2. [20 pts] A society is having a referendum to decide the level of spending p on a certain project. Society consists of several groups of voters with different preferences. In particular, the table below describes the fraction of the population of each group and the indirect utility function of individuals in each group. Voter type | Percent of population Indirect utility function Group 1 25 8p - 2p- Group 2 25 12p - p- Group 3 20 8p -4p2 Group 4 30 16p - 2p2 There are three different options on the ballot: option A is to set p = 1; option B is to set p = 2; option C is to set p = 4 (ignore how or why these options ended up on the ballot). (i) Given a voter's indirect utility function, the voter will have an implied ranking of the three alternatives. For each group of voters, write down this implied ranking in the form "Group r: A > B > C". Use "=" for indifference. 2 (ii) Assuming voters are sincere, which alternative wins if the referendum is conducted using plurality rule, Borda and negative voting (also known as antiplurality - where you vote against your least favorite alternative and the alternative with the least votes against wins)? As usual, assume any ties are broken by randomization (i.e., if a group of voters is indifferent between A and B, then half of those voters will rank A above B and half will rank B above A). (ii) Sequential majority rule with the order A, B, C? (iv) Instant-runoff voting? Recall the procedure is: (1) calculate vote shares under plural- ity; (2) if one alternative has a majority, that alternative wins; (3) if not, the alternative with the lowest vote share is eliminated; (4) voters who voted for the eliminated alter- native vote for their next preferred alternative; (5) recalculate vote shares and repeat steps above, until an alternative receives a majority. (v) Under "best-worst' voting'? Recall that this can be thought of as a points based system where a voter's favorite alternative gets 1 point, and least favorite alternative gets -1 point, with all other alternatives getting 0. The alternative with the most points wins. (vi) Which alternative, if any, is the Condorcet winner? Based purely on your answers above, can we rule out the Condorcet consistency of any of the voting rules considered here