On the nonexistence of non-convergent Nash equilibria in disapproval voting

• Main Authors: Go, Justin Ervin D., Pinto, Jonathan O.
• Format: text
• Language: English
• Published: Animo Repository 2022
• Subjects: Voting; Equilibrium; Game theory; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etdb_math/19; https://animorepository.dlsu.edu.ph/cgi/viewcontent.cgi?article=1020&context=etdb_math
• Institution: De La Salle University

In democratic societies, elections are done to determine rightful candidates to hold public office. In situations where voters are required to rank all the political candidates from most preferred to the least, a scoring rule is used. In an election held under a scoring rule, a candidate receives a score based on their ranking on the voter’s ballot. The highest scoreis awarded to the most preferred candidate while the lowest score is given to the least. In this thesis, we view an election held under a scoring rule as a game where the players are the political candidates and the payoffs are the total scores. We will look into the game’s Nash equilibrium – a state in which any candidate does not gain a higher score by changinghis/her strategy while others keep theirs. A Nash equilibrium is classified as convergent (resp.nonconvergent) whenever players have unanimous (resp. split) strategies in the equilibrium. The aim of this thesis is to contribute to the objective of finding some characterization of scoring rules where non-convergent Nash equilibria (NCNE) exist. Specifically, we focus on a certain type of scoring rule where negative scores are given to the least preferred candidates. For such scoring rules, we shall use the term disapproval voting. In this study, we prove that,in an m-candidate election with m ≥ 4, the disapproval voting whose scoring rule awards negative points to m − k least preferred candidates for k ≥ ⌊ m 2 ⌋ does not have NCNE.