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Equilibrium Analysis of Customer Attraction Games

Xiaotie Deng, Hangxin Gan, Ningyuan Li, Weian Li, Qi Qi

TL;DR

A game model called "customer attraction game" is introduced to demonstrate the competition among online content providers and it is proved the existence of an approximate PNE.

Abstract

We introduce a game model called "customer attraction game" to demonstrate the competition among online content providers. In this model, customers exhibit interest in various topics. Each content provider selects one topic and benefits from the attracted customers. We investigate both symmetric and asymmetric settings involving agents and customers. In the symmetric setting, the existence of pure Nash equilibrium (PNE) is guaranteed, but finding a PNE is PLS-complete. To address this, we propose a fully polynomial time approximation scheme to identify an approximate PNE. Moreover, the tight Price of Anarchy (PoA) is established. In the asymmetric setting, we show the nonexistence of PNE in certain instances and establish that determining its existence is NP-hard. Nevertheless, we prove the existence of an approximate PNE. Additionally, when agents select topics sequentially, we demonstrate that finding a subgame-perfect equilibrium is PSPACE-hard. Furthermore, we present the sequential PoA for the two-agent setting.

Equilibrium Analysis of Customer Attraction Games

TL;DR

A game model called "customer attraction game" is introduced to demonstrate the competition among online content providers and it is proved the existence of an approximate PNE.

Abstract

We introduce a game model called "customer attraction game" to demonstrate the competition among online content providers. In this model, customers exhibit interest in various topics. Each content provider selects one topic and benefits from the attracted customers. We investigate both symmetric and asymmetric settings involving agents and customers. In the symmetric setting, the existence of pure Nash equilibrium (PNE) is guaranteed, but finding a PNE is PLS-complete. To address this, we propose a fully polynomial time approximation scheme to identify an approximate PNE. Moreover, the tight Price of Anarchy (PoA) is established. In the asymmetric setting, we show the nonexistence of PNE in certain instances and establish that determining its existence is NP-hard. Nevertheless, we prove the existence of an approximate PNE. Additionally, when agents select topics sequentially, we demonstrate that finding a subgame-perfect equilibrium is PSPACE-hard. Furthermore, we present the sequential PoA for the two-agent setting.
Paper Structure (7 sections, 12 theorems, 37 equations, 2 figures)

This paper contains 7 sections, 12 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Related Work
  4. Roadmap
  5. Model and Preliminaries
  6. Warm-up: The Symmetric Static Game
  7. The Asymmetric Static Game

Key Result

theorem thmcountertheorem

For any instance $\mathcal{I}$ of symmetric customer attraction game, $\mathcal{I}$ is an exact potential game with respect to the potential function $\Phi(\hbox{\boldmath$S$})$ and the pure Nash equilibrium always exists.

Figures (2)

  • Figure 1: An example of customer attraction game. Each agent is linked to all her available topics. A line from a topic to a customer means this customer is in the attraction range of this topic.
  • Figure 2: An illustration of the construction of $GAD_e$. The nodes are illustrated as blue circles. The grey rectangles represent the dummy agents attrcting each node, with each grey label $/d_k^{e+}$ and $/d_k^{e-}$ representing that the number of dummy agents on the node is $d_k^{e+}$ or $d_k^{e-}$ respectively. The two strategies of $i_1$ are represented as green boxes, and two strategies of $i_2$ are represented as red boxes.

Theorems & Definitions (26)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • proof
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 16 more