In our election, the candidate set will consists of

C = {c, c′} ∪ S
∪ {x
$$\begin{array}{l}{{S}}\\ {{\{x_{i,j}^{1}\mid1\leq i\leq2(m-q)+2n(q+1)+4,1\leq j\leq k\}}}\\ {{\{x_{i,j}^{2}\mid1\leq i\leq2n(q+1)+5,1\leq j\leq k\}}}\\ {{\{x_{i,j,\ell}^{3}\mid1\leq i\leq n,1\leq j\leq2(q+1),1\leq\ell\leq k\}}}\\ {{\{x_{i,j,\ell}^{4}\mid1\leq i\leq m,1\leq j\leq2,1\leq\ell\leq k\}.}}\end{array}$$
∪ {x
∪ {x
∪ {x
For each 1 ≤ i ≤ 2(m − q) + 2n(q **+ 1) + 4 we add a voter with preferences**
· · · ≻ c ≻ x 1 i,1 **≻ · · · ≻** x 1 i,k−1 ≻ c
′
and for each 1 ≤ i ≤ 2n(q **+ 1) + 5 we add a voter with preferences**
· · · ≻ c
′ ≻ x 2 i,1 **≻ · · · ≻** x 2 i,k−1 ≻ c.

For each 1 ≤ i ≤ n and 1 ≤ j ≤ 2(q **+ 1), we add one voter with preferences**

· · · ≻ c ′ ≻ x 3 i,j,1 ≻ · · · ≻ x 3 i,j,k−1 ≻ Ti.
For each 1 ≤ i ≤ m and 1 ≤ j ≤ **2, we add one voter with preferences**
· · · ≻ c ≻ x 4 i,j,1 **≻ · · · ≻** x 4 i,j,k−1 ≻ bi.

We have thus, in essence, interchanged the role of approving c **with disapproving** c
′ **and vice versa, as they**
have the same effect in relation to the candidates. We must now, as in **the above case, add some voters to** prevent our reduction from cheating by deleting buffer candidates.

For each 1 ≤ i ≤ 2(m − q) + 2n(q **+ 1) + 4, we add one voter with preferences**
· · · ≻ c
′ ≻ x 1 i,1 **≻ · · ·** x 1 i,k and for 1 ≤ j ≤ **2, one voter with preferences**

· · · ≻ c
′ ≻ x
$$x_{i,j,1}^{4}\succ\cdot\cdot\succ$$
4
i,j,k.

Using the same argument as above, we see that hitting sets in S **correspond to valid destructive controls**
by either adding or deleting candidates.

## 12.2 Constructive Control

We examine the reduction given in [16] for constructive control by adding or deleting candidates in plurality elections and generalize it to cases of k-approval and k**-veto elections.**
In [16], the construction involved an election with established candidates c, c
′, and d**, and unestablished**
candidates corresponding to the elements of S. The idea was that by adding new candidates helps c **gain**