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Mean-Field-Type
Games for Engineers
Mean-Field-Type
Games for Engineers
Julian Barreiro-Gomez
Hamidou Tembine
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks
does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of
MATLAB® software or related products does not constitute endorsement or sponsorship by The
MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
First edition published 2022

by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press

2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2022 Julian Barreiro-Gomez and Hamidou Tembine
CRC Press is an imprint of Taylor & Francis Group, LLC
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lisher cannot assume responsibility for the validity of all materials or the consequences of their use.
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Trademark notice: Product or corporate names may be trademarks or registered trademarks and are
used only for identification and explanation without intent to infringe.
ISBN: 9780367566128 (hbk)

ISBN: 9780367566135 (pbk)
ISBN: 9781003098607 (ebk)
ISBN: 9781032128047 (eBook+ Enhancements)
DOI: 10.1201/9781003098607
Typeset in CMR10 font

by KnowledgeWorks Global Ltd.
Access the Support Material: www.routeldge.com/9780367566128

To God.
To Mayerly, Margarita, Luis, and Juan
for their permanent support.
To Jacobo for being my biggest motivation.
Julián Barreiro Gómez
To Yandai, Pama, Marie-Claire,

Jean-Pierre and Florence
for their unconditional support
Tembine Hamidou Doumbodo
Contents
List of Figures xiii
List of Tables xxiii
Foreword xxv
Preface xxvii
Acknowledgments xxix
Author Biographies xxxi
Symbols xxxiii
I Preliminaries 1
1 Introduction 3
1.1 Linear-Quadratic Games . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Structure of the Optimal Strategies and Optimal Costs 4
1.1.2 Solvability of the Linear-Quadratic Gaussian Games . 5
1.1.3 Beyond Brownian Motion . . . . . . . . . . . . . . . . 5
1.2 Linear-Quadratic Gaussian Mean-Field-Type Game . . . . . 6
1.2.1 Variance-Awareness and Higher Order Mean-Field
Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 The Role of the Risk in Engineering Applications . . . 8
1.2.3 Uncertainties in Engineering Applications . . . . . . . 10
1.2.4 Network of Networks/System of Systems . . . . . . . . 14
1.2.5 Optimality Systems . . . . . . . . . . . . . . . . . . . 16
1.3 Game Theoretical Solution Concepts . . . . . . . . . . . . . 19
1.3.1 Non-cooperative Game Problem . . . . . . . . . . . . 20
1.3.2 Fully-Cooperative Game Problem . . . . . . . . . . . . 21
1.3.3 Adversarial Game Problem . . . . . . . . . . . . . . . 21
1.3.4 Berge Game Problem . . . . . . . . . . . . . . . . . . 22
1.3.5 Stackelberg Game Problem . . . . . . . . . . . . . . . 23
1.3.6 Co-opetitive Game Problem . . . . . . . . . . . . . . . 24
1.3.7 Partial-Altruism and Self-Abnegation Game Problem . 25
vii
viii Contents
1.4 Partial Integro-Differential System for a Mean-Field-Type Con-

trol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 A Simple Method for Solving Mean-Field-Type Games and
Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.1 Continuous-Time Direct Method . . . . . . . . . . . . 29
1.5.2 Discrete-Time Direct Method . . . . . . . . . . . . . . 30
1.6 A Simple Derivation of the Itô’s Formula . . . . . . . . . . . 30
1.7 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
II Mean-Field-Free and Mean-Field Games 37

2 Mean-Field-Free Games 39
2.1 A Basic Continuous-Time Optimal Control Problem . . . . . 40
2.2 Continuous-Time Differential Game . . . . . . . . . . . . . . 46
2.3 Stochastic Mean-Field-Free Differential Game . . . . . . . . 53
2.4 A Basic Discrete-Time Optimal Control Problem . . . . . . . 57
2.5 Deterministic Difference Games . . . . . . . . . . . . . . . . 60
2.6 Stochastic Mean-Field-Free Difference Game . . . . . . . . . 65
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3 Mean-Field Games 73
3.1 A Continuous-Time Deterministic Mean-Field Game . . . . . 75
3.2 A Continuous-Time Stochastic Mean-Field Game . . . . . . 78
3.3 A Discrete-Time Deterministic Mean-Field Game . . . . . . 86
3.4 A Discrete-Time Stochastic Mean-Field Game . . . . . . . . 91
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
III One-Dimensional Mean-Field-Type Games 103

4 Continuous-Time Mean-Field-Type Games 105
4.1 Mean-Field-Type Game Set-up . . . . . . . . . . . . . . . . . 106
4.2 Semi-explicit Solution of the Mean-Field-Type Game Problem 112
4.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 126
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Co-opetitive Mean-Field-Type Games 139

5.1 Co-opetitive Mean-Field-Type Game Set-up . . . . . . . . . 141
5.2 Semi-explicit Solution of the Co-opetitive Mean-Field-Type
Game Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3 Connections between the Co-opetitive Solution with the Non-
cooperative and Cooperative Solutions . . . . . . . . . . . . 150
5.3.1 Non-cooperative Relationship . . . . . . . . . . . . . . 150
5.3.2 Cooperative Relationship . . . . . . . . . . . . . . . . 151
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Contents ix
6 Mean-Field-Type Games with Jump-Diffusion and Regime

Switching 167
6.2 Semi-explicit Solution of the Mean-Field-Type Game with
Jump-Diffusion Process and Regime Switching . . . . . . . . 172
6.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . 191
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7 Mean-Field-Type Stackelberg Games 199

7.1 Mean-Field-Type Stackelberg Game Set-up . . . . . . . . . . 200
7.2 Semi-explicit Solution of the Stackelberg Mean-Field-Type
Game with Jump-Diffusion Process and Regime Switching . 203
7.3 When Nash Solution Corresponds to Stackelberg Solution for
Mean-Field-Type Games . . . . . . . . . . . . . . . . . . . . 220
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8 Berge Equilibrium in Mean-Field-Type Games 229

8.1 On the Berge Solution Concept . . . . . . . . . . . . . . . . 230
8.2 Berge Mean-Field-Type Game Problem . . . . . . . . . . . . 230
8.3 Semi-explicit Mean-Field-Type Berge Solution . . . . . . . . 234
8.4 When Berge Solution Corresponds to Co-opetitive Solution for
Mean-Field-Type Games . . . . . . . . . . . . . . . . . . . . 243
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
IV Matrix-Valued Mean-Field-Type Games 249

9 Matrix-Valued Mean-Field-Type Games 251
9.1.1 Matrix-Valued Applications . . . . . . . . . . . . . . . 253
9.1.2 Risk-Neutral . . . . . . . . . . . . . . . . . . . . . . . 256
9.1.3 Risk-Sensitive . . . . . . . . . . . . . . . . . . . . . . . 257
9.2 Semi-explicit Solution of the Mean-Field-Type Game Problems:
Risk-Neutral Case . . . . . . . . . . . . . . . . . . . . . . . . 259
9.3 Semi-explicit Solution of the Mean-Field-Type Game Problems:
Risk-Sensitive Case . . . . . . . . . . . . . . . . . . . . . . . 271
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10 A Class of Constrained Matrix-Valued Mean-Field-Type

Games 291
10.1 Constrained Mean-Field-Type Game Set-up . . . . . . . . . 291
10.1.1 Auxiliary Dynamics . . . . . . . . . . . . . . . . . . . 293
10.1.2 Augmented Formulation of the Constrained MFTG . . 294
x Contents
10.2 Semi-explicit Solution of the Constrained Mean-Field-Type

Game Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 295
10.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
V Discrete-Time Mean-Field-Type Games 297

11 One-Dimensional Discrete-Time Mean-Field-Type Games 299
11.1 Discrete-Time Mean-Field-Type Game Set-up . . . . . . . . 299
11.2 Semi-explicit Solution of the Discrete-Time Non-Cooperative
Mean-Field-Type Game Problem . . . . . . . . . . . . . . . . 302
11.3 Semi-explicit Solution of the Discrete-Time Cooperative Mean-
Field-Type Game Problem . . . . . . . . . . . . . . . . . . . 311
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
12 Matrix-Valued Discrete-Time Mean-Field-Type Games 321

12.1 Discrete-Time Mean-Field-Type Game Set-up . . . . . . . . 321
12.2 Semi-explicit Solution of the Discrete-Time Mean-Field-Type
Game Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 325
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
VI Learning Approaches and Applications 359

13 Constrained Mean-Field-Type Games: Stationary Case 361
13.1 Constrained Games . . . . . . . . . . . . . . . . . . . . . . . 361
13.1.1 A Constrained Deterministic Game . . . . . . . . . . . 362
13.1.2 A Constrained Mean-Field-Type Game . . . . . . . . 363
13.1.3 Constrained Variational Equilibrium . . . . . . . . . . 366
13.1.4 Potential Constrained Mean-Field-Type Game . . . . 368
13.1.5 Efficiency Analysis . . . . . . . . . . . . . . . . . . . . 369
13.1.5.1 Variations of the Variance . . . . . . . . . . . 370
13.1.5.2 Variations of the ε-parameters . . . . . . . . 370
13.1.5.3 Variations of the Number of Players . . . . . 370
13.1.5.4 Variations of Connectivity under Graphs . . 370
13.1.6 Learning Variational Equilibria . . . . . . . . . . . . . 371
13.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
13.3 Learning Algorithms . . . . . . . . . . . . . . . . . . . . . . . 375
13.4 Equilibrium under Migration Constraints . . . . . . . . . . . 377
14 Mean-Field-Type Model Predictive Control 379

14.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 379
14.2 Risk-Aware Model Predictive Control Approaches . . . . . . 381
14.2.1 Chance-Constrained Model Predictive Control . . . . 381
14.2.2 Mean-Field-Type Model Predictive Control . . . . . . 381
14.2.3 Chance-Constrained vs Mean-Field Type Model Predic-
tive Control . . . . . . . . . . . . . . . . . . . . . . . . 382
Contents xi
14.2.4 Decomposition and Stability . . . . . . . . . . . . . . 385
15 Data-Driven Mean-Field-Type Games 389

15.1 Data-Driven Mean-Field-Type Game Problem . . . . . . . . 390
15.2 Machine Learning Philosophy . . . . . . . . . . . . . . . . . 392
15.3 Machine-Learning-Based (Linear Regression) Data-Driven Mean-
Field-Type games . . . . . . . . . . . . . . . . . . . . . . . . 394
15.3.1 Availability of Data . . . . . . . . . . . . . . . . . . . 394
15.3.2 Preparation of Data . . . . . . . . . . . . . . . . . . . 394
15.3.3 Machine-Learning Core . . . . . . . . . . . . . . . . . 395
15.4 Error and Performance Metrics . . . . . . . . . . . . . . . . . 398
16 Applications 405
16.1 Water Distribution Systems . . . . . . . . . . . . . . . . . . 405
16.1.1 Five-Tank Water System . . . . . . . . . . . . . . . . 405
16.1.2 Barcelona Drinking Water Distribution Network . . . 410
16.2 Micro-grid Energy Storage . . . . . . . . . . . . . . . . . . . 415
16.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
16.2.2 Numerical Results . . . . . . . . . . . . . . . . . . . . 419
16.3 Continuous Stirred Tank Reactor . . . . . . . . . . . . . . . 420
16.3.1 Linearization-Based Scheduling and Risk-Aware Con-
trol Problem . . . . . . . . . . . . . . . . . . . . . . . 423
16.3.2 Gain-Scheduled Mean-Field-Type Control . . . . . . . 425
16.3.2.1 Design . . . . . . . . . . . . . . . . . . . . . 425
16.3.2.2 Local Stability of the Operating Points . . . 427
16.3.3 Risk-Aware Numerical Illustrative Example . . . . . . 428
16.4 Mechanism Design in Evolutionary Games . . . . . . . . . . 433
16.4.1 A Risk-Aware Approach to the Equilibrium Selection 437
16.4.1.1 Known Desired Nash Equilibrium . . . . . . 437
16.4.1.2 Unknown Desired Nash Equilibrium . . . . . 440
16.4.2 Risk-Aware Control Design . . . . . . . . . . . . . . . 441
16.4.3 Illustrative Example . . . . . . . . . . . . . . . . . . . 443
16.5 Multi-level Building Evacuation with Smoke . . . . . . . . . 445
16.5.1 Markov-Chain-Based Motion Model for Evacuation over
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 446
16.5.1.1 Reaching Evacuation Areas . . . . . . . . . . 449
16.5.1.2 Evacuation of the Whole Area . . . . . . . . 453
16.5.1.3 Jump Intensities for Evacuation . . . . . . . 455
16.5.2 Markov-Chain-Based Modeling for Smoke Motion . . . 456
16.5.3 Mean-Field-Type Control for the Evacuation . . . . . 457
16.5.4 Single-Level Numerical Results . . . . . . . . . . . . . 461
16.6 Coronavirus Propagation Control . . . . . . . . . . . . . . . 466
16.6.1 Single-Player Problem . . . . . . . . . . . . . . . . . . 467
16.6.1.1 Control Problem of Mean-Field Type . . . . 469
xii Contents
16.6.2 Multiple-Decision-Maker Problem . . . . . . . . . . . 470

16.6.2.1 Non-cooperative Games . . . . . . . . . . . . 472
Bibliography 475
Index 489
List of Figures
1.1 Diversification problem with two assets. . . . . . . . . . . . . 8

1.2 Risk vs Return plot in a portfolio problem with two assets. . 9
1.3 Feedback control scheme for the temperature control system. 10
1.4 Evolution of the temperature for two different control scenar-
ios. (a) evolution of the expected temperature. (b) evolution of
the temperature. (c) variance comparison of the two scenarios. 11
1.5 Some engineering applications involving uncertainties. . . . 12
1.6 Network of networks. Interdependence among the water, traf-
fic, energy, district heating and communication systems. . . 15
1.7 Brief recent literature review on different methods to solve
mean-field-type control and game problems with finite number
of decision-makers. . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Two decision-makers illustrating a non-cooperative game
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.9 Two decision-makers illustrating a cooperative game problem. 21
1.10 Two decision-makers illustrating an adversarial game problem. 22
1.11 Two decision-makers illustrating a Berge game problem. . . 23
1.12 Two decision-makers illustrating a Stackelberg game problem. 23
1.13 Three decision-makers illustrating a co-opetitive game prob-
lem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.14 Four decision-makers illustrating a Partial-Altruism game
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.15 Four decision-makers illustrating a Self-Abnegation game
problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.16 Control problem. Equivalently, a single decision-maker
decision-making problem. . . . . . . . . . . . . . . . . . . . . 26
1.17 General scheme corresponding to the Continuous-Time Direct
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.18 General scheme corresponding to the discrete-time direct
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.19 Outline of the book. Arrows show the interdependence among
the chapters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.20 Risk doors exercise. Would you risk for a bigger reward? . . 35
2.1 A basic scalar-valued control scheme. . . . . . . . . . . . . . 40
xiii
xiv List of Figures
2.2 Feedback scheme for the linear-quadratic mean-field-free op-

timal control. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Feedback scheme for the infinite-horizon linear-quadratic
mean-field-free optimal control. . . . . . . . . . . . . . . . . 45
2.4 General scheme of the non-cooperative differential game. Dark
gray nodes represent the n players, whereas the black node
represents the common system state. . . . . . . . . . . . . . 46
2.5 Feedback scheme for the ith decision-maker in the linear-
quadratic mean-field-free differential game. . . . . . . . . . . 49
2.6 Feedback scheme for the infinite-horizon linear-quadratic
mean-field-free differential game. . . . . . . . . . . . . . . . . 53
2.7 Feedback scheme for the linear-quadratic mean-field-free op-
timal control in discrete time. . . . . . . . . . . . . . . . . . 59
2.8 Feedback scheme for the linear-quadratic mean-field-free dif-
ference game. . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 General Mean-Field Game Philosophy in either continuous

or discrete time. Each player (black node) strategically plays
against a population mass of infinite players (dark gray nodes). 74
3.2 Feedback scheme for the linear-quadratic mean-field game in
continuous time. . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Feedback scheme for the linear-quadratic mean-field game in
discrete time. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.1 General scheme of the non-cooperative mean-field-type game.

Dark gray nodes represent the n players, the black node rep-
resents the system state, and the light gray node represents
the stochasticity affecting the system state. . . . . . . . . . . 111
4.2 General scheme of the cooperative mean-field-type game. Dark
gray nodes represent the n players, the black node repre-
sents the system state, and the light gray node represents the
stochasticity affecting the system state. . . . . . . . . . . . . 112
4.3 Feedback scheme for the linear-quadratic mean-field-type non-
cooperative game problem. . . . . . . . . . . . . . . . . . . . 114
4.4 Feedback scheme for the linear-quadratic mean-field-type co-
operative game problem. . . . . . . . . . . . . . . . . . . . . 122
4.5 Evolution of the system state and its expectation for the
scalar-value non-cooperative scenario. . . . . . . . . . . . . . 127
4.6 Evolution of the players’ strategies and their expectation for
the scalar-value non-cooperative scenario. . . . . . . . . . . . 128
4.7 Evolution of the Riccati equations α1 , . . . , α3 for the scalar-
value non-cooperative scenario. . . . . . . . . . . . . . . . . 129
4.8 Evolution of the Riccati equations β1 , . . . , β3 for the scalar-
List of Figures xv
4.9 Evolution of the Riccati equations γ1 , . . . , γ3 for the scalar-

4.10 Evolution of the Riccati equations δ1 , . . . , δ3 for the scalar-
4.11 Optimal cost function for each player for the scalar-value non-
cooperative scenario. . . . . . . . . . . . . . . . . . . . . . . 132
scalar-value fully-cooperative scenario. . . . . . . . . . . . . 133
4.13 Evolution of the players’ strategies and their expectation for
the scalar-value fully-cooperative scenario. . . . . . . . . . . 133
4.14 Evolution of the Riccati equation α0 for the scalar-value fully-
4.15 Evolution of the Riccati equation β0 for the scalar-value fully-
4.16 Evolution of the Riccati equation γ0 for the scalar-value fully-
4.17 Evolution of the Riccati equation δ0 for the scalar-value fully-
4.18 Cost function associated with each player for the scalar-value
fully-cooperative scenario. . . . . . . . . . . . . . . . . . . . 136
−
5.1 Co-opetition scheme λ+ ij (λij ) means λij > 0, (λij < 0). Black
players are altruistic, dark gray players are spiteful, and light
gray players cooperate and compete simultaneously. (a) Fully-
altruistic scenario, (b) Fully-adversarial scenario, (c) Altruistic
and adversarial players, and (d) Mixture of behaviors. . . . 140
5.2 Feedback scheme for the linear-quadratic mean-field-type co-
opetitive game problem. . . . . . . . . . . . . . . . . . . . . 145
5.3 Evolution of the system state and its expectation. . . . . . . 153
5.4 Evolution of the optimal control strategies for the partially
cooperation in a co-opetitive scenario. . . . . . . . . . . . . . 153
5.5 Evolution of the Riccati equations α1 , . . . , α5 for the partially
5.6 Evolution of the Riccati equations β1 , . . . , β5 for the partially
5.7 Evolution of the Riccati equations γ1 , . . . , γ5 for the partially
5.8 Evolution of the Riccati equations δ1 , . . . , δ5 for the partially
5.9 Optimal cost for the five players for the partially cooperation
in a co-opetitive scenario. . . . . . . . . . . . . . . . . . . . . 157
5.10 Co-opetitive parameters for the spiteful behavior in a co-
opetitive scenario. . . . . . . . . . . . . . . . . . . . . . . . . 158
5.11 Evolution of the system state and its expectation for the spite-
ful behavior in a co-opetitive scenario. . . . . . . . . . . . . 159
xvi List of Figures
5.12 Evolution of the optimal strategies and their expectation for

the spiteful behavior in a co-opetitive scenario. . . . . . . . . 160
5.13 Evolution of the Riccati equations α1 , . . . , α5 for the spiteful
behavior in a co-opetitive scenario. . . . . . . . . . . . . . . 161
5.14 Evolution of the Riccati equations β1 , . . . , β5 for the spiteful
5.15 Evolution of the Riccati equations γ1 , . . . , γ5 for the spiteful
5.16 Evolution of the Riccati equations δ1 , . . . , δ5 for the spiteful
5.17 Optimal cost for the five players under the co-opetitive sce-
nario with spiteful behavior. . . . . . . . . . . . . . . . . . . 163
6.1 General scheme of the non-cooperative mean-field-type game

with jumps and regime switching. Dark gray nodes represent
the n players, the black node represents the system state, and
the light gray node represents the stochasticity affecting the
system state. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2 Brownian motion and two jumps. . . . . . . . . . . . . . . . 191
scalar-value non-cooperative scenario with Brownian motion,
Poisson jumps and regime switching. . . . . . . . . . . . . . 192
6.4 Evolution of the optimal strategies u∗1 and u∗2 for the scalar-
value non-cooperative scenario with Brownian motion, Poisson
jumps and regime switching. . . . . . . . . . . . . . . . . . . 193
6.5 Evolution of the Riccati equations α1 and α2 for the scalar-
6.6 Evolution of the Riccati equations β1 and β2 for the scalar-
6.7 Evolution of the Riccati equations γ1 and γ2 for the scalar-
jumps and switching. . . . . . . . . . . . . . . . . . . . . . . 196
6.8 Evolution of the Riccati equations δ1 and δ2 for the scalar-
jumps and switching. . . . . . . . . . . . . . . . . . . . . . . 196
7.1 General scheme of the two-player Stackelberg mean-field-type

game with jump-diffusion and regime switching. . . . . . . . 200
7.2 Hierarchical order in the Stackelberg mean-field-type game. 201
7.3 Brownian motion and two jumps. . . . . . . . . . . . . . . . 221
scalar-value Stackelberg scenario with Brownian motion and
Poisson jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . 223
List of Figures xvii
7.5 Evolution of the optimal control inputs u∗i and u∗j for the
scalar-value Stackelberg scenario with Brownian motion and
Poisson jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.6 Evolution of the differential equations αi and αj for the scalar-
value Stackelberg scenario with Brownian motion and Poisson
jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.7 Evolution of the differential equations βi and βj for the scalar-
jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.8 Evolution of the differential equations γi and γj for the scalar-
jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.9 Evolution of the differential equations δi and δj for the scalar-
jumps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.1 General scheme of the two-player Berge mean-field-type game

with jump-diffusion and regime switching. . . . . . . . . . . 232
8.2 Considered jumps and Brownian for the numerical example. 245
8.3 Evolution of the state x(t) and its expected value E[x(t)] with
initial conditions x(0) = E[x(0)] = 100. . . . . . . . . . . . . 246
8.4 Evolution of the control input u1 (t). . . . . . . . . . . . . . . 246
8.5 Evolution of the control input u2 (t). . . . . . . . . . . . . . . 247
8.6 Evolution of the differential equations α1 (t) and α2 (t). . . . 247
9.1 Example of a matrix-valued application with d = 4. . . . . . 253

9.2 Exchange matrix for six currencies: Euro, US Dollar, Aus-
tralian Dollar, Canadian Dollar, Swiss Franc, and Japanese
Yen, for five different dates. . . . . . . . . . . . . . . . . . . 255
matrix-value continuous-time non-cooperative scenario. . . . 278
9.4 Evolution of the first player strategies and their expecta-
tion for the matrix-value continuous-time non-cooperative sce-
nario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.5 Evolution of the second player strategies and their expecta-
tion for the matrix-value continuous-time non-cooperative sce-
nario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.6 Evolution of the equation P1 for the matrix-value continuous-
time non-cooperative scenario. . . . . . . . . . . . . . . . . . 280
9.8 Evolution of the equation P̄1 for the matrix-value continuous-
xviii List of Figures
9.10 Evolution of the Riccati equations δ1 , and δ2 for the matrix-

value continuous-time non-cooperative scenario. . . . . . . . 283
9.11 Optimal cost function for each player for the matrix-value
continuous-time non-cooperative scenario. . . . . . . . . . . 283
matrix-value continuous-time fully-cooperative scenario. . . 284
9.13 Evolution of the first player strategies and their expectation
for the matrix-value continuous-time fully-cooperative sce-
nario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
9.14 Evolution of the second player strategies and their expec-
tation for the matrix-value continuous-time fully-cooperative
scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
time fully-cooperative scenario. . . . . . . . . . . . . . . . . 286
time fully-cooperative scenario. . . . . . . . . . . . . . . . . 287
9.17 Evolution of the Riccati equation δ0 for the matrix-value
continuous-time fully-cooperative scenario. . . . . . . . . . . 287
continuous-time fully-cooperative scenario. . . . . . . . . . . 288
12.1 Evolution of the (a) system state and (b) its expectation for
the matrix-value discrete-time non-cooperative scenario. . . 345
12.2 Evolution of the (a) first player strategies and (b) their ex-
pectation for the matrix-value discrete-time non-cooperative
scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
12.3 Evolution of the (a) second player strategies and (b) their ex-
pectation for the matrix-value discrete-time non-cooperative
scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
12.4 Evolution of the equation P1 for the matrix-value discrete-time
non-cooperative scenario. . . . . . . . . . . . . . . . . . . . . 349
12.6 Evolution of the equation P̄1 for the matrix-value discrete-time
12.8 Evolution of the Riccati equations δ1 , and δ2 for the matrix-
value discrete-time non-cooperative scenario. . . . . . . . . . 351
discrete-time non-cooperative scenario. . . . . . . . . . . . . 351
12.10 Evolution of the (a) system state and (b) its expectation for
the matrix-value discrete-time fully-cooperative scenario. . . 352
List of Figures xix
12.11 Evolution of the (a) first player strategies and (b) their ex-
pectation for the matrix-value discrete-time fully-cooperative
scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.12 Evolution of the (a) second player strategies and (b) their ex-
pectation for the matrix-value discrete-time fully-cooperative
scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
12.15 Evolution of the Riccati equation δ0 for the matrix-value
discrete-time fully-cooperative scenario. . . . . . . . . . . . . 356
discrete-time fully-cooperative scenario. . . . . . . . . . . . . 356

13.1 Gap between var xVI = xbest−GNE and var xGlobal as dif-
ferent parameters εij = ε, for all i, j ∈ N , and number of
players n change for a fixed variance σ = 1. . . . . . . . . . . 369
13.2 Different topologies for the comparison among δ−parameters
in (13.11a) with εij ≥ 0, for all i, j ∈ N . . . . . . . . . . . . 371
13.3 Evolution of the variables w and x under the learning algo-
rithm in (13.13) for the constrained MFTG presented in (13.4)
with n = 2. Figures correspond to: (a) evolution of w1 and w2 ,
and (b)-(c) evolution of x1 and x2 . . . . . . . . . . . . . . . 372
14.1 Graphical example for the sets X, Dx , and Xc . . . . . . . . . 383
15.1 General scheme for a data-driven mean-field-type game prob-

lem by using machine learning. . . . . . . . . . . . . . . . . 390
15.2 Input/output configuration for the unknown system in a two-
player mean-field-type game problem. . . . . . . . . . . . . . 391
15.3 Machine-learning scheme. . . . . . . . . . . . . . . . . . . . . 392
15.4 Machine-learning-based expected values in comparison with a
particular test. This figure corresponds to Test 7. . . . . . . 400
15.5 Data-based probability measure of x at k = 600. . . . . . . . 401
15.6 Error distribution with ek = xk − yk . . . . . . . . . . . . . . 402
15.7 Machine-learning-based parameters. . . . . . . . . . . . . . . 403
16.1 Illustrative example. Five-tank benchmark involving two play-

ers and two coupled input constraints. Players 1 and 2 corre-
spond to the black and gray colors, respectively. . . . . . . . 406
16.2 Brownian motions. . . . . . . . . . . . . . . . . . . . . . . . 407
16.3 Evolution of the system states. . . . . . . . . . . . . . . . . . 408
16.4 Evolution of the optimal control inputs for the players. . . . 409
16.5 Water distribution network. . . . . . . . . . . . . . . . . . . 412
xx List of Figures
16.6 Control inputs variance comparison for the two different MFT-
MPC controllers. . . . . . . . . . . . . . . . . . . . . . . . . 413
16.7 Results corresponding to the proposed stochastic MFT-MPC
controllers for Scenarios 1 and 2; and behavior of the deter-
ministic MPC controller. . . . . . . . . . . . . . . . . . . . . 414
16.8 General scheme of the micro-grid involving energy storage.
(Adapted from [1].) . . . . . . . . . . . . . . . . . . . . . . . 415
16.9 Evolution of the noise applied to the system. . . . . . . . . . 416
16.10 Evolution of the system state. . . . . . . . . . . . . . . . . . 417
16.11 Evolution of the control input for the first player and its ex-
pectation, i.e., u1,k and E[u1,k ]. . . . . . . . . . . . . . . . . 417
16.12 Evolution of the control input for the second player and its
expectation, i.e., u2,k and E[u2,k ]. . . . . . . . . . . . . . . . 418
16.13 Evolution of the control input for the third player and its
expectation, i.e., u3,k and E[u3,k ]. . . . . . . . . . . . . . . . 418
16.14 Continuous stirred tank reactor. . . . . . . . . . . . . . . . . 420
16.15 Gain-scheduled Mean-Field-Type Control Diagram with n op-
eration points and θ ∈ {1, . . . , n}. . . . . . . . . . . . . . . . 425
16.16 Gain-scheduled mean-field-free control diagram with n opera-
tion points and θ ∈ {1, . . . , n}. . . . . . . . . . . . . . . . . . 426
16.17 Noise Brownian motions for both the reactant concentration
and the reactor temperature. . . . . . . . . . . . . . . . . . . 429
16.18 Performance of the GS-MFTC. Evolution of the reactant con-
centration CA and its expectation E[CA ] tracking the reference
CA ref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
16.19 Performance of the GS-MFTC. Evolution of the reactor tem-
perature TR and its expectation E[TR ] tracking the reference
TRref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
16.20 Evolution of the optimal control input and its expectation. . 430
16.21 (a)–(b) Evolutionary dynamics with imperfect fitness observa-
tion for a population game, and two-strategy population game.
(c)–(d) Closed-loop of the multi-layer game for the equilibrium
selection and two-strategy population game. . . . . . . . . . 436
16.22 Projection dynamics behavior for both the RSP and Zeeman
game with and without noisy fitness functions. . . . . . . . . 438
16.23 Behavior of the risk-aware controller over the projection dy-
namics for both the RSP and Zeeman game with imperfect
fitness observation. . . . . . . . . . . . . . . . . . . . . . . . 444
16.24 Representation of the space B, its respective discretization
into n regions, and an example graph G, and V = {46},
O = {43, 49}, and F = {50}. . . . . . . . . . . . . . . . . . . 447
16.25 Representation of spacial constraints such as walls and obsta-
cles in the graph G f . . . . . . . . . . . . . . . . . . . . . . . 448
16.26 Relationship between possible jump intensities in the Markov
chain and the links E in a connected graph G. . . . . . . . . 449
List of Figures xxi
16.27 Mass of people motion with an initial distribution x(0) evac-

uating the whole space B within time T and a unique exit
area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
16.28 Mass of people motion with an initial distribution x(0) evacu-
ating the whole space B within time T and with two exit areas
O1 and O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
16.29 Smoke spread given a fire source s(0) and covering the whole
space B within time T , i.e., smoke distribution s(T ). . . . . 456
16.30 Evolution of the smoke throughout the area B for 8.3 minutes
and with time steps 0.1 seconds. . . . . . . . . . . . . . . . . 462
16.31 Evacuation comparison with and without mean-field-type con-
troller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
16.32 Evolution of the population mass throughout the area B with-
out risk minimization for 8.3 minutes and with time steps
0.1 seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
16.33 Variance comparison with and without mean-field-type con-
troller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
16.34 Propagation of the virus in Colombia with migration con-
straints among the 32 departments without quarantine pre-
vention. Dots in the map represent infected/dead people: (a)
initial condition, (b) spread of the virus after 400 iterations
(interactions), (c) allowed air traffic. . . . . . . . . . . . . . 468
16.35 Transition rates among the finite states S. . . . . . . . . . . 469
16.36 Transition rates among three different players, i.e., P =
{1, 2, 3} corresponding to the example in (16.55). . . . . . . 471
List of Tables
8.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . 244
9.1 Exchange rates involving six currencies on October 11th, 2018 254
15.1 Summary of available data for machine-learning purposes. . 393

15.2 Summary of prepared data for machine-learning purposes. . 395
15.3 Second summary of prepared data for machine-learning pur-
poses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
15.4 Initial data D for machine-learning purposes. . . . . . . . . . 399
15.5 Correlation among the decision-makers control-inputs. . . . 399
15.6 First preparation of data for machine-learning purposes. . . 401
15.7 Error metrics between a real trajectory and machine-learning-
based estimated trajectories . . . . . . . . . . . . . . . . . . 402
16.1 Description of the variables in the model (16.5). . . . . . . . 421

16.2 Value of the variables for the case study. . . . . . . . . . . . 428
16.3 Summary of results for three different scenarios with and with-
out MFTC/smoke. . . . . . . . . . . . . . . . . . . . . . . . 463
xxiii
Foreword
Mean field was first studied in physics for the behavior of systems with large
numbers of negligible individual particles. Recently mean-field game theory
was introduced in the economics and engineering literature to study the strate-
gic decision-making by small interacting agents of huge populations. Typically
a mean-field game is described by a Fokker-Planck equation, and solved by
a Hamilton-Jacobi-Bellman equation, which requires the number of agents
approaches infinity. This assumption limits the practical usage of mean-field
game theory in engineering fields.
Thanks to my friend, professor Hamidou Tembine, who also mentored my
former Ph.D. student majored in mean-field game theory, his team and collab-
orators including Dr. Julian Barreiro-Gomez have pioneer works to introduce
mean-field-type game theory to engineering scenarios. Mean-field-type games
differ from mean field games since it takes into account higher-order statis-
tics, it can be employed when dynamic programming cannot be applied, the
number of interacting agents is not necessarily large, and it can handle non-
symmetry non-negligible effect of individual decision on the mean field term.
Those significant advantages of mean-field-type game theory open a whole
gate for solving complex engineering problems that cannot be handled by
classic methods.
With such a demand from engineering audiences, this book is very timely
and provides a thorough study of mean-field-type game theory. The strenu-
ous protagonist of this book is to bridge between the theoretical findings and
engineering solutions. The book introduces the basics first, and then math-
ematical frameworks are elaborately explained. The engineering application
examples are shown in detail, and the popular learning approaches are also
investigated. Those advantageous characteristics will make this book a com-
prehensive handbook of many engineering fields for many years, and I will
buy one when it gets published.
Zhu Han, IEEE/AAAS Fellow

John and Rebecca Moores professor
University of Houston
xxv
Preface
If you have picked this book, you are probably already aware about how
powerful and suitable the mean-field-type control and game theory is in order
to solve risk-aware problems in the engineering framework, and that a large
variety of control and dynamic game problems can be set as particular cases
of the mean-field-type games.
Our main goal in this textbook is to provide a quite comprehensive and
simple treatment of the mean-field-type control and game theory, which can
also be interpreted as risk-aware optimal interactive decision-making tech-
niques. To this end, we exclusively focus on the so-called direct method either
in continuous or discrete time. Our experience indicates that other existing
methods reported in the literature to solve the class of stochastic problems
we address in this book, such as partial-differential-equation-based methods,
chaos expansion, dynamic programming, or the stochastic maximum princi-
ple, are not appropriate to start teaching beginner students in the field neither
early-career researchers. We recommend to focus on understanding this book
prior to moving on the study of other research manuscripts using other theo-
retical directions. In this regard, the contents of this book comprises an appro-
priate background to start working and doing research in this game-theoretical
field.
To make the exposition and explanation even easier, we first study the de-
terministic optimal control and differential linear-quadratic games. Then, we
progressively add complexity step-by-step and little-by-little to the problem
settings until we finally study and analyze mean-field-type control and game
problems incorporating several stochastic processes, e.g., Brownian motions,
Poisson jumps, and random coefficients.
This smooth trip, starting with a scalar-valued state optimal control prob-
lem in continuous and discrete time, passes through the scalar-valued deter-
ministic differential games and mean field games, the stochastic state-and-
control-input independent diffusion differential games and mean field games,
until we finally address the mean-field-type games with state-and-control-
input dependent diffusion terms and incorporating Poisson jumps and random
coefficients by means of switching regimes. On the other hand, we go beyond
the Nash equilibrium, which provides a solution for non-cooperative games,
by analyzing other game-theoretical concepts such as the Berge, Stackelberg,
adversarial and co-opetitive equilibria. For the mean-field-type game analysis,
we provide several numerical examples, which are obtained from a MatLab-
xxvii
xxviii Preface
based user-friendly toolbox that is available for the free use of the readers of
this book.
We devote a whole part of the book to discuss about some learning ap-
proaches that guarantee converge to mean-field-type solutions. In particular,
we present the constrained and static mean-field-type games where optimiza-
tion algorithms may be applied such as distributed evolutionary dynamics, the
receding horizon mean-field-type control also know as risk-aware model pre-
dictive technique, and the data-driven mean-field-type games motivating the
use of artificial intelligence tools such as machine learning with either neural
networks or simple linear regression. Finally, we present several engineering
applications in both continuous and discrete time. Among these applications
we find the following: water distribution systems, micro-grid energy storage,
stirred tank reactor, mechanism design for evolutionary dynamics, multi-level
building evacuation problem, and the COVID-19 propagation control.
Julian Barreiro-Gomez
Hamidou Tembine
Acknowledgments
We gratefully acknowledge support from the US Air Force, and the New York
University in the US campus (NYU) and the UAE campus (NYUAD), for
the research conducted at the Learning & Game Theory Laboratory (L&G
Lab) and at the Center on Stability, Instability and Turbulence (SITE). This
material is based upon work supported by Tamkeen under the NYU Abu
Dhabi Research Institute grant CG002.
We also acknowledge our friends, faculty members, and researchers with
whom we have had several scientific discussions about mean-field-type control
and game theory, and also regarding its potential for engineering applications.
We specially thank Prof. Tyrone E. Duncan and Prof. Bozenna Pasik-Duncan
from the mathematics department at Kansas University in the US, and Prof.
Boualem Djehiche from the mathematics department at Royal KTH in Swe-
den. We finally acknowledge all our co-authors with whom we have published
several articles in the mean-field-type field.
xxix
Author Biographies
Julian Barreiro-Gomez received his B.S. degree (cum laude) in Electronics En-
gineering from Universidad Santo Tomás (USTA), Bogota, Colombia, in 2011.
He received the M.Sc. degree in Electrical Engineering and the Ph.D. degree
in Engineering from Universidad de Los Andes (UAndes), Bogota, Colombia,
in 2013 and 2017, respectively. He received the Ph.D. degree (cum laude) in
Automatic, Robotics and Computer Vision from the Technical University of
Catalonia (UPC), Barcelona, Spain, in 2017; the best Ph.D. thesis in control
engineering 2017 award from the Spanish National Committee of Automatic
Control (CEA) and Springer; and the EECI Ph.D. Award from the European
Embedded Control Institute in recognition to the best Ph.D. thesis in Eu-
rope in the field of Control for Complex and Heterogeneous Systems 2017.
He received the ISA Transactions Best Paper Award 2018 in Recognition to
the best paper published in the previous year. Since August 2017, he has
been a Post-Doctoral Associate in the Learning & Game Theory Laboratory
(L&G-Lab) at the New York University in Abu Dhabi (NYUAD), United
Arab Emirates, and since 2019, he has also been with the Research Center
on Stability, Instability and Turbulence (SITE) at the New York University
in Abu Dhabi (NYUAD). His main research interests are: risk-aware control
and games, mean-field-type games, constrained evolutionary game dynamics,
distributed optimization, stochastic optimal control, and distributed predic-
tive control.
Hamidou Tembine received the M.S. degree in applied mathematics from Ecole
Polytechnique, Palaiseau, France, in 2006 and the Ph.D. degree in computer
science from the University of Avignon, Avignon, France, in 2009. He is a
prolific Researcher and holds more than 150 scientific publications including
magazines, letters, journals, and conferences. He is an author of the book on
Distributed Strategic Learning for Engineers (CRC Press, Taylor & Francis
2012), and Coauthor of the book Game Theory and Learning in Wireless Net-
works (Elsevier Academic Press). He has been co-organizer of several scientific
meetings on game theory in networking, wireless communications, smart en-
ergy systems, and smart transportation systems. His current research interests
include evolutionary games, mean-field stochastic games and applications. Dr.
Tembine received the IEEE ComSoc Outstanding Young Researcher Award
for his promising research activities for the benefit of the society in 2014. He
received the best paper awards in the applications of game theory.
xxxi
Symbols
Symbol Description
x Scalar-valued system state ` Running cost (control case)

xi Scalar-valued system state h Terminal cost (control case)
of the ith decision-maker Li Cost functional of the ith
u Scalar-valued control input decision-maker
ui Scalar-valued control input `i Running cost of the ith
of the ith decision-maker decision-maker
X Matrix/vector-valued sys- hi Terminal cost of the ith
tem state decision-maker
U Matrix/vector-valued con- E[·] Expected value
trol input var[·] Variance
Ui Matrix/vector-valued con- cov[·] Co-variance
trol input of the ith decision- H Hamiltonian (control case)
maker f Guess functional (control
b Drift case), or fitness functions in
σ Diffusion a population game
B Standard Brownian motion Hi Hamiltonian of the ith
N Jump process decision-maker
Ñ Compensated jump process Fi Guess functional of the
Θ Set of jump sizes, or set of ith decision-maker (matrix-
operating point in a gain- valued problems)
schedule strategy fi Guess functional of the
ν Radon measure over Θ ith decision-maker (scalar-
s Regime switching valued problems), or the fit-
S Set of regime switching ness of the ith strategy in a
q̃ss0 Jump intensity from regime population game
switching s to s0 BRi Best response of the ith
W Discrete-time noise decision-maker
mx Mean-field term of x N Set of decision-makers
mu Mean-field term of u N0 Set of risk-neutral decision-
m Strategic distribution makers
φ Probability measure of the N+ Set of risk-averse decision-
system state x makers
L Cost functional (control N− Set of risk-seeking decision-
case) makers
xxxiii
xxxiv Symbols
X Feasible set of system state Ui Feasible control strategy of

P(X ) Space of the probability the ith decision-maker
measure of x
hx, yi Inner product for vectors
U Feasible set of control inputs
x, y
U Feasible control strategy
Ui Feasible set of control inputs hA, Bi Trace tr(A, B) for matrices
for the ith decision-maker A, B
Part I
Preliminaries
1
Introduction
We truly live in a more and more interconnected and interactive world. In re-
cent years, we have seen emerging technologies such as internet of everything,
collective intelligence including Artificial Intelligence (AI), blockchains, next-
generation wireless networks, among many others. The quantities-of-interest
in these systems involve both volatilities and risks.
A typical example of risk concerns in the current online market is the
evolution of prices for the digital and cryptocurrencies (e.g., bitcoin, litecoin,
ethereum, dash, and other altcoins (alternatives to bitcoin, etc.). The variance
plays a base model for many risk measures. From random-variable perspective
(probability theory) the volatility can be captured by means of the variance,
which is a mean-field term comprising the second moment and the square of
the mean. Another example concerns the variations of wireless channels in
multiple-input-multiple-output systems. Non-Gaussianity of wireless channels
has been observed experimentally and empirically, and its variability affects
the quality of the communication.
The term mean-field has been referred to as a physics concept that at-
tempts to describe the effect of an infinite number of particles on the motion
of a single particle. Researchers began to apply the concept to social sciences
in the early 1960s to study how an infinite number of factors affect individual
decisions. However, the key ingredient in a game-theoretic context is the in-
fluence of the distribution of states and/or control actions onto the payoffs of
the decision-makers. Notice that there is no need to have a large population
of decision-makers. A mean-field-type game is a game in which the payoffs
and/or the state dynamics coefficient functions involve not only the state
and actions profiles but also the distributions of state-action process (or its
marginal distributions).
Games with distribution-dependent quantity-of-interest such as state
and/or payoffs are particularly attractive because they capture not only the
mean, but also the variance and higher order terms. Such incorporation of
these mean and variance terms is directly associated with the paradigm intro-
duced by H. Markowitz, 1990 Nobel Laureate in Economics. The Markowitz
paradigm, also termed as the mean-variance paradigm, is often characterized
as dealing with portfolio risk and (expected) returns [2–4].
In this book, we address variance reduction problems when several
decision-making entities take place. When the decisions made by the
agents/players/decision-makers influence each other, the decision-making is
DOI: 10.1201/9781003098607-1 3
4 Mean-Field-Type Games for Engineers
said to be interactive (interdependent). Such problems are known as game-

theoretical problems.
“Interactive decision theory would perhaps be a more descriptive

name for the discipline usually called Game Theory”
Robert Aumman
[5, page 47]
In this book we study the mean-field-type game theory, which

can be also named as risk-aware interactive decision-making the-
ory.
Next, we present some basic definitions corresponding to the structure of

a particular class of games, which is addressed throughout this book.
1.1 Linear-Quadratic Games

We start by defining a particular class of either deterministic or stochastic
differential games determined by a specific structure that the system dynamics
and cost functional have.
Definition 1 (Linear-Quadratic Deterministic Games) Game problems,
in which the state dynamics is given by a linear deterministic system and a
cost functional that is quadratic in the state and in the control inputs, are
often called the Linear-Quadratic (LQ) games.
Definition 2 (Linear-Quadratic-Gaussian Games) Game problems, in
which the state dynamics is given by a linear stochastic system with a Brow-
nian motion and a cost functional that is quadratic in the state and in the
control inputs, are often called the Linear-Quadratic Gaussian (LQG) games.
Such games also belong to the family of stochastic linear-quadratic games.
1.1.1 Structure of the Optimal Strategies and Optimal Costs

For generic LQG game problems under perfect state observation, the opti-
mal strategy of the decision-maker is a linear state-feedback strategy, which
Introduction 5
is identical to an optimal control for the corresponding deterministic linear-

quadratic game problem where the Brownian motion is replaced by the zero
process. Moreover, the equilibrium cost only differs from the deterministic
game problem’s equilibrium cost by the integral of a function of time.
However, when the diffusion (volatility) coefficient is state and/or control-
dependent, the structure of the resulting differential system as well as the
equilibrium cost vector are modified. These results were widely known in both
dynamic optimization, control and game theory literature.
In this book, several structures are studied from simple ones up to cases
where the stochastic processes are not only dependent on both the system
states and control inputs, but also on the distribution of the states and/or the
control inputs.
1.1.2 Solvability of the Linear-Quadratic Gaussian Games

For both LQG control and LQG zero-sum games, it can be shown that a simple
square completion method provides an explicit solution to the problem. It
was successfully developed and applied by Duncan et al. [6–11] in the mean-
field-free case (games in the absence of the distribution of the variables of
interest). Moreover, Duncan et al. have extended the direct method to more
general noises including fractional Brownian noise and some non-quadratic
cost functionals such as on spheres and torus.
Here, we follow the same method in order to solve a large variety of mean-
field-type control and game problems in both continuous and discrete time,
making the solution of this complex problem accessible for early-career re-
searchers and engineering students.
1.1.3 Beyond Brownian Motion

Inspired by applications in engineering (e.g., internet connection, battery
state, etc) and in finance (e.g., price, stock option, multi-currency exchange,
etc) where not only Gaussian processes but also jump processes (e.g., Poisson,
Lévy, etc) play important features, the question of extending the framework
to linear-quadratic games under state dynamics driven by jump-diffusion pro-
cesses were naturally posed. Adding a Poisson jump and regime switching
(random coefficients) may allow to capture in particular larger jumps which
may not be captured by just increasing diffusion coefficients. Several examples
such as multi-currency exchange or cloud-server rate allocation on blockchains
are naturally in a matrix form.
Throughout this book, we discuss about several game-theoretical solution
concepts and different structures as it has been pointed out in Section 1.1.1
including the analysis when processes beyond Brownian motion are taken into
consideration, e.g., see Chapter 6 where mean-field-type game problems with
jumps and regime switching (random coefficients) are studied.
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CHAPTER XIX
A single glance at the map to-day—I am writing this in January—is
sufficient to give an idea of the enormous difficulties the Italians have
still to contend against and surmount in forcing their way across the
formidable barrier of stony wilderness between their present position
and their obvious objective—Trieste.
It is said that they will find that apart from the terrible character of
the natural obstacles, which, as I have endeavoured to show, they
are up against, the Austrians have series after series of entrenched
lines, each of which will have to be captured by direct assault.
Although it has been seen that man for man the Italian is far and
away superior in fighting quality to the Austrian, it cannot be denied
that when well supplied with machine guns, and behind positions
which afford him almost complete protection, the Austrian soldier will
put up a very determined resistance before he gives in. This factor
must, therefore, be reckoned with in any commencement of an
attempt at an advance.
The dash and reckless courage of the Italians, whilst thoroughly to
be relied upon under any circumstances however trying, must
always be held, as it were, in leash, otherwise even the smallest
forward move will be only achieved at an awful sacrifice of gallant
lives. The utmost caution, moreover, will have to be exercised, so as
not to fall into a guet apens, and every step forward must be fully
protected.
When success, therefore, can only be reckoned on, as it were, by
yards, it is not surprising to find on examining the map how slow
apparently has been the progress during the past four months; but it
is progress nevertheless, and the most tangible proof of it is
contained in the concluding lines of the brief summary issued by the
Italian supreme command of the operations from September to
December:
“The total of prisoners taken on the Julian Front (i.e., the Isonzo
and the Carso) from August to December was 42,000, and the guns
numbered 60 and the machine guns 200.”
The gradual advance has not been confined to any one particular
sector of this Front, but was part of the general scheme which is
operating like a huge seine net over this part of the Carso. The
interest, therefore, was entirely concentrated in this zone after the
fall of Gorizia, and I never missed an opportunity of going in that
direction on the chance of getting some good subjects for my sketch
book.
I remember some years ago when I was crossing the Gobi desert,
I discovered that the desolation of the scene around me exercised
an inexplicable sort of fascination, and at times I would have a
strange longing to wander away alone into the wilderness.
I experienced somewhat the same sensation on the Carso. It is
most probably what Jack London designated the “call of the wild.” In
this case, however, the fascination was tempered by the knowledge
that one’s wandering fit might be cut short by an Austrian bullet, so
one’s peregrinations had perforce to be somewhat curtailed.
There was, of course, much of great interest to see and sketch in
the area where active operations were in progress, whilst every day
almost there seemed to be something, either in the shape of a
rumour, or in the official communiqué, that formed a good excuse for
getting into a car and heading for the “sound of the guns” again.
They came racing across the stretch of “No man’s land” (see page 294)
To face page 270
On one of these occasions I had as my companion Robert

Vaucher, the correspondent of the Paris Illustration, who had just
arrived at the Front for a short visit.
We decided to make for San Martino del Carso, the first village
captured by the Italians on the Carso, as it was quite close to the
fighting then going on round Oppachiassella.
Our route was via Palmanova, Romans and Sagrado a road one
had got to know by heart, so to speak, but of which one never tired,
for somehow, curiously enough, everything always seemed novel
although you had seen it many times before.
There was also the charm of starting off in a car just after sunrise;
it had a touch of adventure about it that made me feel quite youthful
again. I never tired of the long drives; and in the early morning the air
was like breathing champagne.
One frequently had the road to oneself at this hour, and you could
have imagined you were on a pleasure jaunt till you heard the
booming of the guns above the noise of the engine; for it did not
matter how early one was, the guns never seemed to be silent. With
a sympathetic companion in the car these runs out to the lines were
quite amongst the pleasantest features of one’s life up at the Front.
On this particular occasion the fact of being with someone with
whom I could converse freely made it still more agreeable. We went
off quite “on our own,” as Vaucher speaks Italian fluently; and as our
soldier chauffeur knew the road well, there was not much fear of our
getting lost.
We decided that it was advisable for form’s sake to call on the
Divisional General, and ask for his permission to pass through the
lines. With some little difficulty we succeeded in discovering his
Headquarters. These were, we learned, on the railway, close to the
Rubbia-Savogna Station, on the Trieste-Gorizia line.
It turned out to be about the last place where one would have
expected to unearth a General. The station itself was in ruins, and
presented a pathetically forlorn appearance, with posters and time-
tables hanging in tatters from the walls; no train had passed here for
very many months.
We left the car on the permanent way alongside the platform, and
picked our way along the track through the twisted and displaced
rails to the signalman’s “cabin,” which had been converted into the
Headquarters pro tem.
It was as unconventional and warlike as could well be imagined,
and as a subject for a picture would have delighted a military painter.
The General was a well set up, good-looking man of middle age,
and quite the most unassuming officer of his rank I have ever met.
After carefully examining our military permits to come there, he
received us with the utmost cordiality. He spoke French fluently, and
was apparently much interested in our work as war correspondents.
There was no difficulty, he said, about our going to San Martino,
but we did so at our own risk, as it was his duty to warn us that it was
still being constantly shelled; in fact, he added, the whole
neighbourhood was under fire, and he pointed out a gaping hole a
few yards away where a shell had burst only an hour before our
arrival, and had blown a small hut to atoms.
The railway station was being continually bombarded, and he was
sorry to say that he had lost a good many of his staff here.
No strategic object whatever was attained by this promiscuous
shelling; the only thing it did was to get on the men’s nerves and
make them fidgetty.
“They want to be up and doing instead of waiting about here when
their comrades have gone on ahead.”
We had quite a long talk with him, and gathered some interesting
details of the fighting that had taken place round here. He was most
enthusiastic about the moral of his troops, that no fatigue or pain can
quell.
The only difficulty the officers experienced was in getting them to
advance with caution. “Ils deviennent des tigrés une fois lancés;
c’est difficile de les retenir.”
As we bade him adieu, he asked us as a personal favour not to
mention him in any article we might write, adding modestly: “Je ne
suis qu’un soldat de l’Italie, et ne désire pas de réclame.”
There was a turning off the main road beyond Sdraussina that
passed under the railway embankment, and then went up to San
Martino del Carso.
Here there was an animated scene of military activity. A battalion
of infantry was bivouacing, and up the side of the hill, which was one
of the slopes of Monte San Michele, there was a big camp with tents
arranged in careful alignment. I mention this latter fact as it was an
unusual spectacle to see an encampment so well “pitched.”
Both the bivouac and the tents were quite protected from shell-fire
by the brow of the hill, but they would have made an easy target had
the Austrians had any aeroplanes here; doubtless, though, all
precautions had been taken by the Italian Commander in the event
of this.
Over the crest of the hill the scene changed as though by magic,
and all sign of military movement disappeared.
The desolate waste of the Carso faced us, and we were in the
zone of death and desolation. The road was absolutely without a
vestige of “cover”; it was but a track across the rocky ground, and
now wound over a series of low, undulating ridges, on which one
could trace the battered remains of trenches.
Huge shell-craters were visible everywhere, and the road itself
was so freshly damaged in places that I involuntarily recalled what
the General had told us, and wondered whether we should get back
safely.
The country was so open and uninteresting that one could see all
that there was to see for several miles ahead; it was therefore
certain no scenic surprises were awaiting you.
Meanwhile shells were bursting with unpleasant persistency round
about the road; it was not what one could term an inviting prospect
and recalled an incident that had occurred a few days previously on
this very road.
There was a good deal of firing going on as usual, and the
chauffeur of an officer’s car suddenly lost his nerve and became
completely paralysed with fear, not an altogether unusual case, I
believe.
It was a very awkward situation, as the officer knew nothing about
driving, so he was obliged to sit still for over an hour, when,
fortunately for him, a motor lorry came along, and he was extricated
from his predicament.
It seemed to me to be very purposeless going on further, since
there was absolutely nothing to sketch and still less to write about.
Since, however, we could not be far from our destination now, I
thought it best to say nothing.
But where was the village? I knew by what we had been told that
we must be close to it by now; yet there was no trace of habitation
anywhere.
The chauffeur suddenly turned round and, pointing to what
appeared to be a rugged slope just ahead, said quietly, “Ecco San
Martino del Carso.”
I am hardened to the sight of ruins by now after more than two
years at the war, but I must admit I had a bit of a shock when I
realized that this long, low line of shapeless, dust-covered rubble
actually bore a name, and that this was the place we had risked
coming out to see.
No earthquake could have more effectually wiped out this village
than have the combined Italian and Austrian batteries.
We drove up the slope to what had been the commencement of
the houses. To our surprise a motor lorry was drawn up under the
shelter of a bit of wall; two men were with it. What was their object in
being there one could not perceive, as there was no other sign of life
around.
We left the car here, and I went for a stroll round with my sketch
book, whilst Vaucher took his camera and went off by himself to find
a subject worth a photograph.
The Austrian trenches commenced in amongst the ruins: they
were typical of the Carso; only a couple of feet or so in depth, and
actually hewn out of the solid rock, with a low wall of stones in front
as a breastwork.
So roughly were they made that it was positively tiring to walk
along them even a short distance. To have passed any time in them
under fire with splinters of rock flying about must have been a terrible
ordeal, especially at night.
San Martino was certainly the reverse of interesting, and I was
hoping my comrade would soon return so that we could get away,
when the distant boom of a gun was heard, followed by the ominous
wail of a big shell approaching.
The two soldiers and our chauffeur, who were chatting together,
made a dash for cover underneath the lorry; whilst I, with a sudden
impulse I cannot explain, flung myself face downwards on the
ground, as there was no time to make for the shelter of the trench.
The shell exploded sufficiently near to make one very
uncomfortable, but fortunately without doing us any harm. A couple
more quickly followed, but we could see that the gunners had not yet
got the range, so there was nothing to worry about for the moment.
Vaucher soon returned, having had a futile walk; so we made up
for it all by taking snapshots of ourselves under fire, a somewhat
idiotic procedure.
As we drove back to Udine, we were agreed that, considering how
little there had been to see, le jeu ne valait pas la chandelle, or
rather ne valait pas le petrole, to bring the saying up to date.
A very pleasant little episode a few days later made a welcome
interlude to our warlike energies. The Director of the Censorship,
Colonel Barbarich, received his full colonelcy, and to celebrate the
event the correspondents invited him and his brother officers to a
fish déjeuner at Grado, the little quondam Austrian watering place on
the Adriatic.
We made a “day’s outing” of it; several of the younger men starting
off early so as to have a bathe in the sea before lunch.
It was glorious weather, and we had a “top hole” time. It all went
off without a hitch; the déjeuner was excellent; I don’t think I ever
tasted finer fish anywhere; the wine could not have been better, and,
of course, we had several eloquent speeches to wind up with.
There was just that little “Human” touch about the whole thing that
helped to still further accentuate the camaraderie of the Censorship,
and the good fellowship existing between its officers and the
correspondents.
Grado, though at first sight not much damaged since our visit on
the previous year, had suffered very considerably from the visits of
Austrian aircraft. They were still constantly coming over, in spite of
the apparently adequate defences, and many women and children
had been killed and many more houses demolished.
There was a curious sight in the dining room of the hotel where we
gave the lunch. The proprietor had built a veritable “funk-hole” in a
corner of the room. It was constructed with solid timber, and covered
in with sand-bags in the most approved style.
Inside were a table, chairs, large bed, lamp, food, drink, etc.; in
fact, everything requisite in case a lengthy occupation was
necessary; and there the proprietor and his wife and children would
take refuge whenever the enemy was signalled.
After lunch we were invited to make a trip in one of the new type of
submarine chasers, which are said to be the fastest boats afloat
anywhere, and went for an hour’s run at terrific speed in the direction
of Trieste; in fact, had it not been for a bit of a sea fog hanging about
we should have actually been well in sight of it. Perhaps it was
fortunate for us there was this fog on the water.
Things were a bit quiet in Udine now. Stirring incidents do not
occur every week, and the usual period of comparative inactivity had
come round again whilst further operations were in process of
development; there was but little inducement, therefore, to spend
money on petrol just for the sake of verifying what one knew was
happening up at the lines.
But I had plenty to occupy me in my studio, working on the
numerous sketches the recent doings had provided me with, till
something worth going away for turned up again.
In the interim an event of historic importance occurred.
CHAPTER XX
Declaration of war between Italy and Germany—Effect of declaration

at Udine—Interesting incident—General Cadorna consents to give
me a sitting for a sketch—The curious conditions—Methodic and
business-like—Punctuality and precision—A reminder of old days—I
am received by the Generalissimo—His simple, unaffected manner
—Unconventional chat—“That will please them in England”—My
Gorizia sketch book—The General a capital model—“Hard as
nails”—The sketch finished—Rumour busy again—A visit to
Monfalcone—One of the General’s Aides-de-camp—Start at
unearthly hour—Distance to Monfalcone—Arctic conditions—“In time
for lunch”—Town life and war—Austrian hour for opening fire—
Monfalcone—Deserted aspect—The damage by bombardment—
The guns silent for the moment—The ghost of a town—“That’s only
one of our own guns”—A walk to the shipbuilding yards—The
communication trench—The bank of the canal—The pontoon bridge
—The immense red structure—The deserted shipbuilding
establishment—Fantastic forms—Vessels in course of construction
—A strange blight—The hull of the 20,000 ton liner—The gloomy
interior—The view of the Carso and Trieste through a port-hole—Of
soul stirring interest—Hill No. 144—The “daily strafe”—“Just in
time”—Back to Udine “in time for lunch”—Return to the Carso—
Attack on the Austrian positions at Veliki Hribach—New difficulties—
Dense forest—Impenetrable cover—Formidable lines of trenches
captured—Fighting for position at Nova Vas—Dramatic ending—
Weather breaking up—Operations on a big scale perforce
suspended—Return London await events.
CHAPTER XX
On the 28th August, 1916, Italy declared war on Germany. The
declaration had, however, been so long anticipated that, so far as
one was in a position to judge, it made little or no difference in the
already existing state of affairs; since the two nations had to all
intents and purposes been fighting against each other for months,
and at Udine, at any rate, it scarcely aroused any comment outside
the Press.
However, it settled any doubts that might have existed on the
subject, and henceforth the Italians and the Huns were officially
justified in killing each other whenever they got the chance.
Curiously enough, it was through General Cadorna himself that I
learned that war had been declared. It was under somewhat
interesting circumstances, which I will relate.
I had always desired to make a sketch of the Commander-in-Chief
of the Italian Army, and with this idea had asked Colonel Barbarich at
the Censorship if he would try and arrange it for me.
He willingly agreed, but a few days after he told me that he had
done the best he could for me, but that the General had said that for
the moment he was far too occupied, but perhaps he would accede
to my desire a little later. I must therefore have patience.
This looked like a polite way of putting me off, and I accepted it as
such.
Gorizia and other subjects engaged my attention, and I had
forgotten the incident when, to my surprise, one day Colonel
Barbarich came up to me and said that if I still wished to make the
sketch of the General, His Excellency would be pleased to receive
me on the following Monday morning at eleven o’clock precisely, and
would give me a sitting of exactly half-an-hour.
“But,” added Colonel Barbarich, “you must clearly understand it is
only half-an-hour, and also that the General will not talk to you as he
will not be interviewed.”
The “following Monday” was nearly a week ahead, so this was
methodic and business-like indeed.
Of course, in spite of all the conditions attaching to the sitting, I
was delighted to find that my request had not been overlooked, so I
replied jocularly to the Colonel that failing an earthquake or the ill-
timed intervention of an Austrian shrapnel, I would certainly make it
my duty to keep the appointment.
Well, the auspicious day arrived in due course, and so did I in
good time at the Censorship to meet Colonel Barbarich, who was to
take me on to the General, whose quarters were in a palace
originally intended for the Prefect of Udine, only a short distance
away.
With the knowledge of the punctuality and precision of the
General, at l’heure militaire, that is to say, as the clock was striking
eleven, we made our way up the grand staircase to the first floor
where the General resided and had his offices.
In a large anti-chamber, with a big model of the Isonzo Front
occupying the whole of the centre, we were received by an aide-de-
camp, who evidently expected us exactly at that moment. Colonel
Barbarich briefly introduced me, then to my surprise left at once.
The Aide-de-camp took my card into an adjoining apartment, and
returning immediately, said that His Excellency General Cadorna
was waiting for me, and ushered me in.
A grey-haired officer of medium height, whom I
immediately recognised as the Generalissimo, was
reading an official document
To face page 283
Up till then nothing could have been more matter-of-fact and

business-like. It reminded me of the old days when I sought
journalistic interviews with city magnates. But the business-like
impression vanished as soon as I was inside the door.
I found myself in a very large room, well but scantily furnished.
Standing by a table, which was covered with maps, a grey-haired
officer of medium height, whom I immediately recognised as the
Generalissimo, was reading an official document.
He came forward and cordially shook hands with me in the most
informal way. I began to thank him for his courtesy in receiving me,
and was apologising for not being able to speak Italian, when he cut
me short, saying with a laugh:
“I speak but very leetle English,” but “Peut-être vous parlez
Français.” On my telling him I did he exclaimed genially:
“A la bonheur, then we will speak in French. Now what do you
want me to do? I am at your service.”
His simple and unaffected manner put me at once at my ease and
made me instantly feel that this was going to be a “sympathetic”
interview, and not a quasi official reception.
I must mention that I had asked Colonel Barbarich to explain that I
did not want to worry the General, but would be quite content if I
were permitted to make a few jottings in my sketch book of him at
work and some details of his surroundings. This, as I have
explained, was granted, but with the curious proviso that I was not to
talk whilst I was there.
It came, therefore, as a very pleasant surprise to find myself
received in this amicable fashion, and the ice being thus broken, I
said, I should like to sketch him reading a document, as I found him
on entering the room. He willingly acquiesced, and I at once started
my drawing as there was no time to lose.
With the recollection of the stipulation that I was not to open my
mouth during the sitting, and that I was only allowed half-an-hour I
went on working rapidly and in silence.
But I soon found that the General was not inclined to be taciturn,
and in a few moments we were chatting in the most unconventional
manner as if we were old friends. As a matter of fact, I shrewdly
suspected him of interviewing me.
When he learned how long I had been on the Italian Front he was
much interested, and was immediately anxious to know what I
thought of his soldiers. Were they not splendid? He put the question
with all the enthusiasm and affection of a father who is proud of his
children.
As may be imagined, I had no difficulty in convincing him that I
have a whole-hearted admiration for the Italian Army after what I had
seen of its wonderful doings at Gorizia and elsewhere.
It was then that he gave me the news of the declaration of war
between Italy and Germany; the morning papers had not published it
in the early editions.
“That will please them in England,” he remarked with a laugh. I
agreed with him that it would, although it had long been expected.
The mention of England reminded me that he had just returned
from the war conference in London, so I asked if he had ever been
there before.
“Yes,” he replied with a humorous twinkle in his eye, “forty years
ago; but I do not remember much of it; although my father was
ambassador to England I only lived with him in London for a short
while. It is, of course, much changed since then.”
Whilst thus chatting I was working with feverish haste at my
sketch.
I now noticed he was getting a bit impatient at keeping the same
position, so I suggested a few moments rest. He came over to see
how I had got on, and asked if he might look through my sketch
book.
It happened to be the one I had used at Gorizia, and the sketches
I had made that day pleased him very much.
“You were fortunate, you were able to see something; I never see
anything,” he remarked quite pathetically.
I felt there was no time to lose if I wanted to get finished in the
half-hour, so hinted at his resuming the pose for a few minutes
longer. He did so at once, and I ventured to tell him in a joking way
that he would make a capital model.
“Well, I am as active now as I ever was,” he replied, taking me
seriously, “and I can ride and walk as well now as I could when I was
a young man.”
This I could well believe, because he looks as “hard as nails,” and
chock full of energy and determination, as the Austrian generals
have discovered to their undoing.
I had now completed my rough sketch sufficiently to be able to
finish it in the studio.
The General expressed his gratification at my having done it so
rapidly, so I suggested another ten minutes some other day to put
the finishing touches.
“Come whenever you like, I shall always be pleased to see you,”
replied His Excellency genially.
Although, as I have said, there was no outward evidence of the
declaration of war making any difference in the conduct of the
campaign, rumours soon began to be persistently busy again, and it
became pretty evident that something big was going to happen on
the Carso before the weather broke up and the autumn rains set in
and put a stop to active operations for some time.
There had been a good deal of talk of operations pending in the
vicinity of Monfalcone, so I got permission to accompany a Staff
Officer who was going there one morning. I had always wanted to
see the place and its much talked of shipbuilding yards, but curiously
enough this was the first opportunity I had had of going there.
My companion was one of General Cadorna’s aides-de-camp, so
we went in one of the big cars belonging to Headquarters. We
started at the usual unearthly hour to which one had become
accustomed and which, as I have pointed out, is delightful in the
summer, but is not quite so fascinating on a raw autumn morning
before sunrise.
I was very disappointed when I learned that we should probably be
back in Udine “in time for lunch” unless something untoward
occurred to force us to stay away longer; as I had been looking
forward to an extended run that would last the whole day, but as I
was practically a guest on this occasion I could say nothing. My
companion, like so many Italian officers, spoke French fluently, and
turned out to be a very interesting fellow; and as he had been
stationed for some time at Monfalcone before going on the Staff, he
knew the district we were making for as well as it was possible to
know it.
The distance from Udine to Monfalcone is, roughly, the same as
from London to Brighton, and we went via Palmanova, Cervignano,
and Ronchi.
It was a bitterly cold morning, with an unmistakable nip of frost in
the air, so although I was muffled up to my ears I was gradually
getting frozen, and my eyes were running like taps. It may be
imagined, therefore, how I was envying my companion his big fur-
lined coat.
I had arrived at the Front in the hottest time of the year, so had
taken no precautions against Arctic conditions. Motoring in Northern
Italy in an open car during the winter months must be a very trying
ordeal indeed, if what I experienced that morning was any criterion of
it.
As we sped along I asked the Aide-de-camp if there was any
particular reason for his starting off so early, and if it was absolutely
necessary for us to be back “in time for lunch.” To my mind the very
thought of it took the interest off the trip and brought it down to the
level of an ordinary pleasure jaunt, which was to me particularly
nauseating.
After all these months at the Front I have not yet been able to
accustom myself to the combination of every-day town life and war,
and I am afraid shall never be able to. Doubtless it is a result of old
time experiences.
My companion treated my query somewhat lightly. “You will be
able to see all there is to see in and round Monfalcone in three
hours,” he replied, “so what is the use therefore of staying longer?
Moreover,” he added seriously, “the Austrian batteries have made a
practice of opening fire every morning at about eleven o’clock, and
usually continue for some hours, so there is the risk of not being able
to come away when one wants to.”
There was, of course, no reply possible, and the more especially
as I am not exactly a glutton for high explosives, as will have been
remarked.
Monfalcone is a nice bright little town, typically Austrian, and
before the war must have been a very busy commercial centre.
When I was there it was absolutely deserted, with the exception of
a few soldiers stationed there. The shops were all closed, grass was
growing in the streets, and it presented the usual desolate
appearance of a place continually under the menace of
bombardment.
The damage done to it up till then was really unimportant
considering the reports that had been spread as to its destruction.
Many houses had been demolished, as was to be expected, but I
was surprised to find how relatively undamaged it appeared after the
months of daily gun-fire to which it had been subjected.
We left the car in a convenient courtyard where it was under cover,
and made our way to the Headquarters of the Divisional
Commandant, where, as a matter of etiquette I had to leave my card.
For the moment the guns were silent, and there was a strange
quietude in the streets that struck me as being different to anything I
had noticed anywhere else, except perhaps in Rheims during the
bombardment when there was an occasional lull.
One had the feeling that at any moment something awful might
happen. Even the soldiers one met seemed to me to have a
subdued air, and the drawn expression which is brought about by
constant strain on the nerves.
Instinctively one walked where one’s footsteps made the least
noise, in order to be able to hear in good time the screech of an
approaching shell.
It had turned out a lovely day, and in the brilliant sunshine
Monfalcone should have been a bright and cheerful place, instead of
which it was but the ghost of a town with the shadow of death
continually overhanging it.
The peaceful stillness was not to be of long duration. Silence for
any length of time had been unknown in Monfalcone for many a long
day.
Whilst we were having a talk with the officers at Headquarters
there was a loud detonation, apparently just outside the building. To
my annoyance I could not restrain an involuntary start, as it was
totally unexpected.
“That’s only one of our guns,” remarked, with a smile, a Major with
whom I was chatting, and who had noticed the jump I made. “The
Austrians won’t commence for another couple of hours at least,” he
added.
My companion and I then started off to walk down to the
shipbuilding yard, about a mile and a half from the town, and which
was, of course, the principal sight of the place.
One had not gone far when one had some idea how exposed was
the position of Monfalcone. A deep communication trench
commenced in the main street and continued alongside the road the
whole way down to the port—no one was allowed to walk outside it.
The object of this was to prevent any movement being seen from
the Austrian batteries, which were only a comparatively short
distance away, though it must have been no secret to them what was
going on in Monfalcone.
The Italian guns were now getting busy, and the noise was
deafening, but still there was no response from the enemy; it was
evidently true that he worked to time, and it was not yet eleven
o’clock.
Although only a mile and a half, the walk seemed longer because
one could see nothing on either side, the walls of the trench being
quite six feet high; but at last we came out on the bank of what
looked like a broad canal. This is part of a waterway constructed to
connect up the port with the railway.
The communication trench now took the form of a sunken pathway
winding along the bank under the trees, and was quite picturesque in
places.
At last we reached the end, and facing us was the Adriatic as calm
as a lake, and away on the horizon one could see the hills that guard
Trieste.
We crossed the mouth of the canal by a pontoon bridge, which I
believe had been abandoned by the Austrians when they evacuated
Monfalcone at the beginning of the war.
A short distance ahead, towering above a conglomeration of long
sheds on the low-lying ground, was an immense red structure, the
outlines of which recalled something familiar. As one got nearer one
saw that it was the unfinished steel hull of a gigantic ocean liner, and
that the red colouring was caused by the accumulation of rust from
long exposure.
We soon reached the entrance to a vast shipbuilding
establishment. There were no bolts or bars to prevent our walking in.
The whole place was deserted, and all around us was a spectacle
of ruin and desolation that was more impressive than actual havoc
caused by bombardment.
In the immense workshops the machinery was rotting away; on the
benches lay the tools of workmen; strange metal forms, portions of
the framework of big ships lay here and there on the sodden ground
like huge red skeletons of ante-deluvian animals.
Many vessels had been in the course of construction, mostly for
the Mercantile fleet of Austria, though there were some destroyers
and war-craft on the stocks as well. Rust, of a weird intensity of
colour. I had never seen before, was over everything like a strange
blight.
Alongside the sheds was the hull of the big liner one had seen in
the distance. A 20,000 ton boat, I was told, which was being built for
the Austrian Lloyd Line.

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Mean Field Game and its Applications in Wireless

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My War Memoirs 1939 1945 210 Field Company Royal

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Shrinkage Estimation for Mean and Covariance Matrices

First edition published 2022

and by CRC Press

© 2022 Julian Barreiro-Gomez and Hamidou Tembine

CRC Press is an imprint of Taylor & Francis Group, LLC

ISBN: 9780367566128 (hbk)

ISBN: 9781032128047 (eBook+ Enhancements)

Typeset in CMR10 font

Access the Support Material: www.routeldge.com/9780367566128

To Yandai, Pama, Marie-Claire,

List of Figures xiii

List of Tables xxiii

Author Biographies xxxi

1.4 Partial Integro-Differential System for a Mean-Field-Type Con-

II Mean-Field-Free and Mean-Field Games 37

III One-Dimensional Mean-Field-Type Games 103

5 Co-opetitive Mean-Field-Type Games 139

6 Mean-Field-Type Games with Jump-Diffusion and Regime

7 Mean-Field-Type Stackelberg Games 199

8 Berge Equilibrium in Mean-Field-Type Games 229

IV Matrix-Valued Mean-Field-Type Games 249

10 A Class of Constrained Matrix-Valued Mean-Field-Type

10.2 Semi-explicit Solution of the Constrained Mean-Field-Type

V Discrete-Time Mean-Field-Type Games 297

12 Matrix-Valued Discrete-Time Mean-Field-Type Games 321

VI Learning Approaches and Applications 359

14 Mean-Field-Type Model Predictive Control 379

14.2.4 Decomposition and Stability . . . . . . . . . . . . . . 385

15 Data-Driven Mean-Field-Type Games 389

16.6.2 Multiple-Decision-Maker Problem . . . . . . . . . . . 470

1.1 Diversification problem with two assets. . . . . . . . . . . . . 8

2.1 A basic scalar-valued control scheme. . . . . . . . . . . . . . 40

2.2 Feedback scheme for the linear-quadratic mean-field-free op-

3.1 General Mean-Field Game Philosophy in either continuous

4.1 General scheme of the non-cooperative mean-field-type game.

4.9 Evolution of the Riccati equations γ1 , . . . , γ3 for the scalar-

5.12 Evolution of the optimal strategies and their expectation for

6.1 General scheme of the non-cooperative mean-field-type game

7.1 General scheme of the two-player Stackelberg mean-field-type

8.1 General scheme of the two-player Berge mean-field-type game

9.1 Example of a matrix-valued application with d = 4. . . . . . 253

9.10 Evolution of the Riccati equations δ1 , and δ2 for the matrix-

14.1 Graphical example for the sets X, Dx , and Xc . . . . . . . . . 383

15.1 General scheme for a data-driven mean-field-type game prob-

16.1 Illustrative example. Five-tank benchmark involving two play-

16.27 Mass of people motion with an initial distribution x(0) evac-

8.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . 244

15.1 Summary of available data for machine-learning purposes. . 393