Multi-directional DEA with Win-Rate aggregation: a robust approach to efficiency ranking

This paper proposes a methodology for aggregating multi-directional Data Envelopment Analysis results into a unified efficiency ranking. The directional distance function requires specification of a direction vector, and different directions can yield different efficiency assessments. We address this direction selection problem through win-rate aggregation that counts how often each unit outperforms others across multiple directions sampled from Dirichlet distributions. To distinguish among efficient units with identical win rates, we incorporate super-efficiency scores. The method provides intuitive directional outperformance probabilities for pairwise comparisons and tier classifications grouping units by performance levels. Monte Carlo simulations demonstrate Spearman correlations of 0.97 with standard DEA rankings, discriminatory power of 81 percent, and full coverage of the BCC-efficient frontier within the top tier. Application to 57 Japanese regional banks reveals that efficiency is not determined by bank size but strongly correlates with profitability, with Tier 1 banks outperforming Tier 4 banks in 100 percent of directional comparisons, and that frontier-referenced win rates avoid the scale confounding exhibited by ideal-point rankings such as TOPSIS. The methodology also provides actionable diagnostics identifying each bank’s strengths, weaknesses, and appropriate benchmark institutions.