Submodular mean field games: Existence and approximation of solutions

Abstract

Background. Rectum-preservation for locally advanced rectal cancer has been proposed as an alternative to total mesorectal excision (TME) in patients with major (mCR) or complete clinical response (cCR) after neoadjuvant therapy. The purpose of this study was to report on the short-term outcomes of ReSARCh (Rectal Sparing Approach after preoperative Radio-and/or Chemotherapy) trial, which is a prospective, multicenter, observational trial that investigated the role of transanal local excision (LE) and watch-and-wait (WW) as integrated approaches after neoadjuvant therapy for rectal cancer.

Interpolation results for pathwise Hamilton-Jacobi equations

“…The well-posedness of both strong and weak solutions of MFG master equations in the continuum state space setting has been explored under various sets of monotonicity assumptions [1,8,11,15,16,18,31,32,34,35,47]. The approach in our setting, which involves appealing to Tarski's fixed point theorem for increasing functions on lattices [54], has also been taken in the continuum state space setting, where maximal and minimal solutions were found under related assumptions; see for instance [25][26][27]48]. The partial order used in [48] comes from the notion of stochastic dominance for probability measures.…”

A note on stochastic dominance and compactness

“…Strassen [15]) and, recently, submodular mean field games (cf. Dianetti et al [6]). It is therefore desirable to obtain a more tractable characterization of complete (w.r.t.…”

“…first and second order stochastic dominance in terms of compactness in the weak topology and in the Wasserstein-1 topology, respectively (Theorem 3.5 and Theorem 3.12). Dianetti et al [6] propose a lattice-theoretical approach to mean field games using Tarski's fixed point theorem. The crucial step in their analysis is to show that an appropriately chosen lattice L of flows of probability measures is complete.…”