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Establishing a leader in a pairwise comparisons method

Jacek Szybowski, Konrad Kułakowski, Jiri Mazurek, Sebastian Ernst

TL;DR

This paper investigates how pairwise comparison methods used in MCDM can be manipulated to crown a chosen leader. It introduces two manipulation algorithms—greedy and bubble—built on the EQ procedure that equalizes two alternatives, and grounds the approach in a projection-based formulation with Gram-Schmidt orthogonalization. The authors define manipulation-difficulty metrics via the Ranking Stability Index and Average Ranking Stability Index and validate the methods with Monte Carlo simulations across varying matrix sizes and inconsistency levels. The results show that manipulation becomes harder with more alternatives, inconsistency has limited effect on ease of manipulation, and each manipulation facilitates subsequent ones, with scale preserved by the EQ process. The work provides insights into vulnerabilities of PC-based methods and informs defenses against strategic manipulation in decision support systems.

Abstract

Abstract Like electoral systems, decision-making methods are also vulnerable to manipulation by decision-makers. The ability to effectively defend against such threats can only come from thoroughly understanding the manipulation mechanisms. In the presented article, we show two algorithms that can be used to launch a manipulation attack. They allow for equating the weights of two selected alternatives in the pairwise comparison method and, consequently, choosing a leader. The theoretical considerations are accompanied by a Monte Carlo simulation showing the relationship between the size of the PC matrix, the degree of inconsistency, and the ease of manipulation. This work is a continuation of our previous research published in the paper (Szybowski et al., 2023)

Establishing a leader in a pairwise comparisons method

TL;DR

This paper investigates how pairwise comparison methods used in MCDM can be manipulated to crown a chosen leader. It introduces two manipulation algorithms—greedy and bubble—built on the EQ procedure that equalizes two alternatives, and grounds the approach in a projection-based formulation with Gram-Schmidt orthogonalization. The authors define manipulation-difficulty metrics via the Ranking Stability Index and Average Ranking Stability Index and validate the methods with Monte Carlo simulations across varying matrix sizes and inconsistency levels. The results show that manipulation becomes harder with more alternatives, inconsistency has limited effect on ease of manipulation, and each manipulation facilitates subsequent ones, with scale preserved by the EQ process. The work provides insights into vulnerabilities of PC-based methods and informs defenses against strategic manipulation in decision support systems.

Abstract

Abstract Like electoral systems, decision-making methods are also vulnerable to manipulation by decision-makers. The ability to effectively defend against such threats can only come from thoroughly understanding the manipulation mechanisms. In the presented article, we show two algorithms that can be used to launch a manipulation attack. They allow for equating the weights of two selected alternatives in the pairwise comparison method and, consequently, choosing a leader. The theoretical considerations are accompanied by a Monte Carlo simulation showing the relationship between the size of the PC matrix, the degree of inconsistency, and the ease of manipulation. This work is a continuation of our previous research published in the paper (Szybowski et al., 2023)
Paper Structure (12 sections, 2 theorems, 52 equations, 5 figures)

This paper contains 12 sections, 2 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiplicative and additive pairwise comparisons systems
  4. Ranking stability index
  5. Establishing a leading alternative
  6. Equating two alternatives
  7. The algorithm for establishing a leading alternative in a PC method
  8. The greedy algorithm
  9. The bubble algorithm
  10. Monte Carlo Simulation
  11. Conclusions
  12. Acknowledgments

Key Result

Theorem 1

A family of matrices is a basis of ${\cal A}_{12}$.

Figures (5)

  • Figure 1: Number of iterations vs. number of alternatives
  • Figure 2: Number of iterations and number of tested matrices vs. inconsistency using the bubble algorithm and LBR strategy as example.
  • Figure 3: Frobenius distance between input matrix and its subsequent improvements. Average values for greedy algorithm and LBN strategy.
  • Figure 4: The average RSI (ARSI) value for the matrices studied depending on the size of the matrix.
  • Figure 5: Decreasing (average) value of $\textit{ARSI}$ for modified matrices in successive iterations using the greedy algorithm and LBN strategy as an example. Iteration $0$ shows the average $\textit{ARSI}$ value for the unmodified input matrices.

Theorems & Definitions (7)

  • Theorem 1: Theorem 5,Szybowski2023aomo
  • Example 2
  • Remark 3
  • Remark 4
  • Theorem 5: Theorem 9, Szybowski2023aomo
  • Example 6
  • Example 7