# agreement among decentralized decision makers

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We study the convergence of opinions in a multi-agent system under various assumptionson interaction dynamics. We begin by studying the opinion dynamics of a network withconstant, doubly stochastic transition kernels. A general result is shown which providesconditions of agreement. We extend this model by introducing a finite number of stubbornagents. Convergence is shown to a convex combination of the initial opinions of the stubbornagents. A user-prescribed state dependent update algorithm is introduced to model systemswith limited communication capabilities. We then move to study the convergence of opinionsin a random model that allows for stochastic agreement. Motivated by a cost reductionphenomenon, a transition kernel is designed to force agents to interact more when theiropinions are widely dispersed and interact less when their opinions are tightly spread. Thecomplexity increases with each model to reflect more realistic behaviour. We obtain manyconvergence results from these models, of which the implications are heavily explored.TRANSCRIPT

Mathematics and Engineering 4th Year Project

Agreement Among Decentralized Decision Makers

Ryan Shrott 10016608

Ariel Hubert 10018729

Victor Li 10025708

Payton Karch 10002247

Professor Serdar Yuksel

Thursday 16th April, 2015

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1 Acknowledgments

We would like to extend our sincerest gratitude to Professor Serdar Yuksel for his unwaveringsupport throughout the last eight months. His guidance led us to a better understandingof the background knowledge required to study interaction dynamics. Professor Yukselprovided us with a lot of support and direction and kept us on track throughout the entireprocess. We would also like to thank our classmates of Apple Math 15 for their support.Finally, we would like to thank the rest of the professors in the engineering faculty at Queensfor giving us the tools we needed to obtain our results.

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2 Abstract

We study the convergence of opinions in a multi-agent system under various assumptionson interaction dynamics. We begin by studying the opinion dynamics of a network withconstant, doubly stochastic transition kernels. A general result is shown which providesconditions of agreement. We extend this model by introducing a finite number of stubbornagents. Convergence is shown to a convex combination of the initial opinions of the stubbornagents. A user-prescribed state dependent update algorithm is introduced to model systemswith limited communication capabilities. We then move to study the convergence of opinionsin a random model that allows for stochastic agreement. Motivated by a cost reductionphenomenon, a transition kernel is designed to force agents to interact more when theiropinions are widely dispersed and interact less when their opinions are tightly spread. Thecomplexity increases with each model to reflect more realistic behaviour. We obtain manyconvergence results from these models, of which the implications are heavily explored.

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Contents

1 Acknowledgments 2

2 Abstract 3

3 Introduction 53.1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Opinion Dynamics & Social Learning . . . . . . . . . . . . . . . . . . . . . . . 63.4 Engineering Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4.1 Swarm Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4.2 Problem: Landmine Removal . . . . . . . . . . . . . . . . . . . . . . . 83.4.3 Problem: Forest Fires . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4.4 Problem: Environmental Concerns . . . . . . . . . . . . . . . . . . . . 8

3.5 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5.1 Problem: Load Balancing . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Problem Description & Approach 10

5 Design & Engineering Models 125.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4 Stubborn Agent Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.5 Communication Radius Model . . . . . . . . . . . . . . . . . . . . . . . . . . 185.6 Random Realization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Testing & Results 24

7 Discussion 277.1 Application of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.1.1 Landmine Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.1.2 Forest Fires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.1.3 Environmental Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . 297.1.4 Load Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8 Conclusion 318.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 References 32

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3 Introduction

3.1 General Overview

Decentralized agreement problems can be studied by examining the convergence of opinions.In a general setting, the dynamics of a social network can be described by equations of theform:

xt+1 = xt P (xt) + wtSystems of this form have been extensively covered in [3] by Condello. This paper will extendthese results by studying a complex update algorithm which has applications in engineeringand sociology. Most notably, the modelling has significant energy efficiency applications innumerous interaction systems. Leading up to this model, three other systems will also bestudied. The basic model will be introduced to show agreement to the average initial opinionunder certain conditions. The role and implications of stubborn agents will be studied. Theexistence of clusters will be demonstrated by studying a state dependent update algorithmwhich models interaction dynamics through a prescribed interaction radius.

The thesis is formatted as follows:

In Chapter 4, the problem description will be formulated and the engineering ap-proaches used to solve the problem will be discussed.

In Chapter 5, some notation and necessary definitions needed in order to understandthe modelling will be introduced. The four aforementioned models will also be exten-sively studied.

In Chapter 6, the simulated results for each of the four models will be shown. In Chapter 7, the results will be discussed along with their relation to many engineering

and sociological applications.

3.2 Historical Background

The study of consensus, agreement and social learning has been present in mathematicaland philosophical literature over recent centuries. In 1788, Marquis de Condorcet provedthat truthful reporting of information by a large group of individuals with each holdingan opinion (correlated to some state ) is sufficient to the aggregation of information [1].In 1907, Galton advocated the idea that a large group of relatively uninformed individualswould have significantly more information than each individual separately. To defend histheory, he went to a carnival where people were guessing an animals mass. He found thatalthough most individuals were completely wrong, the median guess was 1197 pounds (theanimal weighed 1198 pounds). Although Galtons claim is not true in general, it can beviewed as a foundational constituent in the advancement of social dynamics [2].

More recently, the mathematics developed by J. Hajnal and M. S. Bartlett in the studyof Weak Ergodicity of non-homogenous Markov chains and by J. Wolfowitz in Productsof Indecompasable, Aperiodic, Stochastic Matrices has been foundational to the study ofmany social learning questions [3]. Will social learning lead to a consensus? Will social

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learning effectively aggregate dispersed information? How can one optimize his/her deci-sions to most effectively spread information? In areas such as coordination and distributedcontrol, it is important to study the consensus processes that a group of agents employ.It is often desirable for a group of agents to reach some sort of agreement and have somecommon knowledge regarding the state of the system [4].

The focus of this project will involve a randomly changing interaction graph. When con-sidering an undirected graph, one may wish to compute the average (or some function) ofthe initial values of the nodes. For example, the DeGroot model uses a weighted average ofthe opinions of the nodes, wherein the weights are the respective trust levels of the agents.Other averaging methods may also be employed. A simple method involves each node ex-changing information with its neighbours using a basic averaging algorithm. Furthermore,under specific connectivity assumptions, one can show that an agreement to the averagevalue will eventually occur.

3.3 Opinion Dynamics & Social Learning

The study of opinion dynamics is important because an individuals actions are controlledby his/her beliefs. But how do these beliefs actually form? We are certainly not bornwith opinions. Our beliefs are formed by our interactions with our social environment. Forexample, our opinions on politics are dependent on our individual values. We vote on thepolitician who we deem to have our own best interests in mind. We look for a politicianthat is in a sense fair. But what exactly is fairness? It is certainly not the same notion inall of us; otherwise everyone would vote for the same politician. In order to answer suchquestions, the spread of information within a community must be studied [5].

Social learning is the process of updating ones beliefs as a function of ones own experiences,others experiences, news from media sources, and propaganda from high-level sources. It isimportant to note that even though each individual is updating his/her beliefs as a functionof his/her own opinions, there is an inherent social character to the process. For example,a holistic view of a society may model the learning process like the spread of a virus. Whena group obtains a certain piece of information, they may immediately change their opinionbased on the new information. Another holistic model of social learning would be to exam-ine social circles instead of individuals. For example, one would need to study the mutualexclusivity and independence of such circles in order to determine the interaction propertiesand the associated probability distributions [5].

In order to investigate the conditions and effectiveness of social learning, a number of ques-tions must be investigated. What

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