Counting Agendas That Elect Alpha

Five voters rank four proposals (best to worst): - Voter 1: Beta, Gamma, Delta, Alpha - Voter 2: Beta, Alpha, Delta, Gamma - Voter 3: Delta, Alpha, Beta, Gamma - Voter 4: Gamma, Delta, Alpha, Beta - Voter 5: Alpha, Gamma, Beta, Delta The chair lists the four proposals in some order $(x_1,x_2,x_3,x_4)$. The votes are held in sequence: $x_1$ against $x_2$, the winner against $x_3$, then the winner against $x_4$; each is a simple majority vote and the last survivor is adopted. Voters are sophisticated: at each vote a voter supports the option that leads to the final outcome he prefers, assuming the later votes are also played that way. Of the $4! = 24$ possible orders, for how many is Alpha the adopted proposal?

• backward induction
• induction

Source: Agenda control in sequential binary voting, after Farquharson (1969) and McKelvey and Niemi (1978). Statement written for AxiomIQ.