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Tight Inapproximability of Nash Equilibria in Public Goods Games

Jérémi Do Dinh, Alexandros Hollender

TL;DR

The paper tackles the computation of equilibria in binary public goods games on directed graphs, showing that finding an $\varepsilon$-well-supported Nash equilibrium remains PPAD-hard for any constant $\varepsilon<\min\{p,1-p\}$ and any price $p\in(0,1)$. The authors achieve a tight hardness result via a direct reduction from the PPAD-complete Pure-Circuit problem, employing NOR and PURIFY gadgets to simulate circuit gates within the public goods framework. This bypasses prior reliance on threshold games and strengthens the hardness bound, clarifying the limits of efficient approximation for these games. The work also raises open questions about extending to $\varepsilon$-Nash equilibria, FIXP-completeness, gadget scalability, and special cases such as $p=1/2$ with bounded-degree graphs, guiding future complexity analyses of local public goods systems.

Abstract

We study public goods games, a type of game where every player has to decide whether or not to produce a good which is public, i.e., neighboring players can also benefit from it. Specifically, we consider a setting where the good is indivisible and where the neighborhood structure is represented by a directed graph, with the players being the nodes. Papadimitriou and Peng (2023) recently showed that in this setting computing mixed Nash equilibria is PPAD-hard, and that this remains the case even for $\varepsilon$-well-supported approximate equilibria for some sufficiently small constant $\varepsilon$. In this work, we strengthen this inapproximability result by showing that the problem remains PPAD-hard for any non-trivial approximation parameter $\varepsilon$.

Tight Inapproximability of Nash Equilibria in Public Goods Games

TL;DR

The paper tackles the computation of equilibria in binary public goods games on directed graphs, showing that finding an -well-supported Nash equilibrium remains PPAD-hard for any constant and any price . The authors achieve a tight hardness result via a direct reduction from the PPAD-complete Pure-Circuit problem, employing NOR and PURIFY gadgets to simulate circuit gates within the public goods framework. This bypasses prior reliance on threshold games and strengthens the hardness bound, clarifying the limits of efficient approximation for these games. The work also raises open questions about extending to -Nash equilibria, FIXP-completeness, gadget scalability, and special cases such as with bounded-degree graphs, guiding future complexity analyses of local public goods systems.

Abstract

We study public goods games, a type of game where every player has to decide whether or not to produce a good which is public, i.e., neighboring players can also benefit from it. Specifically, we consider a setting where the good is indivisible and where the neighborhood structure is represented by a directed graph, with the players being the nodes. Papadimitriou and Peng (2023) recently showed that in this setting computing mixed Nash equilibria is PPAD-hard, and that this remains the case even for -well-supported approximate equilibria for some sufficiently small constant . In this work, we strengthen this inapproximability result by showing that the problem remains PPAD-hard for any non-trivial approximation parameter .
Paper Structure (11 sections, 4 theorems, 13 equations, 1 figure)

This paper contains 11 sections, 4 theorems, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Public Goods Games
  4. Case: $k = 1$.
  5. Case: $k = 0$.
  6. Observation.
  7. Pure-Circuit
  8. Reduction from Pure-Circuit
  9. NOR gates.
  10. PURIFY gates.
  11. Open Problems

Key Result

Lemma 2.1

Given a mixed strategy profile $\vb{s}\xspace \in [0,1]^n$, the expected payoff of player $i$ is $\tilde{u}_i(\vb{s}\xspace) = s_i \cdot \tilde{u}_i(1, \vb{s}_{-i}\xspace) + (1-s_i) \cdot \tilde{u}_i(0, \vb{s}_{-i}\xspace)$, where for $k \in \{0,1\}$:

Figures (1)

  • Figure 1: The gadgets used for the public goods instance in \ref{['thm:pgg-elementary']}.

Theorems & Definitions (14)

  • Definition 1: Game
  • Definition 2
  • Definition 3: Mixed strategy
  • Lemma 2.1
  • proof
  • Definition 4: $\varepsilon\xspace$-best-response
  • Definition 5: Nash equilibria and approximations
  • Definition 6: Utility difference
  • Lemma 2.2
  • proof
  • ...and 4 more