Migration votes calculation

When political polls take place, usually the results are greatly publicized. A certain party won a tot number of votes, another one lost tot other votes, and so on. On the political and mathematical point of view, it is interesting to get the statistical knowledge not only of the simple final results, the percentages obtained by the various parties, but also of the flows of votes from one party to another. By example, if party A got x_A votes in the last poll and in the current poll gained δx_A, it might be useful to know from which parties it got those votes, while, if it lost them, towards which parties they got transferred. Purpose of this article is to determine the formulae that permit to calculate in a poll the flows of votes between parties.

Let N be the number of polling stations where polls take place, n the number of eligible parties, and let p_ij, with 1 <= i,j <= n, the migration probabilities, i.e. the probabilities according to which the voter that in the preceding poll had voted in favour of the i-th party, subsequently votes for the j-th. Finally, let u_k/i be the percentages of votes the i-th party obtained in the k-th polling station in the last poll, and v_k/i those obtained in the current poll. With these definitions we can say that in the k-th polling station of the current poll the i-th party lost in the average a u_k/i ∙ p_ij percentage of votes in favour of the j-th party. Summation over all the parties produces the most probable percentage of votes obtained in the current poll by the j-th party in the k-th polling station:

Σ_l=1ⁿ u_k/l ∙ p_lj = v_k/j.

Here we underlined the symbol v_k/j in order to express the fact that it doesn’t represent the true percentage value, but the one obtained through the use of probabilities. As for the entire poll, which includes all the polling stations, the following relations hold:

u_i = Σ_k=1^N u_k/i, v_i = Σ_k=1^N v_k/i.

It is to be noted that the summations also include the contributions from the voters that remained faithful to their previous party. Now the problem consists in determining the migration probabilities p_ij, which, when multiplied by the percentages of voters of the preceding poll and summed over all polling stations, produce the percentage evaluations of the migratory flows.

Σ_k=1^N u_k/i ∙ p_ij = v_i → j .

For this purpose, let’s consider for each party the half sum  A_i ( p_ji ), made over all the polling stations, of the squares of the differences between the percentages calculated in the current vote by means of the migration probabilities, v_k/i, and those effectively obtained, v_k/i:

A_i ( p_ji ) = Σ_k=1^N ( v_k/i – v_k/i )² / 2 = Σ_k=1^N ( Σ_j=1ⁿ  u_k/j ∙ p_ji – v_k/i )² / 2.

The statistically most probable migration probabilities are then obtained by imposing the condition that the A_i ( p_ji ) be minimal, i.e. that ∂A_i / ∂p_ji = 0 for each i,j:

1)     ∂A_i / ∂p_ji = Σ_k=1^N ( Σ_l=1ⁿ u_k/l ∙ p_li – v_k/i ) ∙ u_k/j = 0.

Condition (1) determines a system of n² non-homogeneous linear equations in the n² unknowns p_ij. Now we must impose the condition that the p_ij really represent probabilities. That corresponds to require that for each i the following relations hold:

2)     Σ_j=1ⁿ p_ij = 1.

Summation is made over the second index, since it must express the totality of votes getting out of the i-th party. Condition (2) requires the addition of n further linear equations in the unknowns p_ij to system (1). Thus the total number of equations to be satisfied becomes n ∙ (n + 1). By assuming that all the equations are linearly independent, the system solvability requires the introduction of n further unknowns. Which ones to introduce? The meaning of the system itself gives us the answer. Since the results we want to determine represent estimates of values subjected to statistical fluctuations, we expect that the migration probabilities p_ij do not necessarily satisfy the n² equations (1). Hence it makes sense to introduce new unknowns δ_i, in order to account for such discrepancies. With these additions, equations (1) get the following expression:

3)     Σ_k=1^N ( Σ_l=1ⁿ u_k/l ∙ p_li – v_k/i ) ∙ u_k/j = δ_i.

Now the system of equations (2) and (3), in the unknowns p_li and δ_j, is made up of n ∙ (n + 1) linear equations with n ∙ (n + 1) unknowns, and is therefore solvable. 

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Tags: Math, migration votes, political parties, Science, statistics, votes