M, military ruler of Logistan, has deferred to international pressure and agreed to hold an election in which he will run against his arch-nemesis, B. Both M and B, being politicians, are liars. You have been engaged as an independent consultant and charged with devising a representative voting procedure, i.e., your mission is to tally the true preference of each citizen who has a consistent, determinable opinion, and no other.

The chief complication relates to the fact that the Logistani electorate is composed of five (to your eyes) indistinguishable ethnic groups, each of which have a distinctive relationship to the truth. When expressing their voting preference:

• Knights respond honestly.
• Liars negate their true view.
• Subversives consider how a a knight with the same views would respond, then say the opposite.
• Revisionists admire knights and liars, and despise subversives. A revisionist will copy the most recent knight or liar to have voted, unless a subversive has voted more recently. In this latter case, the revisionist will vote for the opposite of that subversive.
• Contrarians reverse the answer of the most recent voter.

A contrarian or revisionist would respond randomly if he were the first voter queried.

You are to hold the election at the national stadium, to which the entire Logistani electorate has been invited. After some thought, you decide you can conduct the vote by asking members of the assembled electorate a single yes/no question. This is an open ballot, so each voter will call out his/her answer to the question for all to hear.

Suggest a viable question and any procedural arrangements, explaining how they enable you to fulfill your mission.

There is some confusion about how subversives vote/respond. I've decided the best way to deal with this is with some examples.

1) Suppose that you are a subversive who prefers B as the next leader of Logistani. Suppose further that you were asked:

Q: "Do you prefer B as the next leader of Logistan?"

You quickly decide that a knight who shared your preference/political temperment would answer this straightforward question affirmatively. You therefore respond by saying: "No".

2) Q: Are you a knight?

You realise that a knight who had managed to slip into your subversive body would answer honestly by saying no. Therefore, you respond by answering: "Yes".

One reason why people may find the subversive definition confusing is the misperception that a subversive will always behave as a liar. In the two examples above this is indeed the case. However, there are situations in which the subversive and liar respond differently. One rather trivial example would be when answering the question: "Are you a liar?". There are also other questions or meta-questions to which the liar and subversive respond differently. It is important to find them.

Edited on March 26, 2008, 10:54 am

Posted by FrankM on 2008-03-24 20:35:15