Increasing the Odds of Winning by Inceasing the number of voters?

Factual Questions
1

We’re grading on writing ablity, not logic, but…I want to know.

In a few papers I read today, ones in which the student chose women’s rights as their topic, I saw this argument: By doubling the number of voters, the canidates odds of winning increase.

To me this sounds silly. My conclusion is if 2 candidates are running, and there are 50 voters, their odds are the same as if there were 2 candidates and 100 voters: 50-50. In order to change the odds you’d have to change the number of people running, not double the voters.

Since math has always been my worst subject, I’m asking you if my reasoning leaves something to be desired, or am I right? If I’m wrong, please explain.

2

The argument doesn’t give enough information; for example, is this student saying that one candidate’s odds increase while the others’ decrease? Could be true if the half of the voters that make up the “doubling” have different views from the first half. Other than that, you are mostly correct. The odds don’t have to be 50/50, but if they are 50/50, or any other ratio, for 50 voters, they will be the same ratio for 100 voters (assuming the extra 50 voters think like the first 50.) No matter how many candidates there are, the probabilities of each candidate’s winning must sum to 100%. It is not possible for all the candidates to improve their probability of winning, no matter how many voters you add.

3

It depends on whether or not you want to try to pull a Sore/Loserman type of coup by recounting “damaged” or otherwise questionable ballots with “hanging chads,” “undervotes,” “overvotes,” etc. In that case, the larger the number of voters, and the closer the initial count was before the recounts, the more likely that Sore/Loserman would eventually be declared the “real winner,” since the various “damaged” or “questionable” ballots will most likely represent the votes of morons who would normally vote for that sort of candidate.

In otherwise fair elections, increasing the electorate size should not favor one candidate over another.

I expect to get flamed for this comment, so go ahead. Not like I really care.

4

I suspect that the more people that vote, the less likely it is (in the U.S.) for a third party to win over the Republicrats. Third party voters tend to be more passionate and vote in numbers disproportionate to their actual size. Those on the fence about whether or not to vote would most like vote major party. This is just a guess on my part.

Haj

5

You would think they’d say that, but no. In every case they used the plural.

6

Changing the number of voters doesn’t change the odds of a candidate winning, unless the new voters, as a group, have opinions different from the current population. For example, if, out of a population of 100, 50 are going to vote for candidate X and 50 are going to vote for candidate Y, adding another 100 people to the voting pool with the same opinions would still give X and Y 50%. If, as happened in the Reconstruction South, where blacks were given the right to vote, and southern blacks overwhelmingly favored the Republican party, then there’s an effect on the outcome.

7

There is a statistical way that the claim can be justified BUT I don’t think it is what your students meant. Any way I can come up to describe it both sounds at least a little goofy - and requires a level of statistical sophistication that seems unlikely.

Assume that each voter makes an independent, random decision about how to cast their vote, AND that the odds are not exactly 50-50, but that there is a candidate who is preferred (for this example, I will presume that the odds are 55-45 for the preferred candidate.)

If just one person votes, there is a 55% chance that the preferred candidate will win (the only vote cast.)

If 100 people vote, we can get a (pretty good) approximation of the likelihood that the preferred candidate gets 51 or more votes by using the standard Normal distribution. The relevant values are

Expected value: 55 votes
Standard error of EV: = (p*(1-p))^1/2 / n^1/2

Substituting p = .55, and n = 100, we get

Standard error .4975 * 10 = 4.975

Figuring out the standardized normal value for the test value gives us

Test - EV / SE = 51 - 55 / 4.975 = -0.804

and looking up that value using the normal probability applet at
http://www-stat.stanford.edu/~naras/jsm/FindProbability.html
gives a probability of .789, or 79% that the preferred candidate will win.

Basically, this is a straight forward application of the “law of averages” which indicates your chances of coming close to a true population parameter increase as the number in your sample increases.

Obviously, in the example, I have done much more than just double the number of voters - I went from a VERY small number to a number 100 times as big. The same logic also works for doubling the number of voters, although the differences would become far less dramatic - probably to the point of being unnoticeable for any realistic values of the size of the electorate during the period in which women gained the vote. The difference in probability of the “correct” candidate winning also depends on the exact value chosen for the probabilities - for maximum impact comes when the probability is fairly close to, but not equal to, 50-50.

As I hinted in the opening, to really make this work requires two things - 1 the votes cast are the result of a random process, and 2 there has to be a naturally preferred candidate (or party - or side to the initiative - or …). It is possible to model things this way, but it is a bit abstract and probably not real satisfying to most people.

In addition, and more directly on target to the OP - the notion that this kind of reasoning can serve as an explanation for the motivation of political leaders seems very far fetched.

I suspect that it is more likely that your students were assuming that there is indeed a difference between how women and men vote (a natural assumption for those who have learned about the “gender gap” in modern political life, but which has somewhat less validity for the time period in which women’s sufferage was actually granted.) They (your students) might presume that the party women tend to support finally acheived enough votes from males to take power, and then extended the sufferage to give them a boost in the future. Alternatively, they might be presuming that the fact that a particular party worked to give the women the vote would insure that the grateful suffragettes would therefore support that party and its candidates. In either case, some of the logic of their argument seems to be missing.

8

[Moderator watch ON]
Quoth Neurodoc:

We have ways of making you care. If you want to make comments about the validity of the past election or of the recounts, you’re free to do so… In Great Debates. Posting such material in GQ is a violation of our rules. People can and sometimes do get banned for violation of rules. I’d bet you’d care then, wouldn’t you?

While you may well end up getting flamed for your comments, I presume that most folks doing the flaming would know enough to do it in the appropriate forum, not here.

9

Not true, even if voters are taken from the same population of opinions. If one candidate has a lower probability of receiving a vote, his chances of winning an election will decrease as more people vote.

Take the example a two person election where 40% of the population prefer candidate X and 60% prefer candidate Y. If only one person, chosen at random, is permitted to vote, candidate X has a 40% chance of winning. If three people can vote, candidate X has a 35.2% chance of winning. However, with a large number of votes the chances of X winning will be quite slim, and increasing the number of voters will have less and less significance.