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Nash Equilibria in Reverse Temporal Voronoi Games

Simeon Pawlowski, Vincent Froese

TL;DR

The paper investigates Nash equilibria in reverse temporal Voronoi games on temporal graphs, where players Vie to attract vertices that reach them earlier via the temporal distances td. It establishes that NE existence in rVor can differ markedly from the classic Voronoi game across graph classes, proving NE guarantees for temporally connected trees but not for many monotonically growing graphs, while showing NE existence for monotonically shrinking complete k-partite and threshold graphs. The study uncovers nuanced dynamics where reverse timing removes catch-up effects present in the classic game, enabling both positive NE results and explicit counterexamples. These findings deepen understanding of competitive influence in dynamic networks and suggest avenues for further exploration of growth/shrink patterns and multiagent variants.

Abstract

We study Voronoi games on temporal graphs as introduced by Boehmer et al. (IJCAI 2021) where two players each select a vertex in a temporal graph with the goal of reaching the other vertices earlier than the other player. In this work, we consider the reverse temporal Voronoi game, that is, a player wants to maximize the number of vertices reaching her earlier than the other player. Since temporal distances in temporal graphs are not symmetric in general, this yields a different game. We investigate the difference between the two games with respect to the existence of Nash equilibria in various temporal graph classes including temporal trees, cycles, grids, cliques and split graphs. Our extensive results show that the two games indeed behave quite differently depending on the considered temporal graph class.

Nash Equilibria in Reverse Temporal Voronoi Games

TL;DR

The paper investigates Nash equilibria in reverse temporal Voronoi games on temporal graphs, where players Vie to attract vertices that reach them earlier via the temporal distances td. It establishes that NE existence in rVor can differ markedly from the classic Voronoi game across graph classes, proving NE guarantees for temporally connected trees but not for many monotonically growing graphs, while showing NE existence for monotonically shrinking complete k-partite and threshold graphs. The study uncovers nuanced dynamics where reverse timing removes catch-up effects present in the classic game, enabling both positive NE results and explicit counterexamples. These findings deepen understanding of competitive influence in dynamic networks and suggest avenues for further exploration of growth/shrink patterns and multiagent variants.

Abstract

We study Voronoi games on temporal graphs as introduced by Boehmer et al. (IJCAI 2021) where two players each select a vertex in a temporal graph with the goal of reaching the other vertices earlier than the other player. In this work, we consider the reverse temporal Voronoi game, that is, a player wants to maximize the number of vertices reaching her earlier than the other player. Since temporal distances in temporal graphs are not symmetric in general, this yields a different game. We investigate the difference between the two games with respect to the existence of Nash equilibria in various temporal graph classes including temporal trees, cycles, grids, cliques and split graphs. Our extensive results show that the two games indeed behave quite differently depending on the considered temporal graph class.
Paper Structure (11 sections, 13 theorems, 6 equations, 6 figures, 1 table)

This paper contains 11 sections, 13 theorems, 6 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (Reverse) Temporal Voronoi Games.
  4. Reverse Temporal Voronoi Games ($\mathrm{rVor}$)
  5. Temporally Connected Graphs
  6. Monotonically Growing Graphs
  7. Monotonically Shrinking Graphs
  8. Temporal Voronoi Games ($\mathrm{Vor}$)
  9. Monotonically Growing Graphs
  10. Monotonically Shrinking Graphs
  11. Conclusion

Key Result

Theorem 1

On every temporally connected tree $\mathcal{T}$, there exists a Nash equilibrium in $\mathrm{rVor}(\mathcal{T},2)$.

Figures (6)

  • Figure 1: A monotonically growing temporal cycle without a Nash equilibrium.
  • Figure 2: A monotonically growing temporal grid without a Nash equilibrium.
  • Figure 3: A monotonically shrinking temporal path without a Nash equilibrium.
  • Figure 4: A monotonically shrinking temporal cycle without a Nash equilibrium.
  • Figure 5: A monotonically shrinking temporal split graph without a Nash equilibrium.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • ...and 14 more