Interns retained Payoff-Head of Lab Payoff-Scientist А, В 10 А, С 9 5 А, D 7 3 В, С 8 4 В, D 4 С, D 1 А, В, С А, В, D 10 5 А, С, D В, С, D 3 8 1 7 2. (O
A scientist works in a lab with four summer interns and, as it is the end of the summer, he anticipates the head of the lab will ask him which of them he wants to retain. In terms of quality, the scientist and the head of the lab agree that intern A is better than intern B who is better than intern C who is better than intern D. The scientist would like to keep as many interns as possible and, given any number, the highest-quality ones. However, due to funding restrictions, the head of the lab wants to limit the number of interns that are retained. The head of the lab initially tells the scientist to select two interns to retain. After the scientist chooses two interns, the head of the lab decides whether to allow the scientist to retain a third intern. Thus, the sequence of moves is: 1) the scientist chooses two interns (both of whom are then retained); 2) the head of the lab decides either to allow the scientist to choose a third or not; and possibly 3) if the head of the lab chose to allow the scientist to retain a third intern, the scientist chooses a third intern to retain. There are 10 possible outcomes in terms of the number of interns that are retained at the end of the game (two or three) and who are they. These outcomes and the associated payoffs for the head of the lab and the scientist are shown in the accompanying table.
a. Write down the extensive form of the game (though you can exclude
payoffs) and describe what a strategy looks like for the scientist, and
for the head of the lab.
b. What does SPNE predict as to whether the scientist ends up with two
or three interns? Which interns are selected? (Hint: Watch the episode
entitled “Games” from the fourth season of the television series House.)
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