  Weighted voting games is one of the most important classes of cooperative
games. Recently, Zhang and Grossi [53] proposed a variant of this class, called
delegative simple games, which is well suited to analyse the relative
importance of each voter in liquid democracy elections. Moreover, they defined
a power index, called the delagative Banzhaf index to compute the importance of
each agent (i.e., both voters and delegators) in a delegation graph based on
two key parameters: the total voting weight she has accumulated and the
structure of supports she receives from her delegators.
  We obtain several results related to delegative simple games. We first
propose a pseudo-polynomial time algorithm to compute the delegative Banzhaf
and Shapley-Shubik values in delegative simple games. We then investigate a
bribery problem where the goal is to maximize/minimize the voting power/weight
of a given voter in a delegation graph by changing at most a fixed number of
delegations. We show that the problems of minimizing/maximizing a voter's power
index value are strongly NP-hard. Furthermore, we prove that having a better
approximation guarantee than $1-1/e$ to maximize the voting weight of a voter
is not possible, unless $P = NP$, then we provide some parameterized complexity
results for this problem. Finally, we show that finding a delegation graph with
a given number of gurus that maximizes the minimum power index value an agent
can have is a computationally hard problem.
