Repeated play and Newton's method. (Q2701830)

The paper deals with the problem of solving an equation: (1) \(0 = m(x)\) where \(x\to m(x)\) maps an Euclidean space \(E\) into itself, and the vector \(x\) is composed of finitely many disjoint blocks \(x_i\) which belong to \(E_i\) for all \(i\) form the set \(I\). Thus (1), assumes the system form \(0 = m_i(x)\) for all \(i\) from the set \(I\), (2) \(m_i: E \to E_i\) being the component map of \(m\) that corresponds to \(x_i\). The author considers that the set \(I\) comprises non-communicating, non-informed different agents, or processors, or individuals \(i\), each taking on the task to control and monitor merely his component \(x_i\). In practice, these individuals may lack competence or information, so it takes them some time and adaptation before they can eventually solve (2). The objectives of the paper are to model a relevant sort of adaptation, driven merely by local, idiosyncratic information and to ensure that the process converges, at least locally, to a solution of (1). Although the process is based on Newtons method, it proceeds without matrix inversion and dispenses with the need to exchange information between various blocks. The paper provides both a continuous-time version and a discrete-time version of the procedure.