On Binary Networked Public Goods Game with Altruism
Arnab Maiti, Palash Dey
TL;DR
The paper investigates Binary Networked Public Goods (BNPG) games with altruism, formalizing a model on a base graph $\mathcal{G}$ and an altruistic network $\mathcal{H}$ that can be either directed or undirected. It provides a fine-grained complexity analysis of Pure Strategy Nash Equilibrium (PSNE) existence, showing polynomial-time solvability for asymmetric altruism on trees, complete graphs, and graphs with bounded circuit rank, while establishing $NP$-hardness and $PPAD$-hardness results for related problems and equilibria. It also studies Altruistic Network Modification (ANM), proving NP-hardness for both altruism types on trees, presenting an $FPT$ result for the asymmetric case with respect to maximum degree, and showing para-$\mathsf{NP}$-hardness for the symmetric case; these results are underpinned by reductions from Knapsack and $(3,\text{B}2)$-SAT. Overall, the work reveals that predicting BNPG behavior is substantially easier than strategically modifying the altruistic network to enforce a chosen outcome, with implications for planning interventions in social networks and public goods scenarios.
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.