Democracy, Right to Reject, and Common Knowledge

Should there be a provision to reject all candidates during an election? I will explain why I support such a provision in the ballot using a logical puzzle and introducing the concept of common knowledge.

In India, a new option, None of the Above (NOTA) [link], has been introduced in the ballot in ongoing general elections. This gives the voter a choice of not voting for any of the candidates. Although NOTA is equivalent to an invalid vote, I believe that it can significantly affect the process. The voter exercising this option may not explicitly reject all candidates. Instead, he/she may simply be informing that they are not engaged in the political process. Thus, I believe that another additional option of rejecting all candidates should be present in the ballot. I will explain my reasons in this post even if they are treated being equivalent to invalid votes. First, I will talk about a logical puzzle whose solution is rather tricky. Second, I will present a follow-up question. The answer to this follow up question is explained by the concept of common knowledge. Third, I will relate this concept to dictatorships and revolutions. I will also explain why I believe Facebook, Twitter, etc. are really powerful media. It is not because protests can be organised through these tools, but because of something more fundamental that is also explained by common knowledge. Finally, I will provide my beliefs about how the right-to-reject initiative can help improve a process even in successful democracies.

The Blue-Eyed Islanders Puzzle:

Source [link, link]: There is a tribe residing in an island in which the religion dictates that a person should not know the colour of their eyes. If an individual learns the colour, he/she commits ritualistic suicide just before the next town gathering. People learn of the suicides only after they have taken place; it is announced at the town gathering. The town gathering happens every day at noon. Everyone in the island attends all the town gatherings. It has a small population; every one knows the colors of other people’s eyes, but they do not discuss it for obvious reasons. They do have reflective surfaces in the island so that people do not discover their own eye colour. An outsider enters the island, and he is ignorant about this practice. When he leaves the island, he tells the people at his last town gathering (after the suicide ritual is suppose to have been finished) that he was happy to see other people with blue eyes on the island.  He is telling the truth and everyone trusts him. What happens to the blue eyed people in the island in the following days? Assume that everyone in the town is logical and follow their religious duties sincerely.

This puzzle is tricky to solve. Do take some time to think about it if you are mathematically inclined. I will provide the solution in the next paragraph. If you have clarification about the assumptions in the puzzle, do google for the puzzle for more details or leave me a comment.

The Solution:

Source [link, link]: The trick is to use mathematical induction. If there is only one person with blue eyes in the island, since he know the colour of all others’ eyes in the island, he knows right away that his eyes are blue. He commits suicide just before the next town meeting on the next day. If there are two people with blue eyes, each of them assumes that the other person is only person with blue eyes and waits for the other person to commit suicide next day. When neither of them commit suicide, they both know that they both have blues eyes. Therefore, on the second day, they both commit suicide. This can be extended to n people with blues eyes, i.e, they all commit suicide on the nth day.

The Follow-Up Question:

Source [linklink]: If there are more than one individual with blue eyes, everyone in the island already know that there is at least one person with blue eyes. The outsider only said that there is at least one person with blues eyes in the island, and the islanders already knew that. What additional information did the outsider add that lead to the consequence? If the outsider did not add any new information, isn’t the solution above contradictory to the reasoning presented here?

Common Knowledge:

The answer to the apparent contradiction lies in the concept of common knowledge [link]. If a fact p is known to everyone, it is called mutual knowledge [link]. In this context, p is the fact that there is at least one person with blue eyes. If everyone knows that everyone knows p, and everyone knows that everyone knows that everyone knows p, and so on ad infinitum, then p is common knowledge.

In the context of the islanders puzzle, consider the time when the outsider had not yet visited the island. If there are two people, A and B, with blue eyes, both of them know that there is at least one person with blue eyes. A knows that B has blue eyes, and B knows that A has blue eyes. A, however, does not know that B also knows that there is at least one person with blue eyes. Similarly, if there are 100 people with blues eyes, every one knows that there are at least 99 people with blue eyes. Everyone knows that everyone knows that there are at least 98 people with blue eyes. Every knows that everyone knows that everyone knows that there are at least 97 people with blue eyes. Clearly, this does not continue up to infinity. What the outsider does is to make it a common knowledge that there is at least one person with blue eyes. That triggers the chain of events.

Dictators, Revolutions, and Common Knowledge:

Although the primary reason for the revolutions in the Arab world is believed to be rising food prices [link, link], Wikileaks is considered a major inspiration for the Arab spring [link]. Wikileaks exposed the extent of corruption in the Tunisian and Egyptian dictatorship. It is very unlikely that people of the two countries were unaware that their dictators were corrupt. They may have not known the extent of the corruption, but they all must have known that their dictators were corrupt. Due to the absence of  strong media and public discourse, they probably did not know that everyone knew that the dictators were corrupt. Wikileaks made the mutual knowledge of the corrupt rulers a common knowledge. As the diplomatic cables were published, everyone knew that everyone knew that (and so on) their dictators were corrupt. That kick-started the revolution. I believe that the real power of Facebook and Twitter lies in this phenomenon where mutual knowledge becomes common knowledge more than facilitating the organisation of protests. Dictators hate public gatherings at town squares [link]. A part of the reason for that is the evolution of mutual knowledge to common knowledge happens quickly during public meetings. Of course, common knowledge is not sufficient for revolutions to occur.  The underlying factors such as public dissatisfaction, leadership, unity among the people, etc. have to be given their due credit. A condition stronger than mutual knowledge, however, is necessary for a revolution to take place.

Common Knowledge in Successful Democracies:

The concept of common knowledge is applicable in a democratic setup too. People strategically vote for the party they least hate and has a good chance of winning the elections [link]. It is possible that they may not really want to vote for the party. Some may chose to abstain from voting in such elections. It is very hard to determine how many people have abstained from voting due their lack of faith in the system and how many have abstained due to their ignorance or apathy towards the political process. As a result, the established political parties cyclically exchange power in the electoral cycle. Although people may prefer a fresh face in the elections, no one really knows how many people want a fresh face. Many may not be happy with the choices presented to them, but it is not a common knowledge. The provision of choosing none of the candidates (the NOTA option) or rejecting all candidates makes the displeasure a common knowledge.  That will allow a fresh face, who would have otherwise been reluctant, to enter politics.

Many are apathetic towards the electoral process. They are not going to vote. There are many who are ignorant about the political process, but not apathetic to it. If the NOTA option is provided, such people would cast the NOTA vote. This informs the political parties and candidates that these votes are up for grabs. An incumbent candidate would have the incentive to carry out and advertise their work, and other candidates would have incentive to politicise them for their benefit. The NOTA option can also prevent impersonation-related voter fraud in developing countries.

A vote to reject all candidates informs aspiring politicians of their chances in the subsequent elections. That may enable a dynamic system in which the established political entities cannot be complacent with their support and new people show up when the establishment fails to deliver.

Does it Happen in Practice?

My interest in these options are philosophical/theoretical in nature. They many not actually translate to a significant change in the electoral process, but it does have the potential, and the cost is insignificant. We will have additional information by implementing the system, and it is up to the people to use this information well. It would be interesting to see if there were improvements in people’s satisfaction with their elected representative before and after NOTA was implemented. I am not aware of such studies.

In the Delhi election late last year, a new party, Aam Aadmi Party (AAP), was voted the second largest party. It was agreed upon in the social media that if another elections happens soon (because of the hung parliament), they could become the single largest party. This is an example where “AAP can win” became a common knowledge, and thus, had the potential to change the outcome of the subsequent election. Thus, another party could become a mainstream party. More choice is good, isn’t it [link]?

UPDATE: The NOTA option did affect Russian elections [link].