Puzzle #2: The Soccer Trading Cards Problem

Solution
For the case were there is no overlap between the contents of the packs this problem is easy. In this case it would be guaranteed that each pack would contain cards that you do not yet have. You would have to spend: . Although this is a starting point, it is not the case and we can safely assume that the supermarket wants you to spend more than just € 540,-

The notion of overlap introduces a probabilistic component to this problem. We will call the event in which you get a card you already have O for overlap. There is a probability of overlap , for the time being we will assume that at all times this probability is fixed. This is actually an oversimplification as will become clear later. I also received the cards from the (same) supermarket and based on this I think that an estimate of 10 to 20 percent for is reasonably accurate as a starting point. Let us assume (20 percent) and of course . Now, if we receive a number of cards n, we can say that there should be new cards among these and duplicates. We are really only interested in the new cards, since our `count’ of new cards must eventually equal the entire collection: 270 cards. The question `how many cards do we need to buy before we have 270 unique ones?’ is trivial to answer:

So we need to obtain at least 337.5 cards to have all 270 unique ones. Since half a card does not exist, and cards are always distributed in packets of five, we need to round this up to 340 cards. Since each quintet of cards costs € 10,- this means we must spend at least the amount defined as:

Which is indeed somewhat more. One would need to spend € 680,- to obtain these 340 cards and as a result have the complete collection of 270 cards and an excess of 70 duplicates.

The generic formula is of the form:

Where parameter denotes the probability of obtaining an overlapping card, the number of cards (270) and the ceiling function (denoted with the symbols ) rounds up to the nearest value divisible without remainder by 5.

This in turn can be simplified to the following:

Where the additional parameter denotes the costs of one card, for this case: .

So, what is P(O)?
Let’s assume that the supermarket knows the buying behaviour of its customers and bases largely on that information. Let us further assume that the period in which the cards are `given away’ lasts for about four weeks and that the average family goes to the supermarket once a week spending € 200,-. This means that they will spend € 800,- in the entire period. Based on this the optimal value of can be calculated:

So, based on this simple model the chance of obtaining duplicates for our average family would be 32.5 percent, instead of the 20 percent used before.

Complications & Trading
The previously described model is a bit simplistic. There are many complicating factors. One is that the probability of duplicates changes over time and that this probability is not uniform for all cards: each card has its own . By varying the probabilities one can create highly sought after cards and cards that everyone already has multiple times. I would assume that cards of popular players are more scarce than those of relatively unknown soccer players.

Interestingly this non-uniform distribution of cards causes cards to have a value relative to each other laying the foundations for an economy. A card with a of say 0.10 can be said to be `worth’ twice as much as one with a of 0.20, even though the underlying monetary value for obtaining the cards really is equal. This scarcity principle is seen in many trading card games, where eventually the monetary value changes to reflect the scarcity of the card! For example: some cards in the popular `Magic’ card sets are worth hundreds of Euros.

Another complications arises from the phenomenon of actually trading the cards. Trading is beneficial in the sense that it can reduce the minimum spending needed to obtain all cards. By trading you can also rid yourself of duplicates useless to you. However, people are only willing to trade those cards that they have duplicates of. The previously mentioned scarcity influences the amount of trading that takes place. Hence, if the is equal for all cards: uniform, then there will be maximum trading opportunity. Whereas any deviation from uniformity likely results in less trading, since scarce cards are not easily traded nor likely to be found as duplicate.

The supermarket chain sold the excess cards it still had for € 0,10 per card after the handout ended. Assuming that this reflects about double the true value of a card, after all: the supermarket wants to make a profit, this would imply that all the products bought in the action period had about 2.5 percent soccer trading card tax .

Conclusion
The only reason for a supermarket to give anything to its customers `for free’ is to influence their buying behaviour. If your child is constantly nagging you for more soccer trading cards, you are more likely to round up your spending at the supermarket to the nearest `ten’ to get extra packets of cards. Rounding down, which means: buying less or cheaper products, is probably less common.

Of course trading cards have an entertainment value as well: you can play games with the cards. The trading process also has a certain educational value for children. Especially since this trading appears to mimic normal economic principles. Nevertheless, there is always one party that definitely wins here which is the supermarket itself. Without spending any extra money of their own, they succeed in getting higher profits.