| General
The essence of the method is the use of normal-form linear games to evaluate the distribution of Nash equilibria in spatial data. In many applications in economics, ecology, systems management, etc., the linear game concept, in which players choose strategies that yield the result defined in the payoff table, is modeled 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 provide information about what that state results from. Thus, applying 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 individual-case data are regarded as payoff values for multiple interacting entities and can form a symmetric array of a linear game in 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 corresponding to its position in the symmetric game array. However, spatial data does not provide information on the specific design of this deployment, and it is impossible to infer the structural context of individual cases from their locations within the game array's rows and columns. The proposed solution is therefore stochastic: all effective permutations of a multidimensional symmetric game array that lead to a different NE distribution are evaluated. The results are the values of NE probability occurrences 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, evaluating all effective permutations ensures that all possible formal strategies 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 array, i.e., to individual cases.
Optimality concept
The described approach is based on comparing parameters - evaluated characteristics (type of land use representation, etc.) within individual cases, which represent vectors of values for these parameters in game array structures; the criterion for comparison is NE. However, NE is an equilibrium concept, not a general criterion for system optimality. This equilibrium concept is defined in terms of the equilibrium of opponents' strategies. However, the presented method does not use information about which strategies represent NE, nor does it address the informational content of those strategies. It uses only 2n arrays generated sequentially by permutation, which, however, can always be regarded as a linear game of n players, where NE can be found (in pure strategies) as the intersection of columns and rows (regardless of the informational content of the strategies). In this context, NE identifies this intersection - and thus the vector (case) - that is in equilibrium in terms of (contentless) strategies but can also be considered optimal among the others in the current game array (generated within the complete set of all possible symmetry-nondegenerate permutations of 2n vectors). This vector represents the best compromise of the parameters (the value of each parameter is the highest possible in relation to the others). From this perspective, the method solves only an optimization problem, which vector (case) represents the best compromise among the parameter values (not the equilibrium of the game's strategies). Therefore, optimality is solved about the initial problem, which is an optimization problem.
In the case where cases representing a game array payoff with a non-constant sum are evaluated, it is possible that among thousands of permuted arrays, there may be some in which there is an NE that is not Pareto optimal, such as in the Prisoner's dilemma, etc. However, the described approach does not prioritize Pareto optimality.
Multi-objective optimization (MOO) context
In general, any (non-scalarization) method addressing the problem of finding the case with the optimal trade-off across multiple factors - entities - must be based on comparing individual cases in the evaluated set. The stochastic model applied in this study performs this comparison within the structure of arrays of n-player linear games, where n denotes the number of factors (entities). This comparison, based on finding the NE, is complete for the entire set of evaluated cases, because all game arrays in the form of 2n are permuted (it is not necessary to evaluate larger arrays - e.g., 3n, etc. - because this does not change the spatial trend of the resulting NE probability distribution (Vach 2020)). The number of n elements - the dimension of the compared vectors (i.e., cases with n characteristics - interacting entities) and the dimension n of the permuted symmetric game arrays are logically identical.
The mathematical structure of comparison and the corresponding algorithms are generally justified here by game theory. In this way, the approach applied here differs fundamentally from established MOO methods using MOEA. Another aspect of this difference is that the approach is not based on searching for and evaluating the Pareto front. The resulting NE probability field provides precise information about the spatial distribution of the compromise optimality measure in the representation of individual characteristics (entities).
Reference
Vach, M. (2020). A game-theoretic approach for stochastic estimation of equilibrium in land use data: stochastic estimation of equilibrium in land use data. Stochastic Environmental Research and Risk Assessment, 34(12), 2107–2124. https://doi.org/10.1007/s00477-020-01873-2
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