What Are My Odds? [MATH WARNING]

My wife recently brought up an interesting question that I think might be relevant to you frequent blog readers out there. It concerns participation in a blog contest. For those of you not familiar, a blog contest is a venerable gambling institution of the internet, closely akin to online poker and office bets about who will get the highest score on Bejeweled today. Most blog contests involve participants engaging in various activities in order to earn an entry (essentially a raffle ticket) in a giveaway. The more crazy things readers are willing to do (comment, link to blog, become a fan on the Facebook, StumbleUpon, wear a sandwichboard, etc.), the more entries received. At the end of the contest date, one entry is selected from all those submitted, and somebody wins a nice cookbook or something like that.

All this is well and good. My wife's question, however, revolved around a particular fact/myth/truth claim she had heard regarding these additional entries. She wanted to know, is it possible that increasing your number of entries actually decreases your chances of winning?

Some readers of this blog probably thought "Clearly not, more entries means more chances," and stopped reading. Others thought "Sure, more entries means more chances to not get it," and stopped reading. For those of you still reading, I shall give you the true economist's answer: It depends.

The key is to understand exactly what it depends on, which requires me to provide a brief education in probability and statistics. Don't worry, it won't hurt much. You may even learn something that will help out the next time Grandma wants to make a game of go-fish interesting!

First of all, some labels. Let's call the number of entries you decide to submit 'n,' and the number of entries everyone else decides to submit 'm.' If there is only one winning entry, which is chosen randomly with each entry having an equal chance, then the probability that one of your entries will be picked and you will win is

n / (n + m) x 100%

since (n + m) is the total number of entries. Notice that as you submit more and more entries, both the top (or numerator, in math-speak) and bottom (or denominator, in math-speak) of this fraction get bigger and bigger. The easiest way to see whether your chances go up or down is to find out whether the top grows faster, slower, or at the same speed as the bottom.

The rate of change of the top is always the same, one. If you submit one more entry, you get one more chance out of however many total entries there are. Simple enough.

The rate of change of the bottom is a bit more complex. First off, the bottom always increases by the one from your new entry. Next, we have to add to that the increase in the number of rival contestant entries that are caused by your additional entry. The mathematical notation for this is dm/dn, but don't let that intimidate you. This just means that for your first entry there are some number of people (we'll call that number M) who are participating whether you do or not. Beginning on your second entry, a certain number of rival contestant entries are added, probably because they heard about the blog contest because of your entry. Mathematically, all this can be summarized by saying that for your first entry, m = M, and for each entry after that we have

m = M + (dm/dn) * (n - 1),

where we subtract one from your entries because the first one doesn't count against you. In this case, the rate of change of the bottom is

1 + dm/dn.

So if we want to know what your chances of winning will be as you add more and more entries, we will simply place the rate of change of the top over the rate of change of the bottom,

1 / (1 + dm/dn).

We can then see if increasing your number of entries is good or bad by comparing this to your chances of winning with just the first entry,

1 / (1 + M).

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, dm/dn = 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, dm/dn = 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 dm/dn = 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).