Multi-Agent Average Convergence

Let I = [0, N − 1] denote the set of indices. We define the set of states S to contain all maps of type I → R for fixed positive N ∈ N (representing the number of agents). For each s ∈ S, i ∈ I, s(i) represents the value held by the agent with ID i. We denote the arbitrarily fixed start state s0. We introduce the function g whose domain MR is the set of non-empty finite multisets of real numbers and whose range is the reals. Given such a multiset V ∈ MR, g(V ) = 1 |V | ∑ v∈V v. In other words, g computes the average of a non-empty finite multiset of reals. The function map : (I → R) × P(I) → MR is defined as in the accompanying PVS work’s set map [1] recursively: