by buying 1000 packs from the 2018 world cup sticker series, listing every single sticker they got and the number of times they got it. Since each pack contains 5 stickers, this means that they collected a total of 5000 stickers. They then used the proportion of times they got each sticker as an estimate of the probability of getting each sticker. For example, they were fortunate enough to get the Belgium team photo sticker 18 times, so they estimated the probability of getting such a sticker as

$18/5000 \approx 1/278$

Using this method, they were able to calculate which team collectively were the easiest and most difficult to complete. The results are summarised in the following figure:

On this basis, stickers of players from Senegal and Columbia were easy to obtain, while those of players from Belgium and Saudi Arabia were much harder. So although the Belgium team photo was one of the most frequent stickers in their collection, the individual players from Belgium were among the least frequent.

Now, you won’t need me to tell you that this is pretty much a waste of time. With just 5000 stickers collected, it’s bound to be that some stickers occur more often than others, and we don’t learn anything this way about whether stickers of certain teams or players are genuinely more difficult to find than others. One could try to ascertain whether the pattern of results here is consistent with an equal distribution of stickers, but there’s no mention of such an analysis being done, and a sample of just 5000 stickers would probably be too small to reach definitive conclusions anyway.

One fun thing though: with the 5000 stickers collected, these guys managed to complete the album except for one missing sticker: Radja Nainggolan of Belgium.  But he ended up not being selected for the World Cup anyway 😀.

This doesn’t really bring us any closer to the question of whether rare stickers exist or not. One interesting suggestion is to look at frequencies of requests in sites that handle markets for second-hand stickers. And indeed, this site finds jumps in frequencies of requests for certain types of stickers, suggesting such types are rarer than others. However, it’s not necessarily the case that the rare stickers are the ones with the most requests: Lionel Messi might be a popular trade, because… Messi… and not because his sticker is rare. Still, the post is a fun read, with complete details about how you might approach this type of analysis.

Finally, far be it for me to promote the exploits of one of our competitors, but some of the guys at ATASS pooled their world cup collections to address exactly this issue. A complete description of their findings can be found here. In summary, from an analysis of nearly 11,000 stickers, they found evidence that shiny stickers – of which there were 50 in the 2018 World Cup series – are much rarer than standard stickers.

Moreover, the strength of the evidence is completely overwhelming. This is partly because the number of stickers collected is large – 11,000 rather than 5,000 in the study I mentioned above – but also because there are 50 shiny stickers, and all of them occurred with a lower frequency than an average sticker. In fact, overall, shiny stickers occurred at a rate that’s around half that of a normal sticker. Now, if a single sticker isn’t found, it’s reasonable to put that down to chance; but it’s beyond the realms of plausibility that 50 stickers of a certain kind all occurred at below the average rate.

On this basis, Panini’s claim that all stickers are equally likely is completely implausible. The only way it could really hold true, assuming the ATASS analysis to be correct, is if there were variations in the physical distribution of stickers, either geographically or temporally. So, although Panini might produce all stickers in equal numbers, variations in distribution might mean that some stickers were harder to get at certain times in certain places.

That seems unlikely though, and the evidence does seem to point to the fact that shiny stickers in the World Cup 2018 series were indeed harder to find than the others.

In summary: yes, rare stickers exist.