With a new football World Cup comes a new Panini football sticker album. Shortly after the release of the World Cup 2022 sticker album from Panini, fans and statisticians calculated that it would cost (on average) between £850 and £900 to complete the 2022 edition of the sticker book. This is an increase of around £100 compared to the 2018 edition – a pack of five stickers has risen in price by ten pence from £0.80 to £0.90 (the number of “spots” to complete in the album has actually decreased slightly, from 682 to 670). These are eye-watering sums – so what are the possible strategies for bringing the cost of completion down, and how much can you save under each of them?

1. Form a collective

One of the strategies that we explored in our Got. Got. Need! 2018 bulletin was teaming up with friends to form a collective that continues to buy packs of stickers until all of the members of the collective have a completed album. This involves some serious sticker buying and plenty of sticker swapping – and with that comes some significant savings when it comes to the cost per member of completing an album. Some rough-and-ready simulation modelling using the 2022 World Cup album inputs suggests that a collective with two members could save in the region of £300 per head relative to a single-player “unilateral” strategy; increase the collective to six members and this saving becomes around £500 per member. Moreover, the larger the size of the collective, the lower the variance per member – as shown in the chart below. For example, across 5,000 simulations the difference between the lowest and highest cost of completion for the unilateral strategy is more than £1,000 (!), which falls to around £300 when one is in a collective of six members.

2. The hybrid approach

One of the downsides of the collective strategy is the need to find enough like-minded collaborators that are willing to commit to keep buying stickers until all the members have completed their albums. And even then, you’d need to find a way of enforcing the agreement for everyone to keep buying stickers even when some people have completed their album(s). If you would rather plough your own furrow then I would suggest a hybrid strategy of: (i) completing some spots in the conventional unilateral way; (ii) and infilling your gaps using Panini’s missing sticker service.

The cost of completing one of the Panini sticker albums is incredibly “back-end” loaded – that is, as you complete more spots, the cost of completing the next spot increases (or in economic speak, the marginal cost of completing spots is upward sloping). This is because, as you complete more spots, the probability that the next sticker is a duplicate of an existing sticker increases, and so you need to buy more stickers to not find a duplicate. For example:

– Each pack of five stickers is guaranteed by Panini to not contain any duplicates. This means that your first pack of five stickers will automatically “complete” five spots in your album. So the cost of filling your first five spots is 18 pence a spot (£0.90 divided by five).

– By the time you’ve completed 669 spots and only have one spot left to fill, the probability that the first sticker in your next pack is a duplicate is 669 divided by 670 – i.e. 99.85%. So the probability that that sticker isn’t a duplicate – which is what we’re after – is one minus this, or 0.15%. Which means you have to buy a lot of packs to complete that final spot!

The chart below shows how the incremental cost of completing the final 50 spots of the album evolves as one completes more spots. Completing these 50 spots costs more than £500 on average – around 60% of the total cost of completion, even though these 50 stickers represent only 7% of the total number of stickers in the album. And completing that 670th spot will, on average, cost you more than £100 in stickers…

For those that can’t stomach these incremental costs, Panini offers the option to buy “missing” stickers for £0.28 a sticker (and you can make up to five orders of 50 stickers, for a total of 250 missing stickers). Whilst it might not be as satisfying as completing those final spots yourself, strategically making use of this service can save a significant amount. The chart below shows that adopting this strategy for the final 50 spots could save you in the region of £500 (on average), whilst an even more liberal use of this service (i.e. the full 250 allocation) could save you slightly more than £650 (the latter of these strategies brings the average cost of completion below the £200 mark).

3. Buy one on eBay

At the time of writing, a completed version of the 2022 World Cup sticker album could be snapped up on eBay (in “mint condition”) for between £150 and £175 – the cheapest of the three strategies considered in this article. Interestingly, there appears to be quite an active secondary market for completed versions of the 2022 album; more than 30 have changed hands on eBay UK in the last month. Proof that for some, at least, this is the dominant strategy?