4 Presentations

• 
  01:30 PM - 01:55 PM

  Hilbert Space-Valued LQ Mean Field Games: An Infinite-Dimensional Analysis

  • Hanchao Liu, presenter, HEC Montréal
  • Dena Firoozi, HEC Montréal

  In this talk we present a comprehensive study of Hilbert space-valued Linear-Quadrati (LQ) mean field games (MFGs), generalizing the classic LQG mean field game theory to scenarios where the state equations are driven by infinite-dimensional stochastic equations. In this framework, state and control processes take values in separable Hilbert spaces. Moreover, the state equations involve infinite dimensional noises, namely Q-Wiener processes. All agents are coupled through the average state of the population appearing in their linear dynamics and quadratic cost functional. In addition, the diffusion coefficient of each agent involves the state, control, and the average state processes. We first discuss the well-posedness of a system of coupled infinite-dimensional stochastic evolution equations, which forms the foundation of MFGs in Hilbert spaces. Next, we present the Nash Certainty Equivalence principle and obtain a unique Nash equilibrium for the limiting Hilbert space-valued MFG. Finally, we establish the $\epsilon$-Nash property for the finite-player game in Hilbert space.

• 
  01:55 PM - 02:20 PM

  Sparse Network Mean Field Games: Ring and Lattice Structures and Related Topologies

  • Peter Caines, presenter, GERAD - McGill University
  • Minyi Huang, Carleton University

  In order to study Mean Field Games on large sparse and dense networks, a graph limit concept related to, but distinct from, graphons has been introduced in [Caines, CDC 2022]. Specifically, for sequences of networks modelled as being embedded in the unit cube in some m-dimensional Euclidean space (more generally, compact sets in Riemannian manifolds), there exist (weak) measure limits of sequences of empirical measures of vertex densities (vertexons) on the unit cube and associated (weak) measure limits of sequences of empirical measures of edge densities (graphexons) on a cube in 2m dimensional space. The resulting extension of the Graphon Mean Field Game (GMFG) theory of [Caines-Huang, SICON, 2021] yields the vertexon-graphexon MFG (GXMFG) theory. In particular, for sparse limit graphexons, the existence and uniqueness of solutions for LQG GXMFG examples are presented where the influence between agent populations on neighbouring nodes is modelled via second order PDEs defined on graph limit edges.

• 
  02:20 PM - 02:45 PM

  Large-population risk-sensitive linear quadratic control and asymptotic behaviour

  • Yu Wang, presenter,
  • Minyi Huang, Carleton University

  We study a risk-sensitive linear quadratic optimal control problem where a large number of N agents have mean-field interactions. We derive the centralized optimal control law and the resulting decentralized individual control law by passing to the mean-field limit. The performance difference between the two sets of control laws does not vanish, and instead has an upper bound depending on the noise intensity and the risk sensitivity parameter. This phenomenon has connections with large deviations and is in contrast to a risk-sensitive mean-field game in which the decentralized strategy results in a peroformance loss of O(1/N) for an individual player.

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  02:45 PM - 03:10 PM

  Modelling Early-Stage Venture Investments with Linear Quadratic Gaussian Quantilized Principal-Agent Mean Field Games.

  • Rinel Foguen Tchuendom, presenter, Ecole Polytechnique à Montréal
  • Dena Firoozi, HEC Montréal
  • Michèle Breton, GERAD, HEC Montréal

  In this talk, we propose to model venture investments using quantilized mean field games. We consider a venture investor (typically an incubator/accelerator) who supports a large number of homogeneous early-stage entrepreneurs. These venture investments are typically staged such that after a set finite horizon only a fixed proportion of top ranking entrepreneurs obtain further investments. We formulate a linear quadratic Gaussian quantilized principal-agent mean field game whose solution consists of the principal being in a Stackelberg equilibrium with the agent who is in a mean field game equilibrium. The principal models the venture investor, the agent models a generic early-stage entrepreneur, and the quantilization models the top ranking feature. The choice of linear dynamics, quadratic costs and Gaussian stochasticity is natural and allows for the existence of explicit solutions. We consider two cases, one where the agents are least committed to be top ranking and one where they are most committed. We find that in both cases, the support of the principal is constant and independent of the fixed proportion of top ranking agents obtaining further investments. When agents are least committed, we present a simulation illustrating a negative relationship between the fixed proportion and the global performance of agents. Such a relationship is not available when the agents are most committed.