(15 points) Consider a household of size 6. Four members of the household are teenagers and the two remaining members are the parents. Every de- cision in this household is made through an especial voting system. In the rst round each member announces her/his preferences over the Options. Then the rst choice of the parents gets 2 scores while the rst choice of each teenager gets 1 score. The option with the highest score will be the winner. In case of a tie the option which is the rst choice of the highest number of the voters is the winner. Let us show each of the teenagers with t, while 3' E {1, 2,3, 4} and par ents with pl, 332. Assume the family is deciding about a city to travel for their vacation. Options are {Mtlan(M), Rome(R), Venice(V)]> and the preferences are as follows: 31 32 t3 *4 P1 .92 M M M R V V R R R M M R V V V V R M (a) Which city is the Winner under their voting system? Show your work. (2 pts) (b) Can the teenagers collude to manipulate the voting system by strate- gically reporting their preferences in such a way that regardless of the parents' preferences, the teenagers be better off compared to the re- sults found in part a? How? (4 pts) (c) If instead of their voting system, they use the Borda voting system, which city is the winner? Which city is the Cmdorcet winner? (5 pt s) (d) Now assume that each member also cares about others\" happiness. Instead of voting, the household decides based on the gained utility. The utility of each memberj forj 6 {1,2,- - ,6} is U_,- = 25 + 5(f 1) + 33 if her/his rst choice is chosen, UJ- = 15 + 5f + 3(s 1) if her/his second choice is chosen and Uj = 5 + 5f + 33 if her/his last choice is the chosen location, while 1' is the number of members whose rst choice is the chosen city and s is the number of members whose second choice is the chosen one. If they want to maximize the total utility of the household, which city they will choose to travel? Why? (4 pts)