Regret cross-efficiency evaluation using attitudinal entropy approach

The paper addresses limitations of traditional DEA and cross-efficiency evaluation, notably: (1) DEA’s potential for extreme, DMU-favoring weights and inability to rank efficient DMUs; (2) cross-efficiency’s multiple optimal weight solutions leading to non-unique scores; (3) aggregation commonly using simple averages that ignore weight information and DM preferences; and (4) prevailing assumptions of complete rationality, neglecting bounded rationality in real decisions. The research proposes incorporating regret theory to reflect bounded rational behavior and using attitudinal entropy to aggregate cross-efficiency in a way that embeds information preference. The objective is to develop a comprehensive regret-based cross-efficiency evaluation method (RACE) that (i) constructs a secondary goal considering both endogenous (group mean) and exogenous (ideal/anti-ideal) reference points, and (ii) aggregates cross-efficiencies via attitudinal entropy to reflect preferences for information uncertainty. The study evaluates Chinese high-tech industry regions and university data to demonstrate validity and robustness.

The literature reviewed spans three lines. First, cross-efficiency in DEA: after Sexton et al. (1986) introduced cross-efficiency to combine self- and peer-evaluation, subsequent work addressed multiple optimal weights via secondary goals, including benevolent and aggressive strategies (Doyle and Green, 1994), extensions balancing extremes (Liang et al., 2008; Lim, 2012), neutral strategies (Wang and Chin, 2010), and weight-balancing to avoid extreme weights (Lam, 2010; Jahanshahloo et al., 2011; Ramón et al., 2011; Wang et al., 2012; Wu et al., 2012a). Second, aggregation of cross-efficiency weights: beyond simple averages, studies applied cooperative game theory (Shapley value) (Wu et al., 2009a), order-priority (Angiz et al., 2013), consensus algorithms (Wu et al., 2022), Shannon and grey entropies (Wu et al., 2011; Shuai and Wu, 2011; Song et al., 2017; Song and Liu, 2018). Third, bounded rationality: regret theory (Bell, 1982; Loomes and Sugden, 1982) emphasizes regret/rejoice relative to alternatives and has been used for cross-efficiency modeling and aggregation (Gong et al., 2021; Liu and Chen, 2022; Jin et al., 2022, 2023). Prospect-theory-based models considered ideal/anti-ideal references and risk attitudes (Liu et al., 2019; Chen et al., 2020; Shi et al., 2021; Wu et al., 2022). Existing regret-based approaches often rely on limited reference points (e.g., aggressive/benevolent or ideal/non-ideal) and consider preferences only at isolated stages. The paper identifies gaps in holistic treatment of bounded rationality across the entire cross-efficiency process and comprehensive weight aggregation reflecting subjective information preferences, motivating the proposed RACE method.

The methodology comprises: (A) Preliminaries and core models; (B) Regret-theoretic perceived utilities; (C) Attitudinal entropy; and (D) The RACE procedure. A) DEA cross-efficiency setup: For n DMUs using m inputs x_ij and s outputs y_rj, the CCR model computes self-efficiency for DMU k with nonnegative input and output weights w_i, μ_r. Cross-efficiency uses the optimal weights of each DMU to evaluate peers, forming the n×n cross-efficiency matrix; traditional aggregation uses simple averages but suffers when optimal weights are non-unique, prompting secondary goal models (aggressive/benevolent, neutral, etc.). B) Regret theory and perceived utility: The approach models bounded rationality through regret/rejoice. Utility for inputs (cost) is monotonically decreasing under constant absolute risk aversion φ_in(t)=a(1−e^{−a t}), 0<a<1; for outputs (benefit) it is increasing concave φ_out(t)=b^{−1}(1−e^{−b t}), 0<b<1. Regret-rejoice function R(Δφ)=1−e^{−θ Δφ}, with regret aversion parameter θ≥0; it is steeper for losses (Δφ<0), capturing asymmetry between regret and rejoice. Perceived utility of option A with reference B: U(t_A|t_B)=φ(t_A)+R(φ(t_A)−φ(t_B)). Three reference types are used: exogenous ideal (regret attribute: compare to min input/max output), exogenous anti-ideal (rejoice attribute: compare to max input/min output), and endogenous mean (heterogeneous attribute: compare to geometric means of inputs/outputs). For each attribute, composite perceived utilities U^+ are defined accordingly, using the appropriate reference (min/max/geometric mean) and the regret-rejoice term. C) Attitudinal entropy for aggregation: Standard Shannon entropy H^Sh_k=−∑d E{dk} ln E_{dk} (with E_{dk} the column-normalized cross-efficiency for DMU k) reflects dispersion. To embed subjective information preference, attitudinal entropy H^A_k=log_β(∑d E{dk}^α ln(1/E_{dk})) is adopted, where α>0 encodes preference for low-probability events (uncertainty attitude) and β is the logarithm base controlling gain. As α→1, attitudinal entropy approaches Shannon entropy. Lower entropy implies higher consensus/acceptability. D) RACE secondary goal model and steps: To resolve multiple optimal weights under bounded rationality and to integrate both endogeneity and exogeneity, a secondary goal model maximizes a convex combination of perceived utilities: Maximize λ·I + (1−λ)·( ζ·G + (1−ζ)·L ), subject to the DEA cross-efficiency constraints, weight nonnegativity, and normalization, with λ,ζ∈[0,1]. I aggregates perceived utilities relative to endogenous references (geometric means), G to exogenous anti-ideal (rejoice), and L to exogenous ideal (regret). Parameters: λ reflects emphasis on endogenous perception; ζ on rejoice within exogenous perception; a,b are risk aversion for inputs/outputs; θ is regret aversion; α,β control information preference in entropy aggregation. Algorithm (RACE):

1. Construct the cross-efficiency matrix under the regret-based secondary goal model: solve CCR for self-efficiencies; form peer evaluations using optimal weights; solve the regret-informed secondary model to determine unique weights and cross-efficiency entries θ_{dk}.
2. Column-normalize cross-efficiency: E_{dk}=θ_{dk}/∑d θ{dk}.
3. Compute attitudinal entropy H^A_k and derive aggregation weights w_k proportional to information utility (inverse-entropy concept operationalized as w_k=H_k/∑_k H_k after transforming entropy to information utility as described by the authors).
4. Aggregate final scores: E_i^{RACE}=∑{k=1}^n w_k·θ{ik}. Parameterization for empirical study: typically a=b=0.02, θ=0.3, λ=ζ=0.5, α=1.5, β=2 (by cited references). Sensitivity analyses vary parameters to explore robustness.

Empirical analyses on two datasets demonstrate validity and robustness of RACE.

1. Chinese high-tech industry regions (12 provinces, 2020): Inputs: R&D personnel (FTE) and R&D expenditure; Outputs: inventions in force and sales revenue of new products. Compared methods: CCR, aggressive, benevolent, and RACE. Results (Table 4): RACE fully ranks DMUs and generally yields scores between benevolent and aggressive strategies. RACE rankings and scores: DMU1 (0.9798, rank 1), DMU5 (0.9458, 2), DMU11 (0.8717, 3), DMU3 (0.8365, 4), DMU2 (0.6478, 5), DMU10 (0.6424, 6), DMU4 (0.5709, 7), DMU7 (0.5688, 8), DMU8 (0.5281, 9), DMU6 (0.5201, 10), DMU9 (0.5099, 11), DMU12 (0.4825, 12). Average RACE score: 0.6754. Differences from traditional strategies appear for some DMUs (e.g., DMU3’s RACE score exceeds both aggressive and benevolent; DMU1 shows the opposite direction), reflecting bounded-rational preference integration. Sensitivity analysis: Varying single parameters shows smooth overall trends but local rank shifts. For example, with increasing regret aversion θ, most DMUs slightly decline; for λ (endogenous emphasis), when λ<0.64 DMU1>DMU2, while λ>0.64 flips to DMU2>DMU1; rejoice preference ζ yields intervals where DMU1 and DMU2 alternately lead. Information preference α in (0,2) affects discrimination; DMU1 peaks around α≈0.8 while others dip. Double-parameter surfaces reveal joint effects: e.g., when λ<0.19 and ζ<0.8, DMU5 becomes top-ranked instead of DMU1; ranges of α with a/b or β also shift top rankings, capturing nuanced behavioral ambivalence.
2. Chinese universities (13 DMUs, 2017) compared with PCE (prospect-based), Wu’s entropy-based model, RCEC (regret-based aggregation), and CCR (Table 7): RACE yields higher mean efficiency (0.7642) than PCE (0.7274), Wu (0.7192), and RCEC (0.7521), and closely aligns in ranks with advanced models, with high Spearman correlations: 0.93 with RCEC and 0.86 with Wu. RACE’s IQR is larger than most comparators, indicating stronger discrimination among mid-performing DMUs; its SD (0.1025) is lower than RCEC’s (0.1200), suggesting more consistent evaluations for extremes. One notable exception is DMU11 whose RACE score is slightly below RCEC, attributed to higher loss relative to the endogenous input reference.

The findings show that embedding bounded rationality via regret theory and combining endogenous (geometric mean) and exogenous (ideal/anti-ideal) references generates unique weights and stable cross-efficiency rankings while reflecting realistic decision attitudes. The RACE secondary goal balances regret and rejoice perceptions and allows DMs to tune emphasis through λ and ζ, capturing internal versus external benchmarking. Attitudinal entropy integrates preferences for information uncertainty (α,β), improving aggregation beyond simple averaging or Shannon entropy and preserving linkages between cross-efficiencies and their underlying weight structures. Empirically, RACE delivers scores typically between benevolent and aggressive extremes, avoids ties inherent in CCR, shows robust alignment with strong existing models (high correlation with RCEC and Wu), and enhances discrimination among mid-performing units while maintaining stability for extremes. Sensitivity analyses confirm interpretability of parameters: higher regret aversion dampens scores; shifting endogenous/exogenous emphasis can flip local rankings; information preference modulates discrimination. These results address the research question by demonstrating a comprehensive, psychologically grounded cross-efficiency framework that captures the complexity of DM preferences across the full evaluation process.

The paper proposes RACE, a regret cross-efficiency evaluation method using attitudinal entropy, to address overestimation, incomplete ranking, and multiple optimal weights in DEA cross-efficiency while integrating bounded rationality. Contributions include: (1) a secondary goal model blending endogenous (geometric mean) and exogenous (ideal/anti-ideal) references under regret theory; (2) incorporation of risk and regret attitudes via perceived utilities; (3) an attitudinal entropy-based aggregation reflecting information uncertainty preferences; and (4) empirical validation demonstrating improved ranking capability, robust alignment with advanced models, and enhanced discrimination. Potential applications span finance, business strategy, and public project evaluation. Future research directions: extend fixed reference points to interval or dynamic references; develop multi-stage or dynamic versions of RACE to capture evolving decision processes; further study calibration of preference parameters to mitigate potential overestimation and tailor discrimination levels to context-specific needs.

Identified limitations include: (1) reliance on fixed reference points (ideal/anti-ideal/mean); adopting interval-valued or adaptive references could increase flexibility; (2) a static evaluation framework that may not capture dynamic or multi-stage decision processes; (3) empirical evidence shows slightly higher mean scores and larger IQR under RACE, implying potential overestimation for some DMUs and wider efficiency gaps when uniform assessments are needed; and (4) parameter selection (risk/regret attitudes, endogenous/exogenous emphasis, information preference) influences outcomes and requires careful elicitation or calibration.