In ancient Athens, the philosopher Socrates was on trial for being a heretic. The citizenry would vote, first on guilt and then on penalty. The usual penalties were banishment and death. Suppose there were three voting blocs: 40% wanted him to be acquitted (A),30% considered him guilty, but favored the lesser punishment of banishment (B), and another 30% viewed him as guilty and deserving of the death penalty (C).

a. Socrates was found guilty in a first round of voting and then sentenced to death in the second round (by a greater margin than those who voted guilty in the first). For this to be possible (with sincere voting), how would type A have to rank the three alternatives, acquittal (a) vs. banishment (b) vs. death (c)?

b. What would have to Bís preference ranking, so that it has no reason to vote insincerely?

c. In ancient Rome, the procedure was different: citizens would first vote on execution vs. life, and then on banishment vs. acquittal in the second round if there was no majority for execution. What would have happened in this case, depending on Cís preferences (assuming people vote sincerely)?

d. Supposing C prefers banishment to acquittal, and Aís preferences are as determined in part a). Would A have an incentive to vote sincerely under the Roman procedure? What if C prefers acquittal to banishment?

e. At a similar trial, in Rome, a speaker who supported acquittal proposed to vote directly over all three alternatives, so that the one with the most votes wins. The outcome was actually banishment, rather than acquittal. Assuming the distribution of most-preferred alternatives was as in the Socrates trial, what must have been the full preference rankings for the three types, in order for this to happen? Who voted insincerely?