Inductive Trap of Death

There is a prisoner, an executioner, their families, a king and a mathematician.

The prisoner loves his family more than anything else.

The executioner loves his family more than anything else.

Every morning at 6 the executioner can sign a death warrant E for the prisoner.

Every morning at 6 the prisoner can sign a death warrant P for half of the executioner's family.

If death warrant E is not issued in January, the king signs a death warrant for the executioner's entire family.

Knowing he will die the prisoner wants revenge on the executioner's family, but loves his own family more than revenge.

Every morning at 7 the king checks and if death warrant P is signed but death warrant E is not signed, then he revokes P and issues a death warrant for the prisoner's family.

No one except the king can revoke a death warrant. P and E can each only be issued just once and when being decided each of these actions is done privately. All warrants are carried out 3 hours after being signed.

Both the prisoner and executioner are made aware of all these facts in each other's presence on the night of December 31. A mathematician is also present at this meeting.

Right after the meeting the mathematician is forced to make a prediction on who will live and who will die on a piece of paper that only he sees. On February 1 the king examines the paper. If the prediction is not accurate, the king issues a death warrant for the mathematician.

---------------- What happens?

The mathematician uses induction and concludes that warrants E and P are both issued on January 1 causing casualties on both sides. In reality, warrant E is issued on January 17 and warrant P is never issued, causing the prisoner to be executed and the executioner's family to escape this predicament unscathed. The mathematician is executed on Feb 1.

For simpler but more subjective version see: original formulation of the problem


Questions:

Why does induction fail where common sense succeeds?
Does game theory provide the correct answer?

Solutions from the literature

Want more? Modified prisoner problem