# World Cup and coupon… Panini stickers collecting

The World Cup is under way and Cara and I are collecting World Cup panini stickers. Apparently, Panini stickers were not as popular in he US as they were in Portugal during my childhood – they were giving sticker albums in a soccer match we went to watch and Cara didn’t know what they were! So for those who don’t know, they are collective stickers for the world cup: there is essentially a sticker per player and around 20 players are represented on each team in the world cup, they can be bought in packs of 7 stickers and the objective is to collect all of them (this collection has 639 stickers) to fill the sticker album. We were discussing how many packs we would need to buy (each pack is around a dollar) to complete the collection and decided to actually estimate the numbers – I did it through the coupon collecting estimate and Cara through simulations, it was interesting to see the comparison.

Each sticker has the same rarity on the packs. The album itself already comes with 8 stickers but I am going to ignore that (as these 8 also come in the packs so the benefit of coming in the album is marginal). First of all, the idea of collecting these stickers is to trade with other collectors, and since each sticker has the same rarity only a few packs over $639/7 \approx 91$ could be enough to finish the collection. However, without trading the task becomes much more difficult and, by coupon collecting, around

$\frac{639(\log(639)+\gamma)}7 \approx 642$

packs are needed in expectation, which is quite a bit of money for stickers.

Panini, however, makes the process a bit easier by allowing each collector to order 40 stickers (at roughly 10usd, so the price of 10 packs). Although some more serious collectors think of this as cheating it does lower the price of completion (assuming no trades) by a lot:

$\frac{639(\log(639)-\log(40))}7 + 10 \approx 263$

In fact, it is easy to see that this can be lowered even further by buying multiple new sticker albums and ordering more sets of 40 (there is a sweet spot of number of extra albums, as the price of these extra stickers are around 0.25 and the price of a sticker in a pack is around $1/7 \approx 14$, (besides taking away the fun) it is not optimal to get all the stickers in albums.

Cara ran simulations (see here) and saw that these coupon collecting estimates are quite accurate.

Whether you are collecting panini stickers or not, have fun watching the world cup! And go Portugal!!!!