= 5. Consider a matching model where agents in X = {x1, x2, x3}, Y {y1, 92, 93}, and Z = {21, 22, 23} are to be matched in triples (xi, yj, k). Assume that all agents have strict preferences over pairs of agents from the other groups. A ranking function r: XxYxZ{1, ..., 9} rep- resents the preferences in the following way. r(x1,y2, 2) = (x1 (92, 22), Ty2 (x1, Z2), T22 (x1,y2) = (2, 3, 6), means that x1 ranks (y2.22) as her second best option, y2 ranks (x1, 22) as her third best option and z2 ranks (x1,y2) as her sixth best option (and similarly for all possible triples). (a) How many different matchings are there? (b) Consider preferences over the matchings that satisfy the follow- ing: 21 22 23 Y1 3 T21: 12 Y2 Y3 Ty 21 22 23 21 22 23 21 22 23 Y1 Y1 ,Tx3 2 1 91 Y2 1 3 Y3 ,Tx2: Y2 21 22 21 22 23 I1 1 , Ty2 3 I2 I3 321 21 22 23 I1 , Ty3 12 12 I3 Tz1 1 Y1 y2 y3 Y1 y2 y3 21 3 21 1 21 ,T22: , T23: 312 1 12 x2 x2 4 23 x3 x3 The empty spaces in the tables can be filled with arbitrary rank- ings (consistent with the existing ones though). A matching into triples is blocked by (x', y', ') if x' prefers (y', '), and y' prefers (x', ') and z' prefers (x', y') to their partners in the matching. A Matching is stable if it is not blocked by any triple. Show that with these preferences, stable matchings into triples do not exist. Hint: Show that must get at least her third best option in any stable matching and in fact must be matched with (y1, 21). Show next that y2 must be matched with (2, 2). Finally show that {(x1,y1, 1), (x2, y2, 22), (3, y3, z3) is not stable