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.

翻译：在经典的二元网络公共品（BNPG）博弈中，玩家可以选择投资于公共项目或不投资。根据所有玩家的决策，每位玩家根据其效用函数获得奖励。然而，经典的BNPG博弈模型未考虑玩家经常表现出的利他性，而这种利他性可能显著影响均衡行为。Yu等人（2021）扩展了经典BNPG博弈，以捕捉玩家的利他性方面。在本文中，我们首先研究在具有利他性的BNPG博弈中判断纯策略纳什均衡（PSNE）是否存在的问题。该问题已知为NP-完全问题。我们通过展示当输入网络为树或完全图时，该问题存在高效算法，从而补充了这一困难结果。我们还进一步研究了利他性网络修改问题，其任务是通过添加或删除少量边来判断一个目标策略配置能否成为PSNE。该问题同样已知为NP-完全问题。我们通过展示即使在树上也存在难解性结果，强化了这一困难性结论。本研究中一个可能令人意外的发现是：当利他性网络为无向图时，上述问题即使在有界度图中仍为NP-难问题，但当利他性网络为有向图时，该问题变为多项式时间可解。我们还展示了关于混合策略纳什均衡（MSNE）计算的一些结果以及若干参数化复杂度结果。总而言之，我们的结果表明：预测BNPG博弈中玩家将如何行为，远比让玩家按期望方式行为要容易得多。