? E XERCISE 263.2 (Empty core in roommate problem) An even number of people have to be split into…

? EXERCISE 263.2 (Empty core in roommate problem) An even number of people have to be split into pairs; any person may be matched with any other person. (The matching problem is “one-sided”.) Consider an example in which there are four people, i, j, k, and R. Show that if the preferences of i, j, and k are those given in Figure 264.1 then for any preferences of R the core is empty. (Notice that R is the least favorite roommate of every other player.)

i          j              k

j         k              i

k          i              j

R    R     R

 

Figure 264.1 The preferences of players i, j, and k in the game in Exercise 263.2.