Minimax Multi-District Apportionments

Abstract

The problem of seat apportionment in electoral systems turns out to be quite complex, since no apportionment method exists which succeeds in verifying all the principal fairness criteria. Gambarelli (1999) introduced an apportionment technique which is custom made for each case, respects Hare minimum, Hare maximum and Monotonicity and satisfies other criteria in order of preference. In this chapter a generalization of that method is proposed, in order to extend it to the multi-district election case, where criteria should be respected at a global as well as at a local level. An existence theorem and a generating algorithm are supplied.

Appendices

Appendix 1: Some Notes on the Banzhaf Normalized Power Index

In the Theory of Cooperative Games, a power index is a function which assigns shares of power to the players as a quantitative measure of their influence in voting situations.

For instance, suppose that a system is formed of three parties without particular propensity for special alliances, and that a simple majority is required. If the allotment of seats is (40, 30, 30), any reasonable power index will assign an equal power allotment of (1/3, 1/3, 1/3). If the seat allotment of the three parties is (60, 30, 10), then any reasonable index would give a power share of (1, 0, 0), since the first party attains the majority by itself. Some complications occur if the seat allotment is (50, 30, 20). If A, B, C are the three parties, we can remark that A is crucial for the three coalitions {A, B, C}, {A, B} and {A, C}, i.e. such coalitions attain the majority with party A and lose it without A. On the other hand, party B is only crucial for the coalition {A, B} and party C is only crucial for the coalition {A, C}. In general, the power indices are based on the crucialities of the parties. In particular, the Banzhaf index (1965) assigns to each party the number of coalitions for which it is crucial. In our example, the assigned powers are (3, 1, 1). The Banzhaf normalized power index assigns to each party a quota of the unity proportional to the number of coalitions for which it is crucial. In our example, the assigned powers are (3/5, 1/5, 1/5).

In addition to John F. Banzhaf, several authors independently introduced various indices having the same normalization: Coleman (1971), Penrose (1946) and, according to a particular interpretation, Luther Martin in the XVIII century [see Riker (1986) and Felsenthal and Machover (2005)]. That is the reason why this index should be mentioned as “Banzhaf-Coleman-Martin-Penrose Normalized power index”.

A geometric interpretation is shown in Palestini (2005). For the automatic computation in general cases, we suggest the algorithm by Bilbao et al. (2000). The algorithm by Gambarelli (1996) takes into account previous computations, when the seats vary recursively. Then (with reference to the Appendix 2) it is more suitable for the application of the $\beta_{L}$-criterion, if computed before the $N_{L}$-criterion.

Overviews of further power indices can be found in Gambarelli (1983), Holubiec and Mercik (1994), Gambarelli and Owen (2004).

Appendix 2: An Algorithm Generator of the Solutions

We show an algorithm for the automatic generation of the solutions having as first criteria $F_{G}$ $N_{G}$ $\beta_{G}$ or $F_{G}$ $\beta_{G}$ $N_{G}$. Notice that this procedure can be easily structured for parallel processing, so that the time of computation can be considerably reduced.

INPUT

V the valid votes.

a the seats to be assigned to the districts.

“Global option” of the ordering of criteria at the global level ($F_{G}$ $N_{G}$ $\beta_{G}$ or $F_{G}$ $\beta_{G}$ $N_{G}$).

“Local option” of the ordering of criteria at the local level ($N_{L}$ $\beta_{L}$ or $\beta_{L}$ $N_{L}$).

OUTPUT

S ₁ , S ₂ , …, S _n the set of survived matrices.

WORKING AREAS

$\overline{{\mathop N\nolimits^{V} }}$ the matrix of normalized votes.

$\overline{{\mathop \beta \nolimits^{V} }}$ the matrix of the Banzhaf normalized power indices of votes.

$\overline{{\mathop N\nolimits^{S} }}$ :

the matrix of normalized seats.

$\overline{{\mathop \beta \nolimits^{S} }}$ :

the matrix of the Banzhaf normalized power indices of the seats.

R ₁ , R ₂ , …, R _n :

the set of matrices survived to the first local criterion.

B :

the set of vectors b generated by criteria $F_{G}$ $N_{G}$ $\beta_{G}$ or $F_{G}$ $\beta_{G}$ $N_{G}$.

c _CUR :

the vector c(V, S) at the current step.

c _MIN :

the minimum vector c _CUR of the past steps.

f _d , f _p :

pointers to set S.

S :

the matrix in construction:

PROCEDURE

Read the input data.

Compute $\overline{V}$

Compute $\overline{\beta }$ using Bilbao et al. (2000).

Compute B according to the global option.

Set maximum values to c _MIN.

• For every b of B:

• Set (n _d , n _p ) as first pointers.

• Move n _d to f _d and n _p to f _p

• For all S of the current b:

• Set S (move a and b to the arrays of the totals and move zeroes to all s _dp )

• Call the subroutine “Construction of the next S”.

• Update f _d , f _p .

• Call the subroutine “Generation of solution” using R _k as output.

• Return

Return

Set maximum values to c _MIN.

Move n to m.

Varying t from 1 to m:

• Move R _t to S.

• Call subroutine “Generation of solution” using S _n as output.

Return

End

SUBROUTINE “GENERATION OF SOLUTION”

If the local option is N _L , compute $\overline{{\mathop N\nolimits^{S} }}$

else compute $\overline{{\mathop \beta \nolimits^{S} }}$ using Gambarelli (1996) (case $\beta_{L}$ $N_{L}$) or Bilbao et al. (2000) (case $N_{L}$ $\beta_{L}$).

During the above computation, construct c _CUR and compare it with c _MIN.

Just if c _{CUR >} c _MIN exit.

When the construction of the normalized matrix is over:

If c _{CUR =} c _MIN move n + 1 to n else move 1 to n move c _CUR to c _MIN.

Move S to output.

Exit

SUBROUTINE “CONSTRUCTION OF THE NEXT S ”

If min $(a_{{n_{d} }} ,b_{{n_{p} }} ) = a_{{n_{d} }} ,$ then	move $a_{{n_{d} }}$ to $s_{{n_{d} n_{p} }} ,$
	move 0 to all the other elements of the last row and to $a_{{n_{d} }},$
	move ($b_{{n_{p} }}$ − $a_{{n_{d} }}$) to $b_{{n_{p} }},$
	and iterate the procedure on the submatrix obtained by
	eliminating the last column, i.e. decreasing $n_{d}$ by 1
If min $(a_{{n_{d} }} ,b_{{n_{p} }} ) = b_{{n_{p} }} ,$ then	move $b_{{n_{p} }}$ to $s_{{n_{d} n_{p} }} ,$
	move 0 to all other elements of the last column and to b np
	move ($a_{{n_{d} }}$ − $b_{{n_{p} }}$) to $a_{{n_{d} }},$
	and iterate the procedure on the submatrix obtained by
	eliminating the last column, i.e. decreasing $n_{p}$ by 1
At the end of the procedure we obtain a 1 = b 1; this number will be moved to s 11
Exit

EXAMPLE

In example 3 the construction sequence of the first S is:

The sequence of the other S continues as follows: