Full ordering in the Shorrocks mobility sense of the semiring of monotone doubly stochastic matrices (Q550660)

The transition matrix \(P=[ p_{ij}]_{n \times n}\) defining completely the finite homogeneous Markov chain has elements \(p_{ij}\) which describe the probability of a movement from the state \(s_i\) to the state \(s_j\). The transition matrices are stochastic, i. e., the sum of elements of every transition matrix row equals 1. \textit{A. F. Shorrocks} [Econometrica~46, 1013--1024 (1978; Zbl 0391.90033)] defined mobility indices on a set of transition matrices \({\mathcal T}\) as a continuous function \(M: {\mathcal T} \rightarrow {\mathbb R}\). In this paper, the ordering on a set of monotone doubly stochastic transition matrices by forming a semiring in which mobility measure induces an ordering in the Shorrocks' sense is investigated. It is proved that there is a class of the equilibrium mobility indices which induces the full ordering in the semiring. The introduced ordering is compatible with \textit{V. Dardanoni}'s partial ordering [Soc. Choice Welfare 12, No.~2, 181--192 (1995; Zbl 0821.90023)] on a domain of monotone transition matrices. Mobility measures that induce the full ordering on the semiring satisfy three basic properties of the measuring the equality of life chances in intergenerational mobility: immobility, monotonicity and perfect mobility.