# The Complexity of Online Bribery in Sequential Elections

## TL;DR

This paper explores the complexity of bribery in sequential elections, revealing that online, sequential decision-making can significantly increase computational difficulty, though some systems remain manageable.

## Contribution

It introduces a new model for online, sequential bribery and analyzes its complexity, highlighting cases where complexity increases or remains manageable.

## Key findings

- Sequential bribery can be PSPACE-complete for some systems.
- For certain election systems, bribery complexity does not increase dramatically.
- The model captures realistic voting scenarios with limited information and timing constraints.

## Abstract

Prior work on the complexity of bribery assumes that the bribery happens simultaneously, and that the briber has full knowledge of all votes. However, in many real-world settings votes come in sequentially, and the briber may have a use-it-or-lose-it moment to decide whether to alter a given vote, and when making that decision the briber may not know what votes remaining voters will cast. We introduce a model for, and initiate the study of, bribery in such an online, sequential setting. We show that even for election systems whose winner-determination problem is polynomial-time computable, an online, sequential setting may vastly increase the complexity of bribery, jumping the problem up to completeness for high levels of the polynomial hierarchy or even PSPACE. But we also show that for some natural, important election systems, such a dramatic complexity increase does not occur, and we pinpoint the complexity of their bribery problems.