New algorithm for checking Pareto optimality in bimatrix games

Abstract

Bimatrix games have had theoretical importance and important applications since the very beginning of game theory to date. Pareto optimal strategy vectors represent a reasonable and frequently applied solution concept (basically in the cooperative approach). Particularly, it often arises whether a non-cooperative solution can be improved in cooperative sense, i.e. it is Pareto optimal or not. However, in concrete cases it may be hard to determine the Pareto optimal strategy vectors or at least check the Pareto optimality of a given strategy vector. In the present paper, the Pareto optimality of the strategy pairs in general n × m ($$n = 2,3, \ldots$$ n = 2 , 3 , … ; $$m = 2,3, \ldots$$ m = 2 , 3 , … ) bimatrix games is studied. First of all an elementary proof is provided for a theorem, which makes the proposed Pareto optimality checking algorithm simpler. Then a nonlinear transform is proposed, which makes the algorithm even more convenient in the important case of 2 × 2 bimatrix games (and for certain other cases). Two numerical examples present the practical applicability of the checking algorithm. The problem of 2 × 2 bimatrix games can be solved even by hand. For larger games, numerous computational tools are available in the practice to apply the checking algorithm in exact or at least approximate way.

Case Study on Spatial Mismatch between Multivariate and Student-Teacher Rate in U.S. Public School Districts

“…Although Sutcher et al (2019) proposed teacher mismatch between supply and demand as education reports, they did not mention their methods and navigate on large-scale assessment (Sutcher et al 2019). Moreover, spatial mismatch originally refers to the phenomenon that the spatial allocation of production factors deviates from Pareto optimality due to various reasons, resulting in the loss of economic benefits (Kicsiny and Varga 2022;, Marrero-Vera et al 2022. Previous spatial mismatch analysis intensively focuses on minor educator labor workforce allocation (Holzer 1991;Wasmer and Zenou 2002;Gobillon et al 2007;Hsieh and Moretti 2019;Kicsiny and Varga 2022;Logan et al 2020;Li et al 2016;Silva et al 2021;Wang et al 2022;Yang et al 2022).…”

“…Moreover, spatial mismatch originally refers to the phenomenon that the spatial allocation of production factors deviates from Pareto optimality due to various reasons, resulting in the loss of economic benefits (Kicsiny and Varga 2022;, Marrero-Vera et al 2022. Previous spatial mismatch analysis intensively focuses on minor educator labor workforce allocation (Holzer 1991;Wasmer and Zenou 2002;Gobillon et al 2007;Hsieh and Moretti 2019;Kicsiny and Varga 2022;Logan et al 2020;Li et al 2016;Silva et al 2021;Wang et al 2022;Yang et al 2022). The spatial mismatch hypothesis is not employed in high-need school districts in the U.S.…”