# A Regret-Enhanced DEA Approach to Mapping Renewable Energy Efficiency in Asia's Growth Economies
## 2 Methodology  
### Bounded Rationality and Regret Theory  
Bounded rationality, an idea put forth by [PERSON], posits that individuals arrive at decisions considering their constrained information and cognitive abilities [54]. This counters the prevalent assumption in economic frameworks where people are deemed entirely rational and adept at making reasoned choices [49]. An alternative perspective on rational decision making amidst uncertainty is offered by regret theory [53]. This viewpoint proposes that decisions are driven by the anticipation of regret that might arise from selecting an incorrect option. In the context of decision making in business, the regret minimization framework acts as a guiding principle, encouraging individuals to envision themselves in the future and make choices that minimize potential remorse. **Definition 1**. : _Risk-Adjusted Utility._  
Let \(T\) be an alternative selected by the decision maker and let \(t\) be the consequence (or outcome) that results from choosing \(T\). The risk-adjusted utility of selecting alternative \(T\), denoted by \(u(t)\), is defined as follows:  
\[u(t)=t^{\theta},\ \ 0<\theta<1 \tag{1}\]  
Here, the symbol \(\theta\) represents the decision maker's risk aversion coefficient. It controls the degree of concavity of the utility function and thereby captures the decision maker's attitude toward risk. A smaller value of \(\theta\) corresponds to higher risk aversion, meaning that the decision maker derives less additional utility from larger outcomes. When \(\theta=1\), the decision maker is risk-neutral, and when \(\theta<1\), diminishing marginal utility is observed. In this study, a typical value of \(\theta=0.88\) is used, consistent with empirical findings in the regret theory literature [55]. **Definition 2**. : _Regret-Rejoice Utility Function._  
Let \(T_{1}\) and \(T_{2}\) be two alternatives leading to consequences \(t_{1}\) and \(t_{2}\), respectively. The regret-rejoice utility function, denoted by \(r(t_{1},t_{2})\), is defined as the emotional adjustment to utility when comparing \(t_{1}\) against \(t_{2}\):  
\[r(t_{1},t_{2})=R(\Delta t)=1-e^{-\lambda(u(t_{1})-u(t_{2}))}=1-e^{-\lambda \Delta u},\ \lambda>0 \tag{2}\]  
where \(\Delta u=u(t_{1})-u(t_{2})=u(\Delta t)\), \(R(\Delta t)\) is the regret-rejoice transformation function applied to a single scalar value. The parameter \(\lambda\) expresses the regret aversion coefficient of the decision maker. The larger the value of \(\lambda\), the larger regret aversion of the decision maker. The value of \(r(t_{1},t_{2})\) expresses the regret value when \(u(t_{1})\leq u(t_{2})\) or \((\Delta u\leq 0)\). The parameter \(\lambda\) signifies the decision maker's regret aversion coefficient. A higher \(\lambda\) value corresponds to a greater extent of regret aversion exhibited by the decision maker. The term \(r(t_{1},t_{2})\) denotes the measure of regret when the utility associated with choice \(T_{1}\) is less than or equal to the utility linked with choice \(T_{2}\). Otherwise, it expresses the rejoice value. A value of 0.3 was assumed experimentally [55]. **Definition 3**. : _Final Decision Utility._  
Let \(T_{j}\) be a set of alternatives for \(j=1,\ldots,m\), each resulting in a consequence \(t_{j}\). The final decision utility of alternative \(T_{i}\), denoted by \(l(t_{j})\), is defined as the sum of the risk-adjusted utility and the regret-rejoice utility relative to the best alternative in the set:  
\[l\big{(}t_{j}\big{)}=u\big{(}t_{j}\big{)}+r\big{(}t_{j},t^{*}\big{)} \tag{3}\]  
where  
\[t^{*}=\max_{1\leq j\leq m}t_{j}\text{ and }r\big{(}t_{j},t^{*}\big{)}\leq 0 \tag{4}\]  
**Remark 1**. : _The function \(R(\cdot)\) captures the emotional response of the decision maker to the difference in utilities. It is monotonically increasing and concave, satisfying the following:_  
\[\left\{ \begin{aligned} \mathbf{R}^{\prime}(\Delta u)=\lambda e^{- \lambda\Delta u}>0\\ \mathbf{R}^{\prime\prime}(\Delta u)=-\lambda^{2}e^{-\lambda\Delta u}<0 \\ R(\Delta u)=0\end{aligned} \right.