0-1 Integer Linear Programming can also be used to model bribery, including cases involving prices and weights. In our first theorem, we assume that no symmetry is involved.
Theorem 5.2.1 Bribery and $bribery of the weighted scoring protocols (α1, . . . , αm) of m
candidates and n voters can be formulated as a 0-1 Integer Linear Programming instance consisting of Θ(m2n) variables and Θ(mn) constraints.
Proof: Consider an election of m candidates, C = {c1, . . . , cm} and n voters V = {v1, . . . , vn}.
We wish to ensure the victory of c1 by changing the preferences of voters of total cost at
the total budget of the briber. For each candidate c ∈ C, let s(c) denote its initial score. Further, for v ∈ V and c ∈ C, let 1 ≤ r(v, c) ≤ m denote the ranking position that v initially gives c.
For 1 ≤ i ≤ n, let the variable xi = 1 iff we are to bribe voter vi, and for 1 ≤ i ≤ n and
2 ≤ j, ` ≤ m, let yi,j,`= 1 iff we are to bribe voter vi to rank cj in the `th position. Without
loss of generality, we may assume that all bribed voters are bribed to rank c1 as their first
choice.
If vi is to be bribed, then we must give each candidate exactly one ranking position and
must fill each ranking position exactly once. This corresponds to the equalitiesX
j
yi,j,`= xi
for all i, `, and X
`
yi,j,`= xi for all i, j.
Because each bribed voter will rank c1 as their first preference, the final score of c1 is
given by s(c1) +
X
i
α1− αr(vi,c1) w(vi)xi. For j 6= 1, the final score of cj is given by
s(cj) − X i αr(vi,cj)w(vi)xi+ X i,` α`w(vi)yi,j,`.
The budget of the briber is enforced by the inequality X
i
π(vi)xi ≤ q.
In our next theorem, we show how symmetry may be broken if the scoring protocol vector contains similar weights, as in the case of k-approval elections. This affects how we represent the new preferences to be given to the bribed voters.
Theorem 5.2.2 Bribery and $bribery of the weighted k-approval elections of m candidates and n voters can be formulated as a 0-1 Integer Linear Programming instance consisting of Θ(mn) variables and Θ(m + n) constraints.
Proof: Consider an election of m candidates, C = {c1, . . . , cm} and n voters V = {v1, . . . , vn}.
We wish to ensure the victory of c1 by changing the preferences of voters of total cost at
most q. Let w(v) be the weight of voter v, and π(v) be the cost of bribing voter v. For each candidate c ∈ C, let s(c) denote its initial score. Additionally, for each v ∈ V and c ∈ C, we predefine app(v, c) to be 1 if v initially approves c, and 0 otherwise.
For 1 ≤ i ≤ n, let the variable xi = 1 iff we are to bribe voter vi, and for 1 ≤ i ≤ n and
2 ≤ j ≤ m, let the variable yi,j = 1 iff vi is bribed to approve candidate cj. Without loss of
generality, we may assume that all bribed voters are bribed to approve c1. By the budget
constraint, X
i
π(vi)xi ≤ q, and
X
j
yi,j = (k − 1)xi, as k − 1 additional approvals (besides
The final score of c1 is s(c1) +
X
i
w(vi) [1 − app(vi, c1)] xi, while the final score of cj for
2 ≤ j ≤ m is s(cj) − X i w(vi)app(vi, cj)xi+ X i w(vi)yi,j.
As in the case of manipulation, symmetry can also be broken when the voters are un- weighted. This affects how we represent not only the preferences to be given to the bribed voters, but the initial preferences of the voters we are bribing.
We break symmetry in the initial voters by grouping voters with similar preferences, and counting the number of preferences affected for each case. This is known as succinct preference profile representation, which we described in Section 2.7 (see also [FHH09]).
A bribery can then be represented by counting the number of voters of each distinct preference ordering we are to bribe. This representation is especially useful for problems in which there are many similar voters.
As an example, consider an unweighted Borda election of three candidates a, b, and c, in which 20 voters have a preference of a b c, 13 of b c a, and 9 of c a b. The briber wishes to ensure the victory of c by changing the preferences of at most 10 voters. Using the encoding in the previous theorem, the SAT solver would potentially consider all combinations of at most 10 voters among the set of 42 voters, to bribe. However, it suffices to know how many voters who have each of the three initial preferences are to be bribed, and the preferences to be given to the voters during the bribery. Because this election is unweighted, the same symmetry breaking principles used in the manipulation problem can be applied to the preferences to be given to the bribed voters. We give the general encoding as follows.
Theorem 5.2.3 Bribery and $bribery of the unweighted scoring protocols (α1, . . . , αm) of m
candidates and n voters can also be formulated as a 0-1 Integer Linear Programming instance consisting of Θ(m2n) variables and Θ(mn) constraints.
Proof: Consider an unweighted election of candidates C = {c1, . . . , cm} and voters V =
{v1, . . . , vn} under the scoring protocol (α1, . . . , αm). We wish to ensure the victory of c1
by changing the preferences of voters of total cost at most q. For each candidate c ∈ C, let s(c) be the initial score of c. Suppose there are s distinct preferences among the voters V , P1, . . . , Ps. For each preference P , let N (P ) ≥ 1 denote the number of voters initially having
preference profile P . For 0 ≤ j ≤ N (P ), let Π(P, j) denote the cost of bribing the j cheapest voters of preference P . For 1 ≤ i ≤ s and 0 ≤ j ≤ N (Pi), let variable xi,j = 1 iff we bribe
exactly j voters with initial preference Pi. For 0 ≤ i ≤ q and 2 ≤ j, ` ≤ m, let yi,j,` = 1 iff
Clearly, X
j
xi,j = 1 for all 1 ≤ i ≤ s. In total,
X
i,j
jxi,j voters are bribed, at a cost of
X
i,j
Π(Pi, j)xi,j, and this quantity must be at most q.
The constraints for the preferences to be given to the bribed voters are similar to the reduction for manipulation, and as follows:
∀j, `X i yi,j,` = 1 ∀jX i,` iyi,j,`= X i,j jxi,j ∀`X i,j iyi,j,` = X i,j jxi,j
The final score of c1 is given by s(c1) +
X
i,j
jα1− αr(Pi,c1) xi,j, while the final score of
cj for j 6= 1 is given by s(cj) − X i,j jαr(Pi,cj)xi,j+ X i,j,` iα`yi,j,`.
As in the case of unweighted bribery, we can further break symmetry in scoring protocols with some similar weights.
Consider an unweighted k-approval election of candidates C = {c1, . . . , cm} and voters
V = {v1, . . . , vn}. We wish to ensure the victory of c1 by changing the preferences of voters
of total cost at most q. For each candidate c ∈ C, let s(c) be the initial score of c.
Suppose there are s distinct preferences among the voters V , P1, . . . , Ps. We consider
preference similar if they approve the same set of k candidates. For each preference P , let N (P ) ≥ 1 denote the number of voters initially having preference profile P . For 0 ≤ j ≤ N (P ), let Π(P, j) denote the cost of bribing the j cheapest voters of preference P .
For 1 ≤ i ≤ s and 0 ≤ j ≤ N (Pi), let variable xi,j = 1 iff we bribe exactly j voters with
initial preference Pi. For 2 ≤ i ≤ m and 0 ≤ j ≤ q, we let yi,j = 1 iff the bribed voters give
exactly j approvals to candidate ci. Recall that without loss of generality, all bribed voters
will approve c1. The inequality
X
i,j
Π(Pi, j)xi,j ≤ q tells us that we bribe voters of total price
at most q. For 2 ≤ i ≤ m, candidate ci receives
X
j
jyi,j approvals. This quantity must be at
most the number of voters bribed, or X
i,j
jxi,j. The total number of approvals given, given
by the quantity X
i,j
The final score of candidate c1 can be given by s(c1) −
X
i,j
[1 − app(Pi, c1)] jxi,j while
that of c` for 2 ≤ ` ≤ m by s(c`) − X i,j app(Pi, c`)jxi,j + X j jyi,j.
In this reduction, briberies affecting the same number of voters of each initial preference are represented by the same instantiation of the variables. Both of these symmetry breaking principles can be combined in the case of bribery of k-approval elections.