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The Battling Influencers Game: Nash Equilibria Structure of a Potential Game and Implications to Value Alignment

Young Wu, Yancheng Zhu, Jin-Yi Cai, Xiaojin Zhu

TL;DR

This work formalizes the Battling Influencers Game (BIG), a multi-player, simultaneous-move general-sum model where influencer actions combine at an affine receiver and each influencer aims to steer the receiver toward a personal target. The authors prove BIG is a convex-potential game, showing that pure Nash equilibria coincide with the minima of a convex potential, and that there is either a unique NE or an infinite continuum of NE; at NE, most influencers tend to extreme exaggeration. They extend BIG to alternative loss functions and finite action spaces, preserving a potential structure and revealing stronger equilibrium properties such as weakly dominant equilibria or exponentially many NE in finite settings. The paper also applies BIG to AI value alignment, modeling heterogeneous value providers as strategic data sources whose exaggerations can bias the global alignment outcome, and supports this with an empirical demonstration of exaggerated reporting converging to an NE. Overall, BIG offers a tractable framework to reason about strategic influence, equilibrium behavior, and incentives to misreport in value-alignment pipelines, with implications for mechanism design to promote truthful data provision.

Abstract

When multiple influencers attempt to compete for a receiver's attention, their influencing strategies must account for the presence of one another. We introduce the Battling Influencers Game (BIG), a multi-player simultaneous-move general-sum game, to provide a game-theoretic characterization of this social phenomenon. We prove that BIG is a potential game, that it has either one or an infinite number of pure Nash equilibria (NEs), and these pure NEs can be found by convex optimization. Interestingly, we also prove that at any pure NE, all (except at most one) influencers must exaggerate their actions to the maximum extent. In other words, it is rational for the influencers to be non-truthful and extreme because they anticipate other influencers to cancel out part of their influence. We discuss the implications of BIG to value alignment.

The Battling Influencers Game: Nash Equilibria Structure of a Potential Game and Implications to Value Alignment

TL;DR

This work formalizes the Battling Influencers Game (BIG), a multi-player, simultaneous-move general-sum model where influencer actions combine at an affine receiver and each influencer aims to steer the receiver toward a personal target. The authors prove BIG is a convex-potential game, showing that pure Nash equilibria coincide with the minima of a convex potential, and that there is either a unique NE or an infinite continuum of NE; at NE, most influencers tend to extreme exaggeration. They extend BIG to alternative loss functions and finite action spaces, preserving a potential structure and revealing stronger equilibrium properties such as weakly dominant equilibria or exponentially many NE in finite settings. The paper also applies BIG to AI value alignment, modeling heterogeneous value providers as strategic data sources whose exaggerations can bias the global alignment outcome, and supports this with an empirical demonstration of exaggerated reporting converging to an NE. Overall, BIG offers a tractable framework to reason about strategic influence, equilibrium behavior, and incentives to misreport in value-alignment pipelines, with implications for mechanism design to promote truthful data provision.

Abstract

When multiple influencers attempt to compete for a receiver's attention, their influencing strategies must account for the presence of one another. We introduce the Battling Influencers Game (BIG), a multi-player simultaneous-move general-sum game, to provide a game-theoretic characterization of this social phenomenon. We prove that BIG is a potential game, that it has either one or an infinite number of pure Nash equilibria (NEs), and these pure NEs can be found by convex optimization. Interestingly, we also prove that at any pure NE, all (except at most one) influencers must exaggerate their actions to the maximum extent. In other words, it is rational for the influencers to be non-truthful and extreme because they anticipate other influencers to cancel out part of their influence. We discuss the implications of BIG to value alignment.

Paper Structure

This paper contains 11 sections, 6 theorems, 28 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Definition
  4. Pure NEs of BIG and Their Properties
  5. Extensions of BIG
  6. An Alternative Player Loss Function
  7. Finite Action Space
  8. Implications of BIG to Value Alignment
  9. Heterogeneous Value Alignment as a Game
  10. A Value Alignment Experiment
  11. Conclusion and Future Work

Key Result

Theorem 1

The Battling Influencers Game $G$ is a potential game with the potential function

Figures (9)

  • Figure 1: (left) Both players maximally exaggerate their actions with pure NE $(x_1=0, x_2=6)$. (right) the potential function $\phi$
  • Figure 2: Pure NE $(x_1=0, x_2=4)$. Influencer 2 not at boundary.
  • Figure 3: Examples of infinite (left) and unique (right) pure Nash equilibria in $d=2$
  • Figure 4: Examples of unique wDSE (left) and infinite number of wDSEs (right) in $d = 2$
  • Figure 5: Pairwise preference labels $z$ when both players are truthful. Left: player 1 with $x_1=t_1$, right: player 2 with $x_2=t_2$.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Definition 1: Battling Influencer Game (BIG)
  • Definition 2
  • Theorem 1: Potential Game
  • proof
  • Proposition 2: Pure NEs $\iff$ minima
  • proof
  • Corollary 3: Cardinality of $\mathrm{pNE}(G)$
  • proof
  • Example 1: Two influencers with 1D actions
  • Example 2: 2D actions
  • ...and 12 more