English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Mean field game theory has been developed largely following two routes. Oneof them, called the direct approach, starts by solving a large-scale game andnext derives a set of limiting equations as the population size tends toinfinity. The second route is to apply mean field approximations and formalizea fixed point problem by analyzing the best response of a representativeplayer. This paper addresses the connection and difference of the twoapproaches in a linear quadratic (LQ) setting. We first introduce an asymptoticsolvability notion for the direct approach, which means for all sufficientlylarge population sizes, the corresponding game has a set of feedback Nashstrategies in addition to a mild regularity requirement. We provide a necessaryand sufficient condition for asymptotic solvability and show that in this casethe solution converges to a mean field limit. This is accomplished bydeveloping a re-scaling method to derive a low dimensional ordinarydifferential equation (ODE) system, where a non-symmetric Riccati ODE has acentral role. We next compare with the fixed point approach which determines atwo point boundary value (TPBV) problem, and show that asymptotic solvabilityimplies feasibility of the fixed point approach, but the converse is not true.We further address non-uniqueness in the fixed point approach and examine thelong time behavior of the non-symmetric Riccati ODE in the asymptoticsolvability problem.

Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach