Game-Theoretic minimax optimization of educational resource allocation under cultural diversity constraints

Abstract

Resource allocation in culturally and linguistically diverse systems is a problem in conditions of deep uncertainty, where it is uncertain by how much the performance increases from each additional hour of remediation, mother tongue reading materials, digital devices and community liaisons, depending on the learner’s cultural and linguistic background, which is the most difficult to measure quantity in the evidence base. Formulate the problem as a two-player zero-sum game in which a planner allocates a budget over schools and learner groups with four categories of resources while nature chooses both the learner’s cultural-linguistic background and the resource effectiveness scenario. There are three constraints on cultural diversity in the system, a representation constraint, a mother tongue reading provision constraint and a minimum community liaison requirement, defining the planner’s feasible set. The existence of the Nash equilibrium follows from the Sion’s theorem, the planner’s equilibrium policy is a deterministic plan, and the equilibrium can be found exactly through solving a single exponential cone program, the Lagrange multipliers of which give nature’s equilibrium policy. A multiplicative weights learning algorithm converges to the same optimal solution with an optimality certificate in each round. On a 30-school district with 16,116 students split into five groups with eight effectiveness regimes, the GMDC achieved a better guaranteed minimum-group performance by -0.440 versus -0.278 standard deviations above the worst group's best performance under the utilitarian solution, increased the out-of-sample 5th percentile worst-group performance by 0.133 SD, and narrowed the average between-group gap by 39% in return for a loss of only 0.021 SD in district-mean performance. Diversity floors on top of a utilitarian objective did not change anything, while minimax without diversity floors left out the majority group altogether. All 20 additional districts examined gave identical findings (Wilcoxon p<10 ^ (-5)). The nature's equilibrium strategy selected one key assumption, that mother-tongue material is effective in three highest distance groups, turning the robust plan into an agenda for policy makers.