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Public Goods Games in Directed Networks with Constraints on Sharing

Argyrios Deligkas, Gregory Gutin, Mark Jones, Philip R. Neary, Anders Yeo

TL;DR

This work studies a discrete public goods game on directed networks where contributors can share the good with at most $k$ out-neighbors, creating a directed, capacity-constrained sharing model. It provides a comprehensive analysis of existence, computation, and efficiency of both pure and mixed Nash equilibria, establishing sharp complexity dichotomies that depend on $k$ and network structure, including NP-hardness results for $k\ge 2$ and a linear-time algorithm for $k=1$ in mixed equilibria. It also derives tight bounds on Price of Stability and Price of Anarchy, with PoA exactly $k+1$ in both pure and mixed settings under broad conditions, and identifies structural conditions that guarantee the existence of pure equilibria. The findings illuminate how asymmetry in sharing and directed network topology shape strategic public good provision, offering benchmarks of computational feasibility and welfare loss in constrained sharing environments.

Abstract

In a public goods game, every player chooses whether or not to buy a good that all neighboring players will have access to. We consider a setting in which the good is indivisible, neighboring players are out-neighbors in a directed graph, and there is a capacity constraint on their number, k, that can benefit from the good. This means that each player makes a two-pronged decision: decide whether or not to buy and, conditional on buying, choose which k out-neighbors to share access. We examine both pure and mixed Nash equilibria in the model from the perspective of existence, computation, and efficiency. We perform a comprehensive study for these three dimensions with respect to both sharing capacity (k) and the network structure (the underlying directed graph), and establish sharp complexity dichotomies for each.

Public Goods Games in Directed Networks with Constraints on Sharing

TL;DR

This work studies a discrete public goods game on directed networks where contributors can share the good with at most out-neighbors, creating a directed, capacity-constrained sharing model. It provides a comprehensive analysis of existence, computation, and efficiency of both pure and mixed Nash equilibria, establishing sharp complexity dichotomies that depend on and network structure, including NP-hardness results for and a linear-time algorithm for in mixed equilibria. It also derives tight bounds on Price of Stability and Price of Anarchy, with PoA exactly in both pure and mixed settings under broad conditions, and identifies structural conditions that guarantee the existence of pure equilibria. The findings illuminate how asymmetry in sharing and directed network topology shape strategic public good provision, offering benchmarks of computational feasibility and welfare loss in constrained sharing environments.

Abstract

In a public goods game, every player chooses whether or not to buy a good that all neighboring players will have access to. We consider a setting in which the good is indivisible, neighboring players are out-neighbors in a directed graph, and there is a capacity constraint on their number, k, that can benefit from the good. This means that each player makes a two-pronged decision: decide whether or not to buy and, conditional on buying, choose which k out-neighbors to share access. We examine both pure and mixed Nash equilibria in the model from the perspective of existence, computation, and efficiency. We perform a comprehensive study for these three dimensions with respect to both sharing capacity (k) and the network structure (the underlying directed graph), and establish sharp complexity dichotomies for each.

Paper Structure

This paper contains 16 sections, 11 theorems, 9 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Further Related Work
  4. Model and preliminaries
  5. Questions of interest
  6. The complexity of pure Nash equilibria
  7. Pure equilibria when $k =1$
  8. Pure equilibria when $k \ge 2$
  9. Sufficient conditions for pure Nash equilibria
  10. The complexity of mixed Nash equilibria
  11. Mixed equilibria when $k =1$
  12. Mixed equilibria when $k \ge 2$
  13. The efficiency of Nash equilibria
  14. Efficiency of pure strategy equilibria
  15. Efficiency of mixed strategy equilibria
  16. ...and 1 more sections

Key Result

Theorem 1

Let $D$ be a digraph with $\delta^+(D) \geq 1$. If $R$ is a spanning r-galaxy in $D$ then $\mathrm{Leaves}(R)$ is a $1$-pure-Nash set in $D$. Furthermore, if $S$ is a $1$-pure-Nash set in $D$ then there exists a spanning r-galaxy, $R$, in $D$, such that $\mathrm{Leaves}(R)=S$.

Figures (6)

  • Figure 1: No pure strategy equilibrium when $k=1$.
  • Figure 2: If $k=2$ then $\{x_1^1,x_2^1,x_2^2\}$ is a buyer set but not a pure-Nash set. In fact, the unique pure-Nash set is $\{x_1^1,x_3^1,x_3^2,x_3^3,x_3^4\}$.
  • Figure 3: The digraph $D$ constructed from $H$ in the proof of Theorem \ref{['thm:k>=2']} when $k=2$ and $r=2$. Note that $|Z|=(k-1)\cdot|E|=1 \cdot 3$ and $|X|=k\cdot(|V|-r)= 2\cdot(6-2)=8$.
  • Figure 4: The gadgets added in order to transform the digraph $D$ into $D^*$.
  • Figure 5: An illustration of $Z$, $X_1$, $X_2$, $Y_1$ and $Y_2$ in the proof of Theorem \ref{['priceOfStabilityI']}, where the digraph $H$ is shown above. Note that $S=Z \cup Y_1 \cup Y_2$.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2: DG12
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 13 more