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On the role of network structure in learning to coordinate with bounded rationality

Yifei Zhang, Marcos M. Vasconcelos

TL;DR

This work examines learning to coordinate on networks when agents have bounded rationality, showing the networked coordination game is an exact potential game with a potential $\Phi(a)$. Using Log Linear Learning, it proves that increasing network connectivity raises the probability of converging to the optimal consensus, effectively leveraging the Wisdom of Crowds in the bounded-rationality regime. Moreover, among graphs with a fixed number of edges, regular graphs maximize learning success by promoting a more even distribution of potential values, guiding design toward homogeneous connectivity. The results have practical implications for designing networked systems (e.g., sensors, multi-robot teams) to improve coordinated decision-making under realistic, bounded rationality. Numerical experiments corroborate the theory, and future work includes Bayesian task difficulty, heterogeneity in rationality, and multi-task coordination.

Abstract

Many socioeconomic phenomena, such as technology adoption, collaborative problem-solving, and content engagement, involve a collection of agents coordinating to take a common action, aligning their decisions to maximize their individual goals. We consider a model for networked interactions where agents learn to coordinate their binary actions under a strict bound on their rationality. We first prove that our model is a potential game and that the optimal action profile is always to achieve perfect alignment at one of the two possible actions, regardless of the network structure. Using a stochastic learning algorithm known as Log Linear Learning, where agents have the same finite rationality parameter, we show that the probability of agents successfully agreeing on the correct decision is monotonically increasing in the number of network links. Therefore, more connectivity improves the accuracy of collective decision-making, as predicted by the phenomenon known as Wisdom of Crowds. Finally, we show that for a fixed number of links, a regular network maximizes the probability of success. We conclude that when using a network of irrational agents, promoting more homogeneous connectivity improves the accuracy of collective decision-making.

On the role of network structure in learning to coordinate with bounded rationality

TL;DR

This work examines learning to coordinate on networks when agents have bounded rationality, showing the networked coordination game is an exact potential game with a potential . Using Log Linear Learning, it proves that increasing network connectivity raises the probability of converging to the optimal consensus, effectively leveraging the Wisdom of Crowds in the bounded-rationality regime. Moreover, among graphs with a fixed number of edges, regular graphs maximize learning success by promoting a more even distribution of potential values, guiding design toward homogeneous connectivity. The results have practical implications for designing networked systems (e.g., sensors, multi-robot teams) to improve coordinated decision-making under realistic, bounded rationality. Numerical experiments corroborate the theory, and future work includes Bayesian task difficulty, heterogeneity in rationality, and multi-task coordination.

Abstract

Many socioeconomic phenomena, such as technology adoption, collaborative problem-solving, and content engagement, involve a collection of agents coordinating to take a common action, aligning their decisions to maximize their individual goals. We consider a model for networked interactions where agents learn to coordinate their binary actions under a strict bound on their rationality. We first prove that our model is a potential game and that the optimal action profile is always to achieve perfect alignment at one of the two possible actions, regardless of the network structure. Using a stochastic learning algorithm known as Log Linear Learning, where agents have the same finite rationality parameter, we show that the probability of agents successfully agreeing on the correct decision is monotonically increasing in the number of network links. Therefore, more connectivity improves the accuracy of collective decision-making, as predicted by the phenomenon known as Wisdom of Crowds. Finally, we show that for a fixed number of links, a regular network maximizes the probability of success. We conclude that when using a network of irrational agents, promoting more homogeneous connectivity improves the accuracy of collective decision-making.
Paper Structure (12 sections, 8 theorems, 69 equations, 3 figures)

This paper contains 12 sections, 8 theorems, 69 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Two-agent binary coordination games
  4. Coordination games over networks
  5. Potential Network Games
  6. Network Coordination Games
  7. Learning to Coordinate over a Network
  8. Log Linear Learning
  9. Inductive improvement by increasing connectivity
  10. Optimality of regular graphs
  11. Numerical Results
  12. Conclusions and future work

Key Result

Proposition 1

Consider the bimatrix game in bimatrix, and $\mathcal{S}$ denote its set of pure-strategy Nash equilibria. The following holds:

Figures (3)

  • Figure 1: A coordination game with parameter $\theta$ between two players.
  • Figure 2: Two graphs with equal number of nodes and edges.
  • Figure 3: Most papers should have a block diagram that summarizes the problem we are addressing.

Theorems & Definitions (18)

  • Remark 1: Payoff interpretation
  • Proposition 1
  • proof
  • Definition 1: Exact potential games
  • Lemma 1: Monderer and Shapley
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 8 more