Prisoner's dilemma:

If they both choose to help each other, there is no lynch and no other affect.

If they both choose to kill each other, they both die.

If A chooses to kill B but B chooses to help A, B dies and A is silenced (includes voiding powers, but not night votes) until the next day phase has ended. As such, A is a suspicious person who was just directly responsible for B's death and is unable to defend his action.

The silencing is a key component, since otherwise there would be no dilemma (voting to kill the other player wouldn't hurt you).