Abstract
The Quick Count is a statistical exercise that uses information from the voting in a sample of polling stations and seeks to estimate the voting trends, in a given election, and communicate them to the population on the evening of the same day of the election. The objective of the Quick Count for the Mexican Federal Election of June 6, 2021, was to estimate the conformation of the Chamber of Deputies. That is, of the 500 deputies that integrate the chamber, the number obtained by each political force. In this work, an overview of the statistical considerations to obtain these estimates is presented: definition of the sampling design and estimation under a missing data setting. In addition, an algorithm that describes how to transform votes into seats in the Chamber of Deputies is provided.
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Notes
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A complete register of the Mexican citizens entitled to vote over the polling station.
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Where for any real number b, we have that \(\left\lfloor b\right\rfloor \) is the largest integer less than or equal to b. As a consequence \( b\ge \left\lfloor b\right\rfloor \).
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A Appendix: Transforming Votes into Seats in the Deputy Chamber
A Appendix: Transforming Votes into Seats in the Deputy Chamber
This appendix presents a detailed description of the considerations and procedures for transforming the votes received at the polls into seats in the Deputy Chamber.
1.1 A.1 Relative Majority
The territory of Mexico is divided into 300 Federal Electoral districts (article 53 of the [2]). These districts do not intersect with each other and their union is the entire country. The 300 deputies through the RM principle are obtained by holding independent elections in each of these 300 districts. In each district, there can be only one winner, either from a political party or an independent candidate. Then, the vector of seats that each political force obtains through the RM principle is given by
To obtain the vector RM, it is necessary to consider whether there are coalitions in the districts or not. Under a coalition, to determine the political party to which the seat belongs, the coalition agreements must be taken into account. Also, it is important to consider the INE/CG193/2021 agreement,^{Footnote 6} with regard to effective affiliation and relevance to a parliamentary group. This is described in detail in the next section.
1.1.1 Coalitions
In each of the 300 elections, there may be coalitions between different parties. If a coalition exists, the following considerations must be considered:

On the ballot, voters can validly choose any combination of parties in the coalition, and the vote will be added to the coalition’s candidate. For example, if in the ith district there is the twoparty coalition: \(P_{i,1}\) and \(P_{i,2}\). Valid votes on the ballot result from any of the following combinations: \(\{P_{i,1}\}\), \(\{P_{i,2}\}\) and \(\{P_{i,1}, P_{i,2}\}\). Any of these possible voting combinations would imply one vote in favor of the coalition’s candidate. Similarly, it may happen that in the jth district, there is a coalition between three parties: \(P_{j,1}\), \(P_{j,2}\) and \(P_{j,3}\). Thus, the valid ways to vote would be selecting any of the combinations \(\{P_{j,1}\}\), \(\{P_{j,2}\}\), \(\{P_{j,3}\}\), \(\{P_{j,1}, P_{j,2}\}\), \(\{P_{j,1}, P_{j,3}\}\), \(\{P_{j,2}, P_{j,3}\}\) and \(\{P_{j,1}, P_{j,2}, P_{j,3}\}\). If there are no coalitions in a given district, the valid votes are only the votes for each party directly.

If there is a coalition in the ith district, the sum of the votes obtained by all valid ways of voting for the coalition will be used in favor of the coalition candidate. At this point, it is necessary to consider both coalition agreements and effective membership:

i.
The political parties form a coalition and there is only one candidate for the coalition. As already mentioned, votes in favor of any party in the coalition (or valid combination) are considered in favor of this candidate. In addition, in case the coalition wins, the parties previously agree on the coalition party for which the seat will be counted (see the original coalition agreements^{Footnote 7}).

ii.
Effective membership is a concept that was used for the first time in the 2021 election (see INE/CG193/2021 agreement). The effective affiliation is a validation that the INE makes to the coalition agreements. Here, it is verified that the candidate is a member of the party that indicates the agreement. Otherwise, a correction is made, and the seat is counted to the party in which the coalition candidate is active.

i.

To determine the weight of each party at the national level, it is important to distribute the coalition votes among the parties that made up the coalition. This distribution is done at the Federal District level. In this case, the votes for valid combinations that include two or more parties are divided between each party in the combination. Individual votes for each party are not shared. For example, in the case of a coalition between the parties \(P_{i,1}\) and \(P_{i,2}\) in the ith district. The votes obtained by the combination would have to be distributed \(\{P_{i,1}, P_{i,2}\}\), between parties \(P_{i,1}\) and \(P_{i,2}\). This is achieved through the following procedure:

(a)
The total number of votes of the combination is calculated \(\{P_{i,1}, P_{i,2}\}\), i.e. \(v_1 = \# \text{ votes } \{P_{i,1}, P_{i,2}\}\).

(b)
The quantity \(v_2 = \left\lfloor \frac{v_1}{2}\right\rfloor \)^{Footnote 8} is obtained, as well as the remainder \(v_3 = v_1  2 v_2\) and the votes for each party are calculated considering both direct votes and joint votes, i.e. \(VP_{i, 1} = \#\text{ votes } \{P_{i,1}\} + v_2\) and \(VP_{i, 2} = \#\text{ votes } \{P_{i,2}\} + v_2\).

(c)
The remainder is assigned to the party with the most votes, i.e. if \(VP_{i, 1}\ge VP_{i, 2}\), then \(VP_{i, 1} = VP_{i, 1} + v_3\) and in another case \(VP_{i, 2} = VP_{i, 2} + v_3\).
Assuming that in the jth district there was a coalition between three parties, following an analogous procedure, the votes obtained by the combinations \(\{P_{j,1}, P_{j,2}\}\), \(\{P_{j,1}, P_{j,3}\}\), \(\{P_{j,2}, P_{j,3}\}\) and \(\{P_{j,1}, P_{j,2}, P_{j,3}\}\) would have to be distributed between each political party of the combination.

(a)
Through the previous procedure, the total votes in favor of each political party in each of the 300 federal districts are obtained, i.e. \(VTP_{i,j}\) for \(j=1, \ldots , k\) and \(i=1, \ldots , 300\).
1.2 A.2 Voting Types Considered in the Law
The RM principle is based on the number of votes each candidate obtains. Instead, the PR principle depends on several considerations. Before describing the latter, we define three voting types necessary to implement it.
1.2.1 Total Vote Cast
The total vote cast (VTE by its Spanish abbreviation) is the sum of all the votes cast at the polls; see article 15 of the [6]. To obtain the deputies by the PR principle, the first step is to obtain the VTE for each political party at the national level. Obviously, the coalition agreements and effective membership must have been considered before reaching this point. The VTE in the ith district can be written as follows:
where \(VTP_{i,j}\) is the total number of votes in favor of the jth political party in the ith district, \(VTI_{i,s}\) is the total number of votes in favor of the sth independent candidate in the same district, \(VTNR_i\) are the votes for unregistered candidates and \(VTN_i\) are the null votes.
Finally, the VTE disaggregated by party, independent candidates, votes for unregistered and null candidates is given by
where \(VTP_j = \sum _{h=1}^{300} VTP_{h,j}\); for \(j = 1, \ldots , k\) in the same way, to obtain the total votes for independents, unregistered candidates and null votes, these are added over the 300 districts. Finally, the VTE is obtained simply by summing the components of the vector \(VTE_{d}\), i.e.
With this, the citizen participation in the federal election is calculated via
where NL is simply the total number of people entitled to vote (known as nominal list). In the 2021 election, the size of the nominal list was of \(NL = 93,455,692\).
1.2.2 Valid Vote Cast
The Valid Vote Cast (VVE by its Spanish abbreviation) is given by the VTE minus invalid votes, that is,
see article 15 of the [6] and Sect. II of article 54 of the [2]. The disaggregated VVE is obviously defined as
According to the paragraph added in the Official Gazette of the Federation 02102014 in Sect. I of article 41 of [2], the jth political party will preserve its registry at the National level, only if the percentage of its total number of votes with respect to VVE is greater than or equal to \(3 \% \), i.e. if \( q_j \ge 3 \%, \) then the \(P_j \) party retains its registration at the national level, where
For this reason, one of the objectives of the Quick Count is to estimate \( q_1, \ldots , q_k \). Observe that the sum of these percentages is not \( 100 \% \) since the independent candidates, who are also part of the VVE, are not being considered.
1.2.3 National Vote Cast
The National Vote Cast (VNE by its Spanish abbreviation) aims to include in the PR principle only those political parties that have obtained “enough” votes. Hence, the VNE is defined only by the votes of political parties (without independent candidates) and given by the total votes of those parties that comply with having at least \( 3 \% \) of the VVE. The disaggregated VNE is defined as
see article 15–2 of [6] and Sect. III of article 54 of the [2]. Then, the VNE is obtained as
where \(\mathbbm {1}_{A}(x)\) is the indicator function defined as 1 if \(x \in A\) and 0 in another case.
1.3 A.3 Proportional Representation
The inputs to calculate the 200 deputies by the PR principle are the number of deputies obtained by the RM principle of each political force (A.1), as well as the \(VNE_d\), (A.8). Based on this information, the 200 deputies are distributed according to the PR principle as described below.
1.3.1 First Distribution
Start by calculating the Natural Quotient (NQ) as \(NQ = \frac{VNE}{200} \); see [6], article 16–2. NQ can be interpreted as the number of votes that each seat in the Chamber is worth under the PR principle. To carry out the first distribution, the deputies assigned to each political party will be determined as (see [6], articles 16–3 and 17–1)
Then, it is clear that
Therefore, we need to assign \( \gamma = 200  \sum _ {j = 1} ^ k C_j \) deputies; this is done following the largest remainder method:

1.
The residuals are calculated, i.e. \( \delta _j = VTP_ {j} \mathbbm {1}(q_j \ge 3 \%)  (C_j)(NQ) \), for \( j = 1, \ldots , k \).

2.
The political parties are sorted from highest to lowest according to their residuals.

3.
A seat is assigned to each party according to that order until there are no more deputies to distribute.
It is important to note that the reminder \(\gamma \) is always less or equal to the number of political parties that contribute to the VNE. Therefore, the previous procedure will always lead to the distribution of the 200 deputies, denoting \( L_j \in \{0, 1 \} \) to the places awarded to the party \(P_j\), for \(j = 1, \ldots , k\), by the largest remainder method. The total number of deputies via the PR mechanism, which each party has up to this point, is obtained via \( R_j = C_j + L_j, \ \text{ para } \ j = 1 \ldots , k, \) and clearly \( \sum _ {j = 1}^k R_j = 200 \). Therefore, the total number of deputies that would correspond to the party \( P_j \) is given by the sum of deputies obtained via the principles of RM and PR, i.e. \( NP_j = M_j + R_j, \ \text{ for } \ \ j = 1, \ldots , k. \)
1.3.2 Overrepresentation
In an ideal democracy, the composition of the Chamber of Deputies would faithfully represent each of the sectors of society. In the case of Mexico, the 500 deputies should be distributed proportionally according to the VVE. However, the law allows a limited overrepresentation by establishing two upper bounds:

1.
No political party can have more than 300 deputies by both principles (RM and PR).

2.
No political party can have a greater number of seats than what results from considering \(8 \%\) of its participation in the VNE.
The number of seats for the political party \( P_j \) satisfies the overrepresentation bounds if
where
LEGIPE [6], article 17–2, indicates that the party that does not comply with this limit should lose the necessary seats by the PR principle, until the condition is satisfied.
1.3.3 Adjustment
In the event that any political party exceeds the overrepresentation bound, an adjustment is necessary. First, the political party in excess is assigned the maximum number of seats possible via the PR principle (so that it does not exceed the limit (A.9)). Second, this party is removed from the procedure and the remaining seats are distributed among the parties that are not in excess; see article 18–1 of [6]. It is important to observe that these steps imply an iterative process, since (1) there may be several parties that exceed the limit and (2) when reassigning deputies it is necessary to verify again that the bound (A.9) is satisfied by the remaining parties. The algorithm is as follows:

0.
Initialize the indicator variables \(G_j\) that will tell us if a political party is (or was) in excess; additionally, parties that do not contribute to the VNE are discarded, i.e. \( G_j = 1  \mathbbm {1}(q_j \ge 3 \%), \ \ \text{ for } \ \ j = 1 \ldots , k.\)

1.
Calculate the excess for each party, via
$$ E_j = (NP_j  U_j) (1  G_j) \mathbbm {1}(NP_j> U_j), \ \ \text{ for } \ \ j = 1 \ldots , k. $$Clearly, \( E_j> 0 \), if and only if (1) \( P_j \) is one of the parties that contributes to the VNE and has not been in excess and (2) it is in excess in this iteration, i.e. \( NP_j> U_j \).

2.
Update the \(G_j\) variables, i.e.
$$ \displaystyle G_j = {\left\{ \begin{array}{ll} 1, &{} \text {if} E_j> 0, \\ G_j, &{} \text {otherwise}, \end{array}\right. } \ \ \text{ for } \ \ j = 1 \ldots , k. $$ 
3.
Assign the maximum number of seats possible by the PR principle to the parties that are in excess. This is done simply by removing the surplus, i.e. \( R_j = G_j (R_j  E_j), \ \ \text{ for } \ \ j = 1 \ldots , k. \)

4.
Distribute the remainder of seats, i.e. \(\beta = 200  \displaystyle \sum _ {j = 1}^k R_j \), among the parties that are not (or have not been) in excess. Based on article 18 of [6], the Effective National Cast (\( VNE_f \)) is calculated by subtracting from the VNE the votes of the parties that are or were in excess, as well as those that are not part of the VNE, i.e.
$$ VNE_f = VNE  \sum _ {j = 1} ^ k (G_j) (VTP_ {j}) \mathbbm {1}(q_j \ge 3 \%). $$ 
5.
A new natural quotient is calculated via \( NQ = \frac{VNE_f}{\beta }, \) then a procedure analogous to that carried out in the initial distribution is followed. These \(\epsilon \) seats are distributed, first considering
$$ C_j = \left\lfloor \frac{VTP_ {j} \mathbbm {1}(q_j \ge 3 \%) (1G_j)}{NQ}\right\rfloor , \ \ \text{ for } \ \ j = 1 \ldots , k. $$Noting that \( \gamma = \beta  \sum _ {j = 1} ^ {k} C_j \) remains to be assigned, the residuals
$$ \delta _j = \ VTP_ {j} \mathbbm {1}(q_j \ge 3 \%) (1 G_j)  (C_j) (NQ) , \ \ \text{ for }\ \ j = 1,\ldots , k,$$are calculated. Via the largest remainder method, \( L_j\in \{0, 1\} \) is obtained and the number of seats by the PR principle is updated calculating
$$R_j = R_j + C_j + L_j, \ \ \text{ for }\ \ j = 1,\ldots , k. $$Note for some j, \(C_j=0\) (thus \(L_j = 0\)), hence \(R_j = R_j\) is a consistent step.

6.
Update the number of seats, i.e. \(NP_j = M_j + R_j\).

7.
The excess is calculated according to step 2; here, there are two options: (i) if there is a party in excess, proceed to step 3, and continue with the remaining steps; (ii) in case there are no more parties in excess, the algorithm ends (see article 17–2 of the [6]).
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Rodríguez, C.E. (2022). Estimating the Composition of the Chamber of Deputies in the Quick Count for the 2021 Federal Election in Mexico. In: AntonianoVillalobos, I., FuentesGarcía, R., Naranjo, L., NietoBarajas, L.E., RuizVelasco Acosta, S. (eds) Interdisciplinary Statistics in Mexico. Springer Proceedings in Mathematics & Statistics, vol 397. Springer, Cham. https://doi.org/10.1007/9783031127786_12
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