Probabilistic flows of inhabitants in urban areas and self-organization in housing markets

We propose a very simple probabilistic model to explain the spatial structure of the rent distribution of housing market in city of Sapporo. Here we modify the mathematical model proposed by Gauvin et. al. [1]. In their mathematical modeling, they utilized several assumptions to describe the decision making of each inhabitant in Paris. Namely, they assumed that the intrinsic attractiveness of a city depends on the location and there exists a single peak at the center. They also used the assumption that each inhabitant tends to choose the place where the other inhabitants having the similar or superior income to himself/herself are living. In order to find the best possible (desirable) place to live, each buyer in the system moves from one place to the other according to the transition (aggregation) probability described by the above two assumption, and he/she makes a deal with the seller who presents the best possible condition for the buyer. They concluded that the resultant self-organized rent distribution is consistent with the corresponding empirical evidence in Paris. However, it is hard for us to apply their model directly to the other cities having plural centers (not only a single center as in Paris). Hence, here we shall modify the Gauvin's model to include the much more detail structure of the attractiveness by taking into account the empirical data concerning the housing situation in city of Sapporo. We also consider the competition between two distances, namely, the distance between house and center, and the distance between house and office. Computer simulations are carried out to reveal the self-organized spatial structure appearing in the rent distribution. We also compare the resulting distribution with empirical rent distribution in Sapporo as an example of cities designated by ordinance.