PPA-Game: Characterizing and Learning Competitive Dynamics Among Online Content Creators

Renzhe Xu; Haotian Wang; Xingxuan Zhang; Bo Li; Peng Cui

cs.GTMar 22, 2024

PPA-Game: Characterizing and Learning Competitive Dynamics Among Online Content Creators

Authors: Renzhe Xu, Haotian Wang, Xingxuan Zhang, Bo Li, Peng Cui

Organizations: MoE Key Laboratory of Interdisciplinary Research of Computation and Economics, Institute for Theoretical Computer Science, Shanghai University of Finance and Economics, Shanghai, China · College of Computer Science and Technology, National University of Defense Technology, Changsha, China · Department of Computer Science and Technology, Tsinghua University, Beijing, China · School of Economics and Management, Tsinghua University, Beijing, China

Abstract

In this paper, we present the Proportional Payoff Allocation Game (PPA-Game), which characterizes situations where agents compete for divisible resources. In the PPA-game, agents select from available resources, and their payoffs are proportionately determined based on heterogeneous weights attributed to them. Such dynamics simulate content creators on online recommender systems like YouTube and TikTok, who compete for finite consumer attention, with content exposure reliant on inherent and distinct quality. We first conduct a game-theoretical analysis of the PPA-Game. While the PPA-Game does not always guarantee the existence of a pure Nash equilibrium (PNE), we identify prevalent scenarios ensuring its existence. Simulated experiments further prove that the cases where PNE does not exist rarely happen. Beyond analyzing static payoffs, we further discuss the agents' online learning about resource payoffs by integrating a multi-player multi-armed bandit framework. We propose an online algorithm facilitating each agent's maximization of cumulative payoffs over $T$ rounds. Theoretically, we establish that the regret of any agent is bounded by $O(\log^{1 + η} T)$ for any $η> 0$ . Empirical results further validate the effectiveness of our online learning approach.

Figures & tables

Figure 1: Graphical overview of our algorithm: Utilizes hyper-parameters

c_{1}

c_{2}

c_{3}

, and

\eta

, with counter

s

progressing through natural numbers. Initially, the algorithm enters an Exploration Phase for

c_{1}K

rounds ( Section 4.3.1 ), followed by a Learning and Exploitation phase ( Section 4.3.2 ), where it alternates between Learning PNE and Exploitation Subphases based on the counter

s

. Specifically, in round

s

, the Learning PNE Subphase spans

c_{2}s^{\eta}

rounds, and the Exploitation Subphase spans

c_{3}2^{s}

rounds.

Figure 2: Curves showing the average regret and the number of rounds where players do not follow the most efficient PNE. Note that SelfishRobustMMAB could not be applied to scenarios when

N>K

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: An improvement cycle in a game with no PNE.

Strategy profile	PNE?	Least reward change by deviation if PNE	$\delta^{\prime}$ -Nash equilibrium if not PNE
$(1,1,1)$	✗		$0.518$ . Deviated strategy $(1,1,2)$
$(1,1,2)$	✗		$0.022$ . Deviated strategy $(1,2,2)$
$(1,2,1)$	✓	$0.052$ . Deviated strategy $(1,2,2)$
$(1,2,2)$	✗		$0.052$ . Deviated strategy $(1,2,1)$
$(2,1,1)$	✓	$0.133$ . Deviated strategy $(2,1,2)$
$(2,1,2)$	✗		$0.133$ . Deviated strategy $(2,1,1)$

Appendix

Table 1: A showcase that

\delta^{\text{NE}}>\delta^{\text{NoNE}}

. Set

N=3

and

K=2

. Assume the resources’ payoffs to be

[1,0.7]

. Suppose each player has the same weights on the resources (as per Condition (2) in Theorem 3.2 ), with weights

[1,0.8,0.4]

, respectively. We list all strategy profiles and find that

\delta^{\text{NE}}=\min\{0.052,0.133\}=0.052>0.022=\min\{0.518,0.022,0.052,0.133,0.356,0.873\}=\delta^{\text{NoNE}}

Strategy profile	PNE?	Least reward change by deviation if PNE	$\delta^{\prime}$ -Nash equilibrium if not PNE
$(1,1,1)$	✗		$0.389$ . Deviated strategy $(1,1,2)$
$(1,1,2)$	✓	$0.040$ . Deviated strategy $(1,2,2)$
$(1,2,1)$	✓	$0.068$ . Deviated strategy $(1,2,2)$
$(1,2,2)$	✗		$0.068$ . Deviated strategy $(1,2,1)$
$(2,1,1)$	✓	$0.126$ . Deviated strategy $(1,1,1)$
$(2,1,2)$	✗		$0.215$ . Deviated strategy $(2,1,1)$

Appendix

Table 2: A showcase that

\delta^{\text{NE}}<\delta^{\text{NoNE}}

. Set

N=3

and

K=2

. Assume the resources’ payoffs to be

[1,0.6]

. Suppose each player has the same weights on the resources (as per Condition (2) in Theorem 3.2 ), with weights

[1,2/3,4/9]

, respectively. We list all strategy profiles and find that

\delta^{\text{NE}}=\min\{0.040,0.068,0.126\}=0.040<0.068=\min\{0.389,0.068,0.215,0.360,0.874\}=\delta^{\text{NoNE}}

Game Theory

Game theory studies how interacting decision-makers choose strategies, form agreements, and shape outcomes under competition and cooperation.

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User interactions in online recommendation platforms create interdependencies among content creators: feedback on one creator's content influences the system's learning and, in turn, the exposure of other creators' contents. To analyze incentives in such settings, we model collaboration as a multi-agent stochastic linear bandit problem with a transferable utility (TU) cooperative game formulation, where a coalition's value equals the negative sum of its members' cumulative regrets. We show that, for identical (homogenous) agents with fixed action sets, the induced TU game is convex under mild algorithmic conditions, implying a non-empty core that contains the Shapley value and ensures both stability and fairness. For heterogeneous agents, the game still admits a non-empty core, though convexity and Shapley value core-membership are no longer guaranteed. To address this, we propose a simple regret-based payout rule that satisfies three out of the four Shapley axioms and also lies in the core. Experiments on MovieLens-100k dataset illustrate when the empirical payout aligns with -- and diverges from -- the Shapley fairness across different settings and algorithms.

Abstract

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