| The essence of the method is use of linear games in the normal form for evaluation of Nash equilibria distribution in spatial data. In many applications in economics, ecology, systems management, etc., the linear game concept, where players have the choice of strategies leading to the result defined in the payoff table, is conceived deterministically. The result of the game is often evaluated in terms of the Nash equilibrium (NE). 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. Thus, the application of the deterministic concept of linear games to spatial data seems impossible.
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, and it is impossible to identify any structural context of individual cases for their locations in individual rows and columns of the game matrix. 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.
Each permutation represents a game with a set of formal strategies of individual players or interacting entities. These strategies cannot be specified in any way; however, the evaluation of all effective permutations means that all possible formal strategies that can be derived for a given game configuration are considered. In this context, the proposed concept can be categorized as an evaluation of spatial data based on ``games'' with stochastically derived formal strategies. Then, the calculated NE distribution in the game cannot be assigned to any specifically defined strategies and is therefore added directly to the elements of the game matrix, i.e., to individual cases. This also means that the calculated NE probability values for individual cases are equivalent to the Pareto optimality measure. This method could contribute, for example, to solving the problem of stability in the landscape, etc. | |