International Society of Dynamic Games

Dynamic Games and Applications Seminar

Guaranteed cost equilibrium in infinite horizon linear-quadratic differential games

March 7, 2024 11:00 AM — 12:00 PM (Montreal time)

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In this work, we study infinite-horizon linear-quadratic differential games with an output feedback information structure. Our motivation for studying these games arises from their applications in engineering, where players may lack access to complete state information. For example, in a large-scale networked multi-agent system, agents may only possess information about their neighboring agents. In the literature, sufficient conditions for the existence of output feedback Nash equilibria are closely related to solvability of a set of coupled algebraic Riccati equations, with the requirement that these solutions admit certain structural conditions. fulfilling these conditions poses a significant challenge, even in low-dimensional games. Given these limitations, a natural question arises regarding the existence of a broader class of output feedback strategies that adhere to an equilibrium property. Here, ‘broader’ implies that this expanded set of strategies, if it exists, encompasses the output feedback Nash strategies. To address this problem, we introduce the concept of an output feedback guaranteed cost equilibrium. These strategies not only ensure that individual costs remain bounded by a predefined threshold (a design parameter) but also maintain an equilibrium property. The design of these strategies utilizes techniques developed for suboptimal static output feedback controllers and employs linear matrix inequality-based methods for computation.

Dynamic Games and Applications Seminar

A bottom-up approach to the construction of socially optimal discrete choices under congestion

February 29, 2024 11:00 AM — 12:00 PM (Montreal time)

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We consider the problem of N agents having a limited time to decide on a destination choice among a finite number of alternatives D. The agents attempt to minimize collective energy expenditure while favoring motion strategies which limit crowding along their paths in the state space. This can correspond to a situation of crowd evacuation or a group of micro robots distributing themselves on tasks associated to distinct geographic locations. We formulate the problem as a Min linear quadratic optimal control problem with non-positive definite Q matrices accounting for negative costs accruing from decreased crowding. The solution proceeds in three stages, each one improving on the performance of the previous stage: (i) Mapping optimal paths for an arbitrary agent destination assignment; (ii) Mapping optimal paths for fixed fractions of agents assigned to each destination; (iii) Identifying the optimal fraction of agents’ assignments to each destination. The cost function associated with stage (iii) as N goes to infinity is proven to be convex, leads to simplified computations and to epsilon-optimal decentralized control policies when applied for N large.

(with Noureddine Toumi and Jérôme Le Ny).

Dynamic Games and Applications Seminar

On the Non-Uniqueness of Linear Markov Perfect Equilibria in Linear-Quadratic Differential Games: A Geometric Approach

February 22, 2024 11:00 AM — 12:00 PM (Montreal time)

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Although the possibility of multiple nonlinear equilibria in linear-quadratic differential games is extensively discussed, the literature on models with multiple linear Markov perfect equilibria (LMPEs) is scarce. And indeed, almost all papers confined to a single state (a very large majority of the application of differential games to economic problems) find a unique LMPE. This paper explains this finding and derives conditions for multiplicity based on the analysis of the phase plane in the state and the derivative of the value function. The resulting condition is applied to derive additional examples using pathways different from the (two) known ones. All these examples, more precisely, their underlying pathways, contradict usual assumptions in economic models. However, by extending the state space, we provide an economic setting (learning by doing) that leads to multiple LMPEs.