2. (40 points) Consider a divide-the-dollar game in which five citizens {X, Y, A, B.C} divide a single dollar among them. A policy here is an allocation of the dollar among all five citizens (dx,dy.da,dg.de). where d; is the amount of money that individual gets. We have argued before that no Condorcet winner allocation exists in this game. We now suppose that citizens X and are also candidates, besides being voters. They propose allocations sequentially: X proposes first, then proposes second. All five citizens vote on the two proposed allocations and the outcome is decided based on simple majority rule. (i) Suppose that the two candidates are solely office-motivated. Show that Y wins the election for sure. T (ii) Suppose now that the two candidates are policy-motivated: in particular, they care about the amount of money that they get in the winning allocation, independent of who proposed it. If X proposes (1/2,1/2,0,0,0), what does optimally propose in turn? Is it smart of X to propose to split the dollar equally between the two candidates? (iii) If the candidates are policy-motivated, what does X optimally propose to maximize her share in the winning allocation? Who wins the election? (iv) Is the equilibrium with policy-motivated candidates convergent, i.e., do both candi- dates propose the same allocation? What are the payoffs that the candidates get in equilibrium? What are the equilibrium payoffs of ordinary citizens A, B. C