# Strategic Interactions on Networks with a High Population of Agents

**Tid: **
To 2019-11-14 kl 11.00 - 12.00

**Plats: ** F3, Lindstedtsvägen 26

Speaker: Tamer Başar

Director, Center for Advanced Study (CAS)

Swanlund Endowed Chair

CAS Professor of Electrical and Computer Engineering

Coordinated Science Laboratory

University of Illinois, Urbana, Illinois, 61801 USA

Abstract:

Perhaps the most challenging aspect of research on multi-agent dynamical systems, formulated as non-cooperative stochastic differential/dynamic games (SDGs) with asymmetric dynamic information structures is the presence of strategic interactions among agents, with each one developing beliefs on others in the absence of shared information. This belief generation process involves what is known as second-guessing phenomenon, which generally entails infinite recursions, thus compounding the difficulty of obtaining (and arriving at) an equilibrium. This difficulty is somewhat alleviated when there is a high population of agents (players), in which case strategic interactions at the level of each agent become much less pronounced. This leads, under some structural constraints, to what is known as mean field games (MFGs), which have been the subject of intense research activity during the last 10 years or so.

MFGs constitute a class of non-cooperative stochastic differential games where there is a large number of players or agents who interact with each other through a mean field coupling term—also known as the mass behavior or the macroscopic behavior in statistical physics—included in the individual cost functions and/or each agent’s dynamics generated by a controlled stochastic differential equation, capturing the average behavior of all agents. One of the main research issues in MFGs with no hierarchy in decision making is to study the existence, uniqueness and characterization of Nash equilibria with an infinite population of players under specified information structures and further to study finite-population approximations, that is to explore to what extent an infinite-population Nash equilibrium provides an approximate Nash equilibrium for the finite-population game, and what the relationship is between the level of approximation and the size of the population.

Following a general overview of the difficulties brought about by strategic interactions in finite-population SDGs, the talk will dwell on two classes of MFGs: those characterized by risk sensitive (that is, exponentiated) objective functions (known as risk-sensitive MFGs) and those that have risk-neutral (RN) objective functions but with an additional adversarial driving term in the dynamics (known as robust MFGs). In stochastic optimal control, it is known that risk-sensitive (RS) cost functions lead to a behavior akin to robustness, leading to establishment of a connection between RS control problems and RN minimax ones. The talk will explore to what extent a similar connection holds between RS MFGs and robust MFGs, particularly in the context of linear-quadratic problems, which will allow for closed-form solutions and explicit comparisons between the two in both infinite- and finite-population regimes and with respect to the approximation of Nash equilibria in going from the former to the latter. The talk will conclude with a brief discussion of several extensions of the framework, such as to hierarchical decision structures with a small number of players at the top of the hierarchy (leaders) and an infinite population of agents at the bottom (followers) as well as to games where players make noisy observations.

Sponsored by the Digitalization Platform

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