The right to object is a fundamental cornerstone of democracy. Having this option will increase voter participation. It will also reduce the influence of extremists and enhance world peace. https://www.youtube.com/watch?v=0MYpE0lv4eo
Here is an illustration of how Negative Vote might impact a two-person race:
Under current system
A wins and proudly proclaims majority mandate. What if there are 10 other voters who chose not to vote because they dislike both candidates? Having the Negative Vote option will bring some of them out. Suppose 5 did:
A +34 -4 (net 30)
B +33 -1 (net 32)
B wins and must admit (s)he does NOT have majority support, even though (s)he won in a two-candidate race! A humbler winner in a two-person race( or any other race) is also good for democracy. More voters participated and clearer messages are delivered. I also submit A's winning the election with fewer participation means the wrong person got elected. Similar simple math illustrations can be demonstrated with 3 or more candidates.
Voters should not have the right to cast a negative vote because this would really make people feel that their votes do not matter. People already have a problem with voting because they feel they cannot make a difference. If people could basically cancel out another person's vote with a negative vote it would really hurt voting attendance.
Expressing voter discontent with all available candidates is appealing, but A) we already have the option of omitting a vote or simply not voting at all, which is generally interpreted as a negative vote, and B) if negative votes exceed votes for any single candidate would we reject the candidate or merely disempower that candidate by electing without mandate?