Sorry for yet another blogging drought. My recreational internet time has been devoted to trying to participate in a contest in which I could win eight skeins of yarn and some other stuff (needles, stitch markers, a pattern, pins, that kind of stuff).
You wouldn’t be too surprised about the form of the contest: Interact with the brand’s partners in various ways to get codes and submit the codes on the contest web page. The more codes you submit, the more likely you are to win the yarn.
A discussion has come up on the internet: Should you collaborate with others to get more codes. If I could find some codes and you could find some other codes, should we share our codes with each other? And here is where the structure of the contest comes into play. So far most of the discussion that I’ve seen has been about the ethics of collaborating and people’s intuitions about how much this affects their chances of winning.
This is not your standard type of raffle in which you put all the tickets into a hat and then keep drawing until you have awarded all the prizes. I haven’t taught our combinatorics classes, but I wouldn’t be surprised if we had a problem about standard raffles. In this case, if we collaborate, I may lose my edge over you, but we both have an advantage over the people who can not find as many codes as our team can. And based on what I have seen about the world, the distribution of who has how many codes is likely to be a power distribution, so only a few people will be out at the front of the pack with a lot of codes.
If one reads the Terms of Service of the contest, the way it works is that there is going to be one drawing for each code. Everyone who found a particular code will be in the running for the prize that is attached to that particular code. If you have already won, then you can not win again.
The math has to work out different here. In the standard raffle example, if you were the only person to find the rarest code in the whole world (and no other codes), you have almost zero chance of winning. However, in the raffle being used in this internet contest, you would be guaranteed to win.
If I were the sort of person who had an independent study student looking for a math project, I’d have my student work on it. There has got to have been a lot of work done already on the various ways to run raffles, but who knows how much of it is from the standpoint of determining whether you should collaborate with others to gain entry into this particular type of raffle. If there is existing work, that would give a student something to build off of. As is the trend these days, I would imagine that the student would learn enough Python to run some simulations of various numbers of codes, entrants, collaborators, etc. I would not be surprised at all if one of the flavors of this problem had an optimal solution when the ratio of some important quantity to some other important quantity were 1/e.
This problem also reminds me of an old problem about the Green Card diversity lottery, back when it was done on paper. In the old lottery, the government would draw out paper applications at random, and if your application was chosen, you were eligible for a Green Card. However, if your application was chosen more than once (only one entry per person!), then you did not get anything (and were probably banned from ever coming to the United States). In this case, the question was to find the optimal number of applications to send in. Now that everything is done on computer, this question is obsolete.
Since I’m not particularly likely to win the contest, I decided to buy some more yarn that I don’t need and to knit myself a sweater that looks kind of like the ones that the models are wearing in the advertisements for the contest. I think that this means that the real winner here is the yarn store.