ERVD - Election based on Relative Value Distances

Overview

φ_i ∈ [0,1]. Higher = better (closer to PIS). Behaviorally grounded: losses are weighted more heavily than gains (λ=2.25 default). Reference point strongly affects results - choose carefully.

Data

Crisp

Weights

Needs a weight source

How it works

1. 1

   Sum normalization (direction-unaware): r_ij = x_ij / Σ_k x_kj. Reference point also normalized: φ_j = μ_j / Σ_k x_kj.

   $$r_{ij}=\frac{x_{ij}}{{\sum }_k{x}_{kj}};{\phi }_j=\frac{{\mu }_j}{{\sum }_k{x}_{kj}}$$

   Shyur et al. 2015 §3

2. 2

   Prospect-theory value function v_ij relative to normalized reference φ_j. For benefit (type=max): v_ij = (r_ij − φ_j)^α if r_ij > φ_j, else −λ(φ_j − r_ij)^α. For cost (type=min): v_ij = (φ_j − r_ij)^α if r_ij < φ_j, else −λ(r_ij − φ_j)^α.

   $$v_{ij}=\{\begin{array}{ll}(r_{ij}-{\phi }_j{)}^{\alpha }& \text{benefit, }{r}_{ij}>{\phi }_j\\ -\lambda ({\phi }_j-r_{ij}{)}^{\alpha }& \text{benefit, }{r}_{ij}\le {\phi }_j\\ ({\phi }_j-r_{ij}{)}^{\alpha }& \text{cost, }{r}_{ij}<{\phi }_j\\ -\lambda (r_{ij}-{\phi }_j{)}^{\alpha }& \text{cost, }{r}_{ij}\ge {\phi }_j\end{array}$$

   Shyur et al. 2015 §3

3. 3

   Positive ideal A+_j = max_i v_ij; negative ideal A-_j = min_i v_ij per criterion. Separation measures S+_i = Σ w_j |v_ij − A+_j|; S-_i = Σ w_j |v_ij − A-_j|.

   $$A_j^{+}=\underset{i}{\max }{v}_{ij};A_j^{-}=\underset{i}{\min }{v}_{ij};S_i^{+}=\sum _j{w}_j|v_{ij}-A_j^{+}|;S_i^{-}=\sum _j{w}_j|v_{ij}-A_j^{-}|$$

   Shyur et al. 2015 §3

4. 4

   Relative closeness φ_i = S-_i / (S+_i + S-_i). Rank descending (higher = better).

   $${\varphi }_i=\frac{S_i^{-}}{S_i^{+}+S_i^{-}},\text{rank descending}$$

   Shyur et al. 2015 §3

Edge cases and pitfalls

• •if r_ij > φ_j, else −λ(φ_j − r_ij)^α. For cost (type=min): v_ij = (φ_j − r_ij)^α if r_ij < φ_j, else −λ(r_ij − φ_j)^α.

• Sum normalization is direction-unaware - all columns summed as-is. Ref point normalized by same column sum. Cost direction handled in value function F2, not in normalization.

• Alpha default is 0.88 (Kahneman-Tversky), NOT 2.25. Lambda default is 2.25.

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