Since each bribery can be described with 2m! variables, and each variable is bounded 0 ≤ bi, di **≤ ||**V ||, this algorithm tries at most (||V || + 1)2m! **possible briberies. This quantity is polynomial with respect to the**
size of the problem instance. Thus, α**-$bribery is polynomial-time computable.**
Also owing to the constant number of possible preference orderings, unweighted scoring protocols of a fixed number of candidates are also computationally vulnerable to manipulation as well as control by adding or deleting voters.

Theorem 3 All scoring protocols α = (α1, . . . , αm) **are computationally vulnerable to constructive control**
by either adding or deleting voters, as well as to manipulation for a fixed number of candidates m.

It is important to note that despite the theoretical polynomial-time **computability in this case, in practice,**
this problem is significantly difficult for moderate-sized m, due to the exponential nature of m**!. It is often**
the case that the brute-force algorithm, despite being faster in the worst case, is not the best approach to this problem empirically.

In [11, 21] it is shown that most nontrivial manipulation and bribery problems in weighted scoring protocol elections are NP-hard even for a fixed number of candidates. We will **discuss these cases in Section 13.**
It should be noted that control by adding or deleting candidates in elections of a constant number of candidates (participating and additional candidates) is easy, as there are a constant number of different sets of candidates to add or delete, and whether or not a distinguished candidate will win or lose the election can be checked in polynomial time.

## 8 Destructive Misuses

To demonstrate the vast differences between the problems of constructive misuse and that of destructive misuse, we prove that unweighted approval-based elections are computationally vulnerable to most forms of destructive misuse. This is true even for families of scoring protocols, in which the size of the candidate set is unbounded. Consider the problem of bribery below.

Theorem 4 Unweighted f(m)**-approval elections are computationally vulnerable to destructive bribery.** Proof: As a reminder, we assume that f(m**) is computable in time polynomial with respect to** m.

In a destructive bribery, we want some candidate p
′ 6= p to beat p**. Thus, we want to have** p
′ **gain as**
many points as possible relative to p.

For each p
′ 6= p**, we separate the voters into 3 categories:**
1. Those approving of p **but not of** p
′.

2. Those approving either both of p and p
′ or neither of p nor p
′.

3. Those approving of p
′ **but not of** p.