@MISC{Kolokoltsov_nonlinearmarkov, author = {Vassili N. Kolokoltsov}, title = {NONLINEAR MARKOV GAMES ∗}, year = {} }

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Abstract

I am going to put forward a program of the analysis of a new class of stochastic games that I call nonlinear Markov games, as they arise as a (competitive) controlled version of nonlinear Markov processes (an emerging field of intensive research, see e.g. [3], [4],[5]). This class of games can model a variety of situation for economics and epidemics, statistical physics, and pursuit- evasion processes. The discussion below is mostly taken from the author’s monograph in preparation [1]. I shall start by introducing the (yet not very well known) concept of nonlinear Markov chains. 1 Nonlinear Markov chains A discrete in time and space nonlinear Markov semigroup Φ k, k ∈ N, is specified by an arbitrary continuous mapping Φ: Σn → Σn, where the simplex Σn = {µ = (µ1,..., µn) ∈ R n +: n∑ µi = 1} represents of course the set of probability laws on the finite state space {1,...,.n}. For a measure µ ∈ Σn the family µ k = Φ k µ can be considered as an evolution of measures on {1,...,.n}. But it does not yet define a random process (finite-dimensional distributions are not specified). In order to get a process one has to choose a stochastic representation for Φ, i.e. to write it down in the form Φ(µ) = {Φj(µ)} n j=1 = { i=1 n∑ Pij(µ)µi} n i=1, (1.1) where Pij(µ) is a family of stochastic matrices ( ∑d j=1 Pij(µ) = 1 for all i), depending on µ (nonlinearity!), whose elements specify the nonlinear transition probabilities. For any given Φ: Σn ↦ → Σn a representation (1.1) exists, but is not unique. For instance, there exists a unique representation (1.1) with an additional condition that all matrices Pij(µ) = ˜ Pij(µ) are one dimensional: ˜Pij(µ) = Φj(µ), i, j = 1,..., n. (1.2)