Heuristic and exact solutions to the inverse power index problem for small voting bodies

Abstract

Power indices are mappings that quantify the influence of the members of a voting body on collective decisions a priori. Their nonlinearity and discontinuity makes it difficult to compute inverse images, i.e., to determine a voting system which induces a power distribution as close as possible to a desired one. The paper considers approximations to this inverse problem for the Penrose-Banzhaf index by hill-climbing algorithms and exact solutions which are obtained by enumeration and integer linear programming techniques. They are compared to the results of three simple solution heuristics. The heuristics perform well in absolute terms but can be improved upon very considerably in relative terms. The findings complement known asymptotic results for large voting bodies and may improve termination criteria for local search algorithms.

Appendices

Appendix A: ILP formulation for the inverse Penrose-Banzhaf index problem

Even though stating the inverse power index problem as an optimization problem is trivial (see (3)), coming up with an implementable formulation for finding an exact solution is not. Voting systems are discrete objects, and so some kind of discrete optimization is needed. Exhaustive enumeration (see Keijzer et al. 2010) is limited at best to n≤9 (see Table 1). A much more tractable alternative is to describe the set of feasible binary voting systems by integer variables and to use some of the available optimization software packages. These allow significantly larger numbers of variables when dealing with linear rather than non-linear (mixed) integer optimization problems. So it is unfortunate that problem (3) cannot directly be translated into a linear problem for the Penrose-Banzhaf index.²² The “work-around”, which has first been suggested by Kurz (2012b) and is adopted here, is to use ILP techniques in order to merely find out whether some binary voting system v exists whose PBI vector B(v) is at most a specified distance α apart from the target β. This feasibility problem can be solved much more easily than the underlying minimization problem. Still, one can iteratively determine the exact solution of (3) by varying α.

We will mostly confine our description to the case of measuring distance by the d ₁-metric. Adaptations to the $d_{1}'$ or d _∞-metric are straightforward. They involve heterogeneous coefficients in inequality (19) below for $d_{1}'$, and neither i-subscripts in (16)–(18) nor a summation in (19) for d _∞.

The PBI vector (1,0,…,0) of a dictator has at most a d ₁-distance of 2 from any normalized power distribution (summing to 1). This is in fact the worst case, and the minimal achievable deviation α ^∗ must lie inside the interval [l ₁,u ₁], where l ₁=0 and u ₁=2. In each iteration t=1,…,T of the algorithm we will check whether α=(u _t −l _t )/2 is a feasible distance between target β and the PBI values generated by the considered class of voting systems. If so, we set u _t+1=α and leave l _t+1=l _t unchanged; otherwise we update l _t+1=α and leave u _t+1=u _t unchanged. In each iteration the length of the interval [l _t ,u _t ] shrinks by a factor of 2. Since the total number of swings in an n-player voting game lies between n and $m\binom{n}{ m}<n2^{n}$ where $m= \lfloor\frac {n}{2} \rfloor+1$ (see, e.g., Felsenthal and Machover 1998, Sect. 3.3), two distinct PBI vectors differ, both in the d ₁ and the d _∞-metric, by at least $(\frac{1}{n2^{n}} )^{2}$. A finite number T of iterations are, therefore, sufficient for obtaining a solution. More specifically, O(n) bisections on α are needed before $u_{t}-l_{t}\le (\frac {1}{n2^{n}} )^{2}$ and further improvements become theoretically impossible.

A pseudo-code description of this bisection approach reads as follows:²³

• Input: desired power index vector β, class of binary voting systems Γ, metric d(⋅)

• Output: minimum d-distance α ^∗ between β and PBI vectors induced by Γ

• l ₁=0

• u ₁=2

• α ^∗=2

• $\varepsilon= (\frac{1}{n2^{n}} )^{2}$

• t=1

• while u _t −l _t >ε

  • $\alpha=\frac{u_{t}-l_{t}}{2}$

  • t=t+1

  • solve feasibility problem 〈β,Γ,d(⋅),α〉

  • if v∈Γ such that d(B(v)−β)≤α exists

  • then

    • u _t =d(B(v)−β), α ^∗=u _t

    else

    • l _t =α

    end if

• end while

• return α ^∗

The feasibility problem 〈β,Γ,d(⋅),α〉 consists of verifying whether there exists a voting system v∈Γ such that d(B(v),β)≤α. The following ILP formulation describes it for $\varGamma=\mathcal{S}$ and the d ₁-metric. Adaptations to $\mathcal{C}$ or $\mathcal{W}$ and $d_{1}'(\cdot)$ or d _∞(⋅) involve further variables and (modified) constraints, but are otherwise very similar:

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

The binary variables x _S define a Boolean function v via v(S)=x _S ; inequalities (7)–(10) ensure that they represent a simple game. The binary auxiliary variables y _i,S =x _S∪{i}−x _S which are introduced in (11)–(12) for all i∈N and ∅⊆S⊆N∖{i} satisfy y _i,S =1 if and only if coalition S is a swing for voter i, i.e., contributes 1/2ⁿ⁻¹ to $B_{i}'(v)$. They are used in order to determine the number of swings $s_{i}=2^{n-1}\cdot B_{i}'(v)$ for each player i in Eq. (14). The total number of swings $s=\sum_{i=1}^{n}s_{i}$ is defined in Eq. (15). Based on this total number, the individual deviation δ _i =|s _i −β _i ⋅s| from the target number of swings is captured by inequalities (17) and (18). The feasibility of a d ₁-distance α is then finally checked by introducing constraint (19). Namely, a simple game $v\in\mathcal{S}$ whose PBI has d ₁-distance of α or less exists if and only if the feasible set defined by (7)–(19) is non-empty.

The answer to whether this is the case—and, as a by-product, some $v\in\mathcal{S}$ with distance at most α—can be obtained by feeding (7)–(19) into a standard ILP software package in the required format. We have used IBM ILOG CPLEX 12.4 and the hardware described in Sect. 5.

Appendix B: Analytical PBI calculations

This appendix presents some technical details on the PBI computations for the sequence of weighted voting games {v ⁿ} _n∈ℕ with

$$v^n=\bigl(q^n;w^n\bigr)=(2n-a-4; \underbrace{3,\ldots, 3}_{a \,\mathrm{threes}}, \underbrace{2,\ldots, 2}_{n-a-1 \,\mathrm{twos}},1), $$

which is considered in the final paragraphs of Sect. 5.3. Our first lemma determines the number of swings in v ⁿ for each voter i=1,…,n, that is, the cardinality of set {S⊆N∖{i}:v ⁿ(S∪{i})−v ⁿ(S)=1}, as a function of a and n.

Lemma 1

The numbers of swings in v ⁿ are 2n−a−2, 2n−a−4, and a for all voters with weight 3, 2, and 1, respectively.

Proof

It is convenient to exploit the fact that for any $v\in \mathcal{S}$ the number of voter i’s swings in v and in the dual game $v'\in\mathcal{S}$ which is obtained by setting v′(S)=1−v(S) for all S⊆N must coincide. So instead of v ⁿ consider the game v ⁿ′ which involves identical weights but quota q′=4 instead of 2n−a−4. Referring to winning and losing coalitions in v ⁿ′ we have:

• (i) A voter i with w _i =3 renders a losing coalition S⊆N∖{i} winning by joining if either |S|=1 or S={j,k} with w _j =2 and w _k =1. There are n−1 coalitions of the former and n−a−1 coalitions of the latter type, amounting to 2n−a−2 swings altogether.

• (ii) A voter i with w _i =2⊆N∖{i} renders a losing coalition S winning by joining if either S={w _j } with w _j =3 or 2, or S={j,k} with w _j =2 and w _k =1. There are a+(n−a−2) coalitions of the former and n−a−2 coalitions of the latter type, amounting to 2n−a−4 swings altogether.

• (iii) Voter n with w _n =1 renders a losing coalition S⊆N∖{i} winning by joining if S={j} with w _j =3. There are a such coalitions.

 □

Writing ⌊x⌋ to denote the largest integer not greater than x, and x mod y to denote the integer remainder when x is divided by y, we have the following finding for distances in the d ₁-metric:

Lemma 2

Choose

and consider

$${v^n} = (2n-a-4;\underbrace{3,\ldots, 3}_{a(n) \,\mathrm{threes}}, \underbrace{2,\ldots, 2}_{n-a(n)-1 \,\mathrm{twos}},1). $$

Then

$$\lim_{n\to\infty} n\cdot \bigl(d_1\bigl(B\bigl({v^n} \bigr),\beta^n\bigr) \bigr)=\frac{1}{2}. $$

Proof

Suppose that n mod 7=0, i.e., n=7k+0 for some k∈ℕ. Then a=6k−1 and Lemma 1 yields swing numbers of 2⋅7k−(6k−1)−2, 2⋅7k−(6k−1)−4, and 6k−1 for the three voter types, respectively. This implies a total number of

$$(6k-1) (8k-1)+(n-6k) (8k-3)+(6k-1)=56k^2-11k $$

swings, and hence a PBI vector of

$$B\bigl(v^{7k}\bigr)=\frac{1}{56k^2-11k} (\underbrace{8k-1, \ldots, 8k-1}_{6k-1 \,\mathrm{times}}, \underbrace{8k-3, \ldots, 8k-3}_{k \, \mathrm{times}}, 6k-1 ). $$

This yields

and summing the absolute values of these figures up for the 6k−1 voters with weight 3, the k voters with weight 2, and the final voter n one obtains

$$\bigl\Vert B_i\bigl(v^n\bigr)-\beta^n \bigr\Vert_1= \frac{2(28k-3)}{(14k-1)k(56k-11)} $$

in case of n=7k. This number and results of the similarly tedious computations when n mod 7=1,…,6 are summarized in Table 11. For each of the seven cases one easily sees that the deviations tend to $\frac{1}{14k}$, which is equivalent to $\frac{1}{2n}$.

Table 11 d ₁-distances between β ⁿ and PBI of game $v^{n}\in \mathcal{W}$ in Lemma 2 (with k∈ℕ)

 □

In case of the d _∞-metric, choose

$$a(n)= \bigl\lfloor(n+1)/3 \bigr\rfloor+ \lfloor n/3 \rfloor-1. $$

In each corresponding game v ⁿ (see Lemma 2) roughly two thirds of the players have weight 3 each, roughly one third have weight 2, and a single player has weight 1. The games v ⁿ result in very good solutions of the inverse problem for n<8 and the best ones we could find for n≥8. Using this in order to obtain an upper bound one can verify the following result in perfect analogy to Lemma 2:

Lemma 3

The weighted voting game $v^{***}\in\mathcal{W}$ whose PBI minimizes d _∞-distance to β ⁿ satisfies

for n≥8.

Note that the indicated bound tends to $\frac{1}{n^{2}}$, i.e., $\lim_{n\to\infty} {b(n)} /{ \frac{1}{n^{2}}}=1$.