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.

Hi Dej Mar,

1) You've misunderstood the nature of subversives. Please review the revised version of comment 7, Further clarification: Subversives.

2) You are right that some Logistanis prefer M and others prefer B. Preferences differ even within each ethnic group, so that some knights prefer M, others would rather have B.

3) While you may be right that each revisionists may have her own opinion, you should consider whether that opinion is consistent and determinable. The revisionist's response to any given question can change depending on the expressed views of whoever has responded recently. Who can say what is the true opinion of such a voter?

Nevertheless, your remarks include some insights worth pursuing.

4) The only procedural limitations I foresaw were those mentioned in the problem text: The ballot is to be "open", with each person calling out their vote in response to a single yes/no question.

I suppose I should add that the voters are to be left to vote there conscience without coercion, as well as adding a caveat against anything too weird (say, M shooting everyone and re-electing himself). Please feel free to use your imagination.

Edited on March 26, 2008, 10:58 am

Posted by FrankM on 2008-03-25 12:29:18