PANINI, an Italian firm, has produced sticker albums for World Cups since the tournament in Mexico in 1970; in 2014, there were 640 stickers to collect. The market for the stickers is not just for kids, however; it is also for micro-economists. Getting every slot filled delivers an early lesson in probability; the value of statistical tests; the laws of supply and demand; and the importance of liquidity. When you start an album, your first sticker (they come in packs of five) has a 640/640 probability of being one you don’t have already. As the spaces get filled, however, the odds of opening a pack and finding a sticker you want lengthen. According to Sylvain Sardy and Yvan Velenik, two mathematicians at the University of Geneva, the number of sticker packs that you would have to buy on average to fill the album by mechanically buying pack after pack would be 899. That assumes there is no supply shock to the market: the theft of 300,000 stickers in Brazil in April 2014 left many collectors fearful that Panini would run short of cards.

It also assumes that the market is not being rigged. Panini insists that each sticker is printed in the same volume and randomly distributed. But many collectors will be haunted by a single recurrent card. In a 2010 analysis, Messrs Sardy and Velenik played the role of “regulator” by checking the distribution of stickers for a 660-sticker album sold in Switzerland for that year’s World Cup. Out of their sample of 6,000 stickers, they expected to see each sticker 9.09 times on average (6000/660). They tested to see whether the actual fluctuations around this number were consistent with the expected distribution of stickers, and found that it was. Such statistical tests are increasingly being applied to spot price-fixing and anti-competitive behaviour in financial markets, too.

Even in a fair market, though, it is inefficient to buy pack after pack as an individual (not to mention hugely expensive for parents). The answer is to create a market for collectors to swap their unwanted stickers. The playground is one version of this market, where a child who has a card prized by many suddenly understands the power of limited supply. Sticker fairs are another. As with any market, liquidity counts. The more people who can be attracted into the market with their duplicate cards, the better the chances of finding the sticker you want. Messrs Sardy and Velenik reckon that a group of ten people, swapping stickers efficiently and taking advantage of Panini’s practice of selling the final 50 missing stickers to order, would need only 1,435 packs between them to complete all ten albums. Internet forums, where potentially unlimited numbers of people can swap stickers, mean that this number falls even further. The idea of a totally efficient market should dismay Panini, which would sell fewer packs as a result. Fortunately, as in all markets, behaviour is not strictly rational. Despite entreaties from parents and economists, younger football fans will always be prepared to tear out most of their stickers to get hold of Lionel Messi.

Dig deeper:

Brazil's World Cup preparations have shown off the improvisation for which the country is nearly as famous as its footballers (May 2014)

England's football clubs are destroying their roots as they grow (May 2014)

Mass production and modern communication bring out the collector in all of us (December 2005)

Update: This blog post has been updated to remove the news peg.