[Paper Review] The Complexity of Probabilistic Lobbying

Daniel Binkele-Raible, Gábor Erdélyi+4 more|arXiv (Cornell University)|Jun 24, 2009

Game Theory and Voting Systems52 references4 citations

TL;DR

This paper introduces probabilistic lobbying models where a lobbyist influences voters' probabilistic preferences to pass multiple issues, using microbribery and voter bribery under strict or average majority evaluation. It establishes the computational complexity of these problems, showing polynomial-time solvability, NP-completeness with fixed-parameter tractability, and W[2]-completeness, along with PTAS results for weighted variants.

ABSTRACT

We propose models for lobbying in a probabilistic environment, in which an actor (called "The Lobby") seeks to influence voters' preferences of voting for or against multiple issues when the voters' preferences are represented in terms of probabilities. In particular, we provide two evaluation criteria and two bribery methods to formally describe these models, and we consider the resulting forms of lobbying with and without issue weighting. We provide a formal analysis for these problems of lobbying in a stochastic environment, and determine their classical and parameterized complexity depending on the given bribery/evaluation criteria and on various natural parameterizations. Specifically, we show that some of these problems can be solved in polynomial time, some are NP-complete but fixed-parameter tractable, and some are W[2]-complete. Finally, we provide approximability and inapproximability results for these problems and several variants.

Motivation & Objective

• To model lobbying in a stochastic environment where voter preferences are probabilistic rather than deterministic.
• To formalize two evaluation criteria—strict majority and average majority—for determining whether a referendum passes.
• To define two bribery methods—microbribery (per-issue targeting) and voter bribery (uniform funding per voter)—to influence voter probabilities.
• To analyze the classical and parameterized complexity of these lobbying problems under various parameterizations.
• To investigate approximability and inapproximability of the resulting optimization problems, especially with issue weighting.

Proposed method

• Proposes two evaluation criteria: strict majority (majority of voters exceed a threshold probability) and average majority (average probability across voters exceeds threshold).
• Introduces two bribery mechanisms: microbribery (selective issue-by-issue influence) and voter bribery (uniform funding per voter across issues).
• Reduces probabilistic lobbying problems to known combinatorial optimization problems, such as Knapsack, to derive approximation schemes.
• Applies parameterized complexity techniques to classify problems by tractability, using parameters like budget, number of issues, or number of voters.
• Transfers pseudo-polynomial algorithms into polynomial-time approximation schemes (PTAS) for weighted variants by reinterpreting problems as knapsack-type optimization.
• Uses reductions to establish W[2]-completeness and NP-completeness for specific problem variants, especially under strict majority and issue weighting.

Experimental results

Research questions

• RQ1 What is the computational complexity of probabilistic lobbying under microbribery and voter bribery with strict majority evaluation?
• RQ2 How does issue weighting affect the complexity of probabilistic lobbying problems under different evaluation criteria?
• RQ3 Which of the proposed lobbying problems are fixed-parameter tractable, and which are W[2]-complete?
• RQ4 Can approximation schemes be constructed for the hard optimization variants, particularly under issue weighting?
• RQ5 What is the approximability of the minimization and maximization versions of the weighted lobbying problems?

Key findings

• Several probabilistic lobbying problems are solvable in polynomial time, particularly under specific parameterizations or without issue weighting.
• Some problems are NP-complete but fixed-parameter tractable when parameterized by the number of issues or voters.
• A number of problems, especially under strict majority and voter bribery, are W[2]-complete, indicating high parameterized intractability.
• Both Min-MB-SM-PLP-WIW and Min-MB-AM-PLP-WIW admit a polynomial-time approximation scheme (PTAS), derived via reduction to the Knapsack problem.
• The maximization variants Max-MB-SM-PLP-WIW and Max-MB-AM-PLP-WIW also admit a PTAS, confirming approximability for bicriteria optimization under issue weights.
• The problems are strongly NP-hard, meaning their hardness persists regardless of input encoding (unary or binary), complicating parameterized analysis.