Recent advances in mean-field games and its multi-population variants enable large-scale heterogeneous multi-agent systems to be modeled through representative agents and their associated mean-field distributions. However, existing approaches do not explicitly account for uncertainty in the behavior of other populations. To this end, we introduce a new paradigm: risk-averse multi-population mean-field games, where each population optimizes a worst-case expected reward over dynamically feasible ambiguity sets of mean-field flows of a subset of the other populations. Employing an occupation-measure formulation along with tools from set-valued analysis, we establish, under mild assumptions, several theoretical properties of the multi-population game, including the geometric properties of the ambiguity sets and the existence of a novel risk-averse multi-population mean-field equilibrium. Further, we derive contractivity results of the fixed-point operator under entropy regularization and show that it can be utilized to learn the equilibrium. Finally, we propose a risk-averse fictitious-play scheme and show that exploitability decays to zero, despite the additional nonlinearity introduced by the worst-case objective. We report several numerical experiments to illustrate convergence and risk-averse behavior.
Figures & tables
MP-
ER-MP-
RA-MP-
ER-RA-MP-
MFE
MFE
MFE
MFE
Grid
Nominal
7.67
7.60
0.35
0.37
Navigation
ΔEQ
7.65
7.58
0.29
0.35
3 Population
Nominal
12.13
11.73
12.05
11.77
Chasing
ΔEQ
1.16
1.91
1.10
2.0
4 Population
Nominal
11.91
11.59
11.78
11.58
Table 1 . Expected rewards under nominal mean-field flow (μtj,EQ,μt−j,EQ) and sensitivity of corresponding equilibrium policies as defined in ( 21 ).
Figure 1 . (a) RA-FPI; (b) RA-FP; (c)-(f) population distributions of Navigation game.
Figure 2 . (a)-(b) RA-FPI; (c)-(d) RA-FP; (e) population distributions of 4 Population Chasing.
Figure 3 . Exploitability for Stock Market Trading.
Figure 4 . (a) Varying ε under σ=1 ; (b) Varying σ under ε=0.1 ; (c) Cumulative rewards.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5 . Our primary theoretical developments.
Figure 6 . A roadmap of the proof steps for Theorem 1 .
Center for Data Science, New York University and NYU Shanghai, New York, United States of America · Shanghai Center for Data Science; NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai; NYU Shanghai, Shanghai, People’s Republic of China.
Department of Electrical and Computer Engineering, University of Central Florida, Orlando, Florida, USA. · Department of Computer Science, University of Central Florida, Orlando, Florida, USA.