Best way to represent a two-rounds election

[Appiani]

I would like to represent a two-round election and I would like to understand which is the best way to represent it. I was thinking of a differential equation, or perhaps also linear algebra could be used. Can anybody link something with a formal mathematical representation of a two round election, or maybe we can just discuss it here?
Thank you

 

[mfb]

What do you want to represent, and what do you want to study?

 

[Appiani]

Let us take a real example. In France they have a two rounds election law for the presidential elections. If a candidate gets 50%+1 votes at first round, he or she is elected president. Otherwise, there is a second round. I would like to represent this with an equation.

 

[mfb]

In general, "represent something with an equation" does not make sense, unless you have some specific application in mind.

 

[Appiani]

Honestly I do not understand why it should not make sense. Anyway the problem above is simple. Can it be described by a differential equation, or perhaps it is better linear algebra?

 

[mfb]

I walk along the street. Try to express that as equation.

If I specify the speed and the distance, you can set up an equation that finds the time.
If I specify the step size and the distance, you can set up an equation that finds the number of steps.
... and so on, but all those things are much more specific questions.

 

[Appiani]

In this case we have:

time (every round of elections is an occurrency of time)
number of votes per candidate
candidates

In the first round there can be n candidates. In the second round only 2 candidates
I woulk also like to express the equation in a way that there could be n rounds of elections (even though this is not realistic since normally there are no more than 2 rounds)

 

[Samy_A]

In this case we have:

time (every round of elections is an occurrency of time)
number of votes per candidate
candidates

In the first round there can be n candidates. In the second round only 2 candidates
I woulk also like to express the equation in a way that there could be n rounds of elections (even though this is not realistic since normally there are no more than 2 rounds)

An equation should establish some relation between variables.
Here all you have is a n*r matrix, where n is the number of candidates and r the number of rounds.
What do you want to achieve with this matrix?

One interesting application could be to try to predict the result of the second round based on the result of the first round. Polling organizations surely have complex databases with detailed results of previous elections that allow them to do that (more or less accurately, of course).

 

[Appiani]

Let us say that the relation between two variables is between numbers of votes and number of seats obtained. In this case the number of seats obtained can be 1 or 0. Every candidate can obtain 1 "seat" at first round to be a candidate in the second turn or 0. At the second turn 1 seat and the candidate becomes president and 0 and does not become president. As for the chances of being elected etc. I think what you said can be done with Markov chains

 

[mfb]

The number of seats is given by $s_i = floor\left(\frac{v_i-1}{N}+\frac{1}{2}\right)$ with the floor function, the number of votes v_i of candidate i and the total number of valid votes N.
You see? That is a specific relation, that has a formula. It is an overcomplicated way of saying "candidate i needs more than N/2 votes".

 

[Appiani]

I think this is exactly what I was looking for...Thanks!
Now, how would you express the same idea but with iteration. So "candidate i needs more than N/2 votes at first round or time t1 and also more than N/2 votes at second round or time t2 in order to become president"
All in one formula

 

[mfb]

Why would you do that? I don't see an application.
The general approach: define a function X(v) which does exactly what you described, then the formula is simply s_i = X(v,i). Done. If you actually want to calculate it, write a computer program.

There is a logical error in your description, the candidate doesn't need more than N/2 votes in the first round (if he gets that, there is no second round).