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Likes, Budgets, and Equilibria: Designing Contests for Socially Optimal Advertising

Sayantika Mandal, Harman Agrawal, Swaprava Nath

TL;DR

The paper addresses designing socially optimal advertising contests in social networks by modeling a two-timescale diffusion where firms allocate budgets across nodes to maximize awareness via a CSF-driven process. It establishes BRD convergence to a pure Nash equilibrium but shows that such equilibria can be welfare-inefficient, then proposes a CSF form that guarantees a unique symmetric NE that optimizes social welfare, achieving a PoA of 1 under suitable assumptions. Theoretical results are supported by experiments on synthetic networks, demonstrating stable convergence and welfare gains for concave CSFs, while non-concave CSFs can lead to suboptimal and unstable outcomes. Overall, the work provides mechanism-design guidance for regulators and platforms to align competitive advertising with social welfare in networked environments.

Abstract

Firms (businesses, service providers, entertainment organizations, political parties, etc.) advertise on social networks to draw people's attention and improve their awareness of the brands of the firms. In all such cases, the competitive nature of their engagements gives rise to a game where the firms need to decide how to distribute their budget over the agents on a network to maximize their brand's awareness. The firms (players) therefore need to optimize how much budget they should put on the vertices of the network so that the spread improves via direct (via advertisements or free promotional offers) and indirect marketing (words-of-mouth). We propose a two-timescale model of decisions where the communication between the vertices happen in a faster timescale and the strategy update of the firms happen in a slower timescale. We show that under fairly standard conditions, the best response dynamics of the firms converge to a pure strategy Nash equilibrium. However, such equilibria can be away from a socially optimal one. We provide a characterization of the contest success functions and provide examples for the designers of such contests (e.g., regulators, social network providers, etc.) such that the Nash equilibrium becomes unique and social welfare maximizing. Our experiments show that for realistic scenarios, such contest success functions perform fairly well.

Likes, Budgets, and Equilibria: Designing Contests for Socially Optimal Advertising

TL;DR

The paper addresses designing socially optimal advertising contests in social networks by modeling a two-timescale diffusion where firms allocate budgets across nodes to maximize awareness via a CSF-driven process. It establishes BRD convergence to a pure Nash equilibrium but shows that such equilibria can be welfare-inefficient, then proposes a CSF form that guarantees a unique symmetric NE that optimizes social welfare, achieving a PoA of 1 under suitable assumptions. Theoretical results are supported by experiments on synthetic networks, demonstrating stable convergence and welfare gains for concave CSFs, while non-concave CSFs can lead to suboptimal and unstable outcomes. Overall, the work provides mechanism-design guidance for regulators and platforms to align competitive advertising with social welfare in networked environments.

Abstract

Firms (businesses, service providers, entertainment organizations, political parties, etc.) advertise on social networks to draw people's attention and improve their awareness of the brands of the firms. In all such cases, the competitive nature of their engagements gives rise to a game where the firms need to decide how to distribute their budget over the agents on a network to maximize their brand's awareness. The firms (players) therefore need to optimize how much budget they should put on the vertices of the network so that the spread improves via direct (via advertisements or free promotional offers) and indirect marketing (words-of-mouth). We propose a two-timescale model of decisions where the communication between the vertices happen in a faster timescale and the strategy update of the firms happen in a slower timescale. We show that under fairly standard conditions, the best response dynamics of the firms converge to a pure strategy Nash equilibrium. However, such equilibria can be away from a socially optimal one. We provide a characterization of the contest success functions and provide examples for the designers of such contests (e.g., regulators, social network providers, etc.) such that the Nash equilibrium becomes unique and social welfare maximizing. Our experiments show that for realistic scenarios, such contest success functions perform fairly well.

Paper Structure

This paper contains 36 sections, 4 theorems, 65 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. A Two-Timescale Communication Model
  6. Awareness levels of each firm
  7. Limiting Value of Awareness Vector
  8. Payoff of a firm
  9. Design Desiderata
  10. Existence of Nash Equilibrium
  11. Best Response Dynamics
  12. Convergence of Best Response Dynamics
  13. Welfare and Nash Equilibrium
  14. The Optimal Contest Success Function
  15. Experiments
  16. ...and 21 more sections

Key Result

Theorem 1

Under Assumption ass:utility, the best response dynamics (Algorithm alg:BRD) converges to a Nash Equilibrium $\mathbf{b}^*$. More precisely, for any given $\epsilon > 0$ and any initial point $(\mathbf{b}(0), \boldsymbol{\mu}(0))$ in Algorithm alg:BRD, the gradients for $K > \frac{2B^2}{\lambda^2} \

Figures (5)

  • Figure 1: The two-timescale concept
  • Figure 2: Convergence behavior of the welfare ratio $R(k)$ across all three types of $p$-mean welfare for (a) concave Tullock CSF ($q=1$), (b) non-concave Tullock CSF ($q=2$), (c) log CSF and (d) exp CSF.
  • Figure 3: Phase 1: Synthetic data generation pipeline.
  • Figure 4: Social network generated for 100 users (25 tiles).
  • Figure 5: Phase 2: Social network generation pipeline.

Theorems & Definitions (10)

  • Definition 1: Nash equilibrium
  • Definition 2: $p$-mean social welfare
  • Definition 3: Price of Anarchy
  • Theorem 1: Convergence to a Nash Equilibrium
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Proposition 1
  • proof