Four students are choosing colleges to go to. There are four colleges and each collegecan accept only one student. Students’ preferences are represented by the followinglists:s1s2s3s4c1c2c1c1c2c4c4c3c3c3c2c4s1s2s3s4c4c1c3c2Each college has preferences (rankings) over students which is represented by thefollowing lists:c1c2c3c4s4s2s1s1s3s1s2s4s2s3s3s3s1s4s4s2c1c2c3c4(a) Suppose the allocation of student to schools is implemented using serial dicta-torship mechanism with the following ranking of students:s1,s2,s3,s4. Findthis allocation.(b) Suppose the students are initially allocated the seats in the following way:(s1,c4),(s2,c1),(s3,c3) and (s4,c2), and after initial allocation the governmentruns the top trading cycle mechanism where only students’ preferences aretaken into account. Find the final allocation of students to colleges generatedby the mechanism.(c) Find a stable students-optimal allocation by running deferred acceptance al-gorithm.ECON0027 Game Theory, HA51TURN OVER
2. Consider a cooperative game in characteristic form with three players. The value ofa grand coalition isv({1,2,3}) = 10.The value of smaller coalitions arev({1,2}) =v({1,3}) =v({2,3}) =z, wherezis a parameter. Finallyv({1}) =v({2}) =v({3}) = 1.(a) Supposez= 3. Find the core of this game.(b) What happens to the core of the game whenzgets larger.(c) Find values ofzfor which the core of the game is empty