Solution
Sophisticated Voting on an Agenda
Show the problem again
Five voters choose among four proposals: Alpha, Beta, Gamma, Delta. Their strict rankings, best to worst, are - Voter 1: Gamma, Beta, Alpha, Delta - Voter 2: Beta, Gamma, Delta, Alpha - Voter 3: Alpha, Beta, Delta, Gamma - Voter 4: Alpha, Beta, Delta, Gamma - Voter 5: Gamma, Alpha, Beta, Delta The agenda is: first, Alpha against Beta; second, the winner against Gamma; third, that winner against Delta. Each vote is decided by simple majority and the final survivor is adopted. Voters know all preferences and the agenda, and each votes strategically: at every vote he supports the option that leads to the final outcome he prefers, found by backward induction. Which proposal is adopted? Answer with its name.
Worked solution
The answer is Beta. Work backward from the last vote. Delta loses to Alpha (4-1), to Beta (5-0) and to Gamma (3-2), so whichever proposal reaches the final vote wins it, and the third vote is a formality. The second vote is therefore decisive. Alpha against Gamma: voters 1, 2 and 5 prefer Gamma, so Gamma wins 3-2. Beta against Gamma: voters 1 and 5 prefer Gamma, voters 2, 3 and 4 prefer Beta, so Beta wins 3-2. So in the first vote, choosing Alpha leads to the final outcome Gamma and choosing Beta leads to Beta. Voters 2, 3 and 4 prefer Beta to Gamma, so Beta wins the first vote 3-2 and is adopted. A naive voter who just compares the two options on the table would instead elect Alpha over Beta (3-2), then lose it to Gamma, ending with Gamma, which shows that foresight changes the result.
Source: Sequential agenda voting with sophisticated voters, after Farquharson (1969) and McKelvey and Niemi (1978). Statement written for AxiomIQ.