Comparing the best-reply strategy and mean-field games: The stationary case

Abstract

Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

A mean field game model of firm--level innovation

“…Proof. The existence part of the proof uses the theory of upper and lower solutions, specifically Theorem 4.3. in [20], and follows along similar lines to the proof of Proposition 3.12 in [6]. This shows that a solution V k ∈ W 1,p (Ω) to the auxiliary HJB PDE exists, for some p ≥ 1, provided the following hold true:…”

“…As noted in the introduction, this proof is based on the proof of existence and uniqueness in [6]. The proof presented here has some technical differences compared with the one in [6], hence it is reproduced in full. However, it follows a similar framework and so we do not claim the proof to be new.…”

“…Therefore, a solution to the auxiliary Fokker-Planck equation (3.6) exists, it is given by m k , and m k ∈ C 2 (Ω) ∩ C 1 Ω . To prove uniqueness we follow the same proof in [6]. For brevity we only outline the argument here.…”

“…However, they rely heavily on the ability to write a stationary Fokker-Planck equation in the form of an exponential. This characteristic has previously been used in [6] to prove existence and uniqueness in MFG and BRS models in a slightly different framework.…”