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Bribery's Influence on Ranked Aggregation

Pallavi Jain, Anshul Thakur

Abstract

Kemeny Consensus is a well-known rank aggregation method in social choice theory. In this method, given a set of rankings, the goal is to find a ranking $Π$ that minimizes the total Kendall tau distance to the input rankings. Computing a Kemeny consensus is NP-hard, and even verifying whether a given ranking is a Kemeny consensus is coNP-complete. Fitzsimmons and Hemaspaandra [IJCAI 2021] established the computational intractability of achieving a desired consensus through manipulative actions. Kemeny Consensus is an optimisation problem related to Kemeny's rule. In this paper, we consider a decision problem related to Kemeny's rule, known as Kemeny Score, in which the goal is to decide whether there exists a ranking $Π$ whose total Kendall tau distance from the given rankings is at most $k$. Computation of Kemeny score is known to be NP-complete. In this paper, we investigate the impact of several manipulation actions on the Kemeny Score problem, in which given a set of rankings, an integer $k$, and a ranking $X$, the question is to decide whether it is possible to manipulate the given rankings so that the total Kendall tau distance of $X$ from the manipulated rankings is at most $k$. We show that this problem can be solved in polynomial time for various manipulation actions. Interestingly, these same manipulation actions are known to be computationally hard for Kemeny consensus.

Bribery's Influence on Ranked Aggregation

Abstract

Kemeny Consensus is a well-known rank aggregation method in social choice theory. In this method, given a set of rankings, the goal is to find a ranking that minimizes the total Kendall tau distance to the input rankings. Computing a Kemeny consensus is NP-hard, and even verifying whether a given ranking is a Kemeny consensus is coNP-complete. Fitzsimmons and Hemaspaandra [IJCAI 2021] established the computational intractability of achieving a desired consensus through manipulative actions. Kemeny Consensus is an optimisation problem related to Kemeny's rule. In this paper, we consider a decision problem related to Kemeny's rule, known as Kemeny Score, in which the goal is to decide whether there exists a ranking whose total Kendall tau distance from the given rankings is at most . Computation of Kemeny score is known to be NP-complete. In this paper, we investigate the impact of several manipulation actions on the Kemeny Score problem, in which given a set of rankings, an integer , and a ranking , the question is to decide whether it is possible to manipulate the given rankings so that the total Kendall tau distance of from the manipulated rankings is at most . We show that this problem can be solved in polynomial time for various manipulation actions. Interestingly, these same manipulation actions are known to be computationally hard for Kemeny consensus.

Paper Structure

This paper contains 10 sections, 17 theorems, 5 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Kemeny Consensus vs Kemeny Score
  3. Related Work.
  4. Road Map.
  5. Preliminaries
  6. Possible Kemeny Score
  7. $-Bribery-Kemeny Score and Ranking Deletion Kemeny Score
  8. Swap Bribery-Kemeny Score
  9. Candidate Deletion Kemeny Score
  10. Conclusion

Key Result

Lemma 1

Let $\hat{R}$ be an optimal extension of $R$. Then $\hat{R}$ and $X \xspace$ agree over $C"$.

Figures (1)

  • Figure 1: $\Pi(i),t_i$, and $t_f$ are as discussed in above paragraph. Circles represent the candidates of $C'$ and squares denote the candidates of $C"$. Big squares denote a chunk of $C"$ that are consecutive in the ranking.

Theorems & Definitions (34)

  • Example 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Claim 3
  • proof
  • Lemma 4
  • proof
  • Claim 5
  • ...and 24 more