Yule Ball Dance Victor Ask Do not ask Hermione Yes No Ron Do not Do not Do not Ask Ask Ask ask ask ask Victor 8 7 2 Hermione Hermione 4 Ron 3 6 Hermione 6 3 1 Yes No Yes No 1 3 4 8 2 7 8 4 7 2
It is the week before the Yule Ball Dance, and Victor and Ron are each contemplating whether to ask Hermione. As portrayed above, Victor moves first by deciding whether or not to approach Hermione. (Keep in mind that asking a girl to a dance is more frightening than a rogue bludger). If he gets up the gumption to invite her, then Hermione decides whether or not to accept the invitation and go with Victor. After Victor (and possibly Hermione) have acted, Ron decides whether to conquer his case of nerves (perhaps Harry can trick him by making him think he’s drunk Felix Felicis) and finally tell Hermione how he feels about her (and also invite her to the dance). However, note that his information set is such that he doesn’t know what has happened between Victor and Hermione. Ron doesn’t know whether Victor asked Hermione and, if Victor did, whether Hermione accepted. If Ron does invite Hermione and she is not going with Victor— either because Victor didn’t ask, or he did and she declined—then Hermione has to decide whether to accept Ron’s invitation. At those decision nodes for Hermione, she knows where she is in the game, since she is fully informed about what has transpired. The payoffs are specified so that Hermione would prefer to go to the dance with Ron instead of with Victor. Both Ron and Victor would like to go with Hermione, but both would rather not ask if she is unable or unwilling to accept.
For above on the Yule Ball Dance, now assume that, before he decides
whether to ask Hermione, Ron observes whether or not Victor asked her.
However, if Victor does invite Hermione, Ron does not know her answer to
Victor when he decides whether to invite her himself.
a. Write down the extensive form game.
b. Derive all subgame perfect Nash equilibria.
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