On Binary Networked Public Goods Game with Altruism

TL;DR

This paper investigates the computational complexity of finding equilibria and modifying strategies in altruistic binary networked public goods games, revealing tractability in specific network structures and highlighting the difficulty of strategic influence.

Contribution

It provides complexity results for equilibrium existence and network modification problems in BNPG games with altruism, including efficient algorithms for trees and complete graphs.

Findings

PSNE existence is NP-Complete but solvable in polynomial time for trees and complete graphs.

Network modification problems are NP-Complete, even for trees, but become easier with directed altruism networks.

Predicting player behavior is computationally easier than designing strategies to influence it.

Abstract

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.