).
First, let's consider the case in which additional entries on your part never bring in new rivals. This would be the case if additional entries are things like commenting, becoming the author's Facebook friend, and anything else that doesn't really let anyone new know about the blog contest. In this case, d
m/d
n = 0, so as you add more and more of these entries your chances of winning get closer and closer to 1 / (1 + 0), or 100%. You can only make yourself better off by adding these entries.
Next, what if submitting extra entries advertises the blog contest, so that every additional entry of yours brings in the same number of new rival contestant entries. In this case, d
m/d
n = a, some constant number, and your chances of winning approach 1 / (1 +
a). This improves your probability of winning whenever 1 / (1 +
a) > 1 / (1 +
M), or whenever the number of extra contestant entries from each of your submissions,
a, is less than the number of contestant entries there would be even if you never showed up,
M. So even if in total your extra entries bring in more new people than would have entered otherwise, as long as each entry brings in fewer than
M, you are better off submitting the entry. (Note to Jessie: this is a bit different than the way I explained it to you. I was assuming that
a is always less than
M.) If
a >
M, however, you should just stick with your original entry.
Finally, what if some extra submissions will bring in more rival entries than others? Obviously you'll want to start with the ones that do the worst job advertising (why generate more competition than you have to?), and work your way up to the most effective ones. In this case, we can say d
m/d
n =
a *
n, just as a convenient example. This means that your chances of winning approach 1 / (1 +
a *
n), which is shrinking as
n gets bigger and bigger. So at some point,
a *
n has to become bigger than
M, and you definitely don't want to submit any entries beyond that point. However, if you start out with
a *
n <
M, then the first few entries will actually help your chances of winning. In order to make your chances as good as possible, you will want to submit extra entries until
n =
M /
a, which is a bit difficult to interpret (especially since you won't know exactly what
M and
a are, anyway), but suffice it to say that if there are a lot of contestants entering no matter what you do, you'll want to try a few extra submissions, but the more effective entries are at advertising for the contest the fewer entries you want to submit.
So, to sum up what we've learned from the probability accounting above: (1) if nobody is going to join the contest because of your entries, enter as much as possible; (2) if you have several options for earning entries that will all bring in about the same number of new contestants, do them all if they don't bring in too many, or do none of them if the advertising is too effective; and (3) if you have several options for earning entries that vary in how effectively they advertise the contest, do a few of the least effective ones only (how many depends on the contest).
Some of you will no doubt be saying at this point "It's so obvious! Why do we always have math and science people telling us to do things we already know?" I agree that these guidelines are pretty obvious at the end of the blog entry. But not everyone will think it's obvious, and having something reliable tell them why it makes sense could be the difference between constantly losing contests and the occassional win for these people. Even for those of us with strong statistical instincts, would we have kept all three of these points in mind while making our decisions if the statistics didn't remind us to? For a great many of us brilliant, educated people, the answer is a resounding no. Just because we get it immediately when we think about it doesn't mean we would have thought about it without prompting (man, just what tense
was that last verb?).
Keeping people thinking and remembering the "obvious" is pretty much what economics and statistics education is all about. Some people have better instincts than others, but everyone can improve their thinking processes with a little bit of rigorous thinking (just not
too much).
Posts
