Clearly, briberies of votes of the first type take priority over those of the second type, as p
′**can gain two**
points relative to p**. Voters of the third type should not be bribed at all. We check whether** p
′**can potentially**
beat p by first bribing voters of type one, and then those of type two, up to a total of our quota q**. We** always bribe voters to approve of p
′ **but not** p.

We find such a bribery for each other candidate p
′ 6= p by bribing q **voters using the procedure above. If**
there exists no candidate p
′ 6= p that can be made to beat p**, we reject.**
Intuitively, the reason such briberies are easy is that we only need to ensure that p
′ beats p**. It does not**
matter if other candidates will beat both p
′ and p as a result of the bribery, as p **will lose. This principle is**
also demonstrated in [9] for the case of Borda elections and [21] for the case of scoring protocols in general.

Theorem 5 Unweighted f(m)**-approval elections are computationally vulnerable to destructive manipulation**
and destructive control adding or deleting voters.

Proof: **The concept of attempting to make each candidate** p
′ 6= p beat p **can be applied to manipulation,**
as well as control by adding and deleting voters. In manipulation, each manipulator approves of p
′ **but not**
of p**. Similarly, in control by adding voters, we only add voters that approve of** p
′ but not of p**. In control**
by deleting voters, we only delete voters that approve of p **but not of** p
′.

We conclude that all unweighted approval-based scoring protocols, as well as families of scoring protocols, f(m**)-approval, are computationally vulnerable to destructive bribery, manipulation, and control by adding**
or deleting voters. The case of destructive control by adding or deleting candidates will be considered in Section 12. The remainder of this section will thus focus on constructive problems.

## 9 Manipulation Of Approval-Based Scoring Protocols

It is shown in [55] that manipulation in the f(m)-approval families of scoring protocols (recall that f(m**) is** computable in polynomial time with respect to the quantity of m **in this case) is polynomial-time computable,**
by a greedy algorithm. The greedy algorithm operates by iteratively **assigning the manipulators' preferences**
as follows: each manipulator approves p, the distinguished candidate and the current f(m)−**1 candidates with**
the lowest scores. It can be shown that this algorithm is correct by **induction on the number of manipulators.** Such elections are thus computationally vulnerable to insincere voting from a coalition of agents for their collective benefit, whether their goal is to elect a preferred candidate (constructive manipulation) or to prevent the election of a despised candidate (destructive manipulation).

## 10 Bribery In Approval-Based Scoring Protocols

In unweighted and unpriced families of scoring protocols, the goal of the bribery problem is to determine the minimum number of voters one needs to change the preferences for (i.e., bribe) to achieve a desirable outcome. As we have seen earlier, if our goal is to exclude a candidate from winning, as in destructive bribery, this problem is easy and can be computed in polynomial time for all approval-based elections. We thus now focus on constructive bribery cases, where we aim to elect a distinguished candidate.