By Sydney Afriat
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Additional info for Market: Equilibrium, Stability, Mythology (Routledge Frontiers of Politicaleconomy, 44)
If a is a distribution matrix, and ν ∈ , then also νa ∈ . From aI = I we have (νa)I = ν(aI ) = νI. So from νI = 0, also (νa)I = 0, and so we can consider also |νa|. Theorem If a is a distribution matrix and ak is the smallest element in column k, and if ν is any vector for which νI = 0, then |νa| ≤ |ν|(1 − ak ). It can be taken that i i νi aik ≥0 since otherwise ν can be replaced by −ν. Now we have ii a ≥ 0 iii aI = I and definitions of |ν| and ak iv νI = 0 so that “chap02” — 2002/8/9 — 10:37 — page 23 — #15 24 Bibliography |νa| = ≤ = j =k j =k i j νi aik + i i ≤ |ν| − i |νi |aij + |νi |aij − i i i νi aik ∵ (i) νi aik ∵ (ii) (|νi | − νi )aik (|νi | − νi )ak = |ν|(1 − ak ) ∴ ∵ (iii) ∵ (iv) |νa| ≤ |ν|(1 − ak ) Corollary If also g 1 , .
A Sperner labelling for a simplicial dissection T of S is a function L which assigns a Sperner label L(x) to every vertex x of T . Then a Sperner simplex is a simplex of T whose vertex labels describe all the vertices of S. Theorem (Sperner) For any simplex S and any simplicial dissection T of S, and any Sperner labelling of the vertices of T , the number of Sperner simplices is odd, and consequently there exists at least one. First we will see what it means for a 1-simplex S, with vertices 0, 1.
In this section we deal with a square row-distribution matrix. With a Markov process the distribution matrix involved is formed by the transition probabilities, and is square. The original interest of distribution matrices derives from this context, and so does the theory. In such a probability context they are called stochastic matrices. By a transition matrix here we can mean a square distribution matrix, free of the probabilistic connection. For a typical Markov process, individuals in a population have n possible states, and from one period to the next an individual in state i will make a transition to state j with probability aij , so that aij ≥ 0, aij = 1 for all i.