A game-theoretic approach for stochastic estimation of equilibrium in land use data: stochastic estimation of equilibriu
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ORIGINAL PAPER
A game-theoretic approach for stochastic estimation of equilibrium in land use data: stochastic estimation of equilibrium in land use data Marek Vach1 Accepted: 31 August 2020 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This work is focused on the use of linear games in the normal form for evaluation of Nash equilibria probability distribution in spatial data eg land use. Spatial data with varied content represent the observed state and, as a rule, do not automatically offer information about what that state is the result of. However, individual cases can be defined in such data, representing, for example, different sizes or degrees of representation of selected characteristics. The essence of the proposed approach is that such data pertaining to individual cases are regarded as payoff values of multiple interacting entities and can form a matrix of a symmetric linear game in the normal form. For this game, an NE representing a particular distribution of strategies of interacting entities can then be determined. This distribution assigns to each case the share of NE depending on the respective location in the symmetric game matrix. However, spatial data does not provide any information for the specific design of this deployment. The proposed solution is therefore stochastic: all effective permutations of the game matrix or a multidimensional symmetric game configuration leading to a different result for the NE distribution are evaluated. The result are values of the occurrence of NE probability for individual evaluated cases. The properties of the method are tested on the example of a practical application. The results show that the method is applicable for evaluating the spatial distribution of the land use stability. Keywords Nash equilibrium Permutation analysis Spatial data Land use stability Stochastic model
1 Introduction The Nash equilibrium concept (Nash 1950) in linear games is an effective tool that is very often applied in many contexts not limited to economics. Each game has at least one Nash equilibrium (NE) in pure or mixed strategies. Furthermore, any bimatrix or multidimensional regular configuration containing in its elements any payoff values corresponding to the number of dimensions can be interpreted as a game. The use of game theory in spatial data analysis does not arise automatically, possibly due to the fact that the concept of the game is generally always deterministic: primarily,
& Marek Vach [email protected] 1
Department of Water Resources and Environmental Modeling, Faculty of Environmental Sciences, Czech University of Life Sciences Prague, Kamycka 129, 165 00 Prague - Suchdol, Czech Republic
players and their strategies are defined, and payoff functions determine the expected outcome that follows the strategy choice. Of course, the identification of players and strategies is not self-evident in spatial data. However, the concept of NE was used in cluster analysis of spatial data in Gupta and Ranganathan (2010), where a game consist
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