Assessing the local equilibrium of land use in three European Capitals

Data set and computational and data matching Fortran codes for calculation spatial distribution of the local landscape equilibrium among various land use types. This is an application of a novel game-theoretic method to assess the spatial distribution of the probability of Nash equilibrium among four land use types within the functional urban areas of three European Capitals: Prague, Vienna, and Budapest. The calculated spatial distribution of local equilibrium shows trends consistent with the Shannon index, characterizing heterogeneity. However, the approach used provides additional information on the relative degree of equilibrium between land use types and is significantly more sensitive to small land-use types representation optimality changes. The method employed is an autonomous stochastic model grounded in the structure of linear games in normal form. It facilitates the calculation of the distribution of Nash equilibrium probabilities within data represented by spatial cases characterized by multiple attributes. This approach offers a novel perspective, applicable to various environmental data in different contexts.

Data and code description

Default data are raster maps of land use - Prague, Vienna, Budapest (Copernicus Urban Atlas 2012, 2018) in 4m pixel resolution converted to ASCI - see "input_raster_asc_maps". 

These data are converted into final input files - see "input cases" using Fortran code "readmap" - see "auxiliary codes for data conversion". The readmap.f90 code requires a specific coordinate setting for the loaded ASCI map and allows the selection of the size of the squares (here chosen 3, 4, and 6 km)-cases and their diagonal offset from the upper left corner of the raster map. The input case files contain in each line the identifier of the square (in the raster map) and the number of pixels found for each evaluated land use type (for the subsequent Nash Equilibrium - NE probability evaluation, the absolute values for payoff are not relevant and the found pixel counts are directly usable values). The files come in two versions - with squares defined from the top corner of the raster maps - see "primary_squares" and with squares shifted diagonally - see "shifted_squares".

Input cases (for each situation to be evaluated always two connected sets of three cities for the primary and shifted squares) are the input data to the computational code "equs" (as inp.dat). The equs.f90 code published here to compute the NE probabilities of each case is a practical version (compilable in GFortran) containing only the most computationally efficient version - generating 3D and 4D game arrays with two formal strategies for each entity - player and computing the NE in pure strategies. Multiple code versions justifying the concept and its structure are available at github.com/vachm/equsis. A detailed description is formulated in (Vach, 2020), and for limited calculations (up to 32 dedicated cases and a maximum of 5 interacting entities) the website equsis.com is available. 

The output of equs is an outst.dat file containing the identifier of the square - case on the map and the received NE probability value on each line. 
The back-conversion of the calculated data (outst.data must be split back into three cities) into an ASCI raster map suitable for display (in ArcGIS) is provided by the code "creatmap" - see "auxiliary codes for data conversion", which requires specific settings (the same as "readmap"). For one situation under evaluation (one city) creatmap averages the data from two equs outputs - primary and shifted squares.