Simple model of market share dynamics based on clients' firm-switching decisions

Firms compete for clients, creating distributions of market shares ranging from domination by a few giant companies to markets in which there are many small firms. These market structures evolve in time, and may remain stable for many years before a new firm emerges and rapidly obtains a large market share. We seek the simplest realistic model giving rise to such diverse market structures and dynamics. We focus on markets in which every client adopts a single firm, and can, from time to time, switch to a different firm. Examples include markets of cell phone and Internet service providers, and of consumer products with strong brand identification. In the model, the size of a particular firm, labelled $i$, is equal to its current number of clients, $n_i$. In every step of the simulation, a client is chosen at random, and then selects a firm from among the full set of firms with probability $p_i = (n_i^\alpha + \beta)/K$, where $K$ is the normalization factor. Our model thus has two parameters: $\alpha$ represents the degree to which firm size is an advantage ($\alpha$ > 1) or disadvantage ($\alpha$