  In the classical Binary Networked Public Goods (BNPG) game, a player can
either invest in a public project or decide not to invest. Based on the
decisions of all the players, each player receives a reward as per his/her
utility function. However, classical models of BNPG game do not consider
altruism which players often exhibit and can significantly affect equilibrium
behavior. Yu et al. (2021) extended the classical BNPG game to capture the
altruistic aspect of the players. We, in this paper, first study the problem of
deciding the existence of a Pure Strategy Nash Equilibrium (PSNE) in a BNPG
game with altruism. This problem is already known to be NP-Complete. We
complement this hardness result by showing that the problem admits efficient
algorithms when the input network is either a tree or a complete graph. We
further study the Altruistic Network Modification problem, where the task is to
compute if a target strategy profile can be made a PSNE by adding or deleting a
few edges. This problem is also known to be NP-Complete. We strengthen this
hardness result by exhibiting intractability results even for trees. A perhaps
surprising finding of our work is that the above problem remains NP-Hard even
for bounded degree graphs when the altruism network is undirected but becomes
polynomial-time solvable when the altruism network is directed. We also show
some results on computing an MSNE and some parameterized complexity results. In
summary, our results show that it is much easier to predict how the players in
a BNPG game will behave compared to how the players in a BNPG game can be made
to behave in a desirable way.
