One of the critical moments early on in the The Lord of the Rings trilogy is the meeting in Rivendell to decide who should take the One Ring to Mordor. Gimli the Dwarf won’t hear of an Elf doing it, whereas Legolas (who is an Elf) feels similarly about Gimli. Boromir (who is a Man) is opposed to either of them taking charge of the Ring. And then there is Frodo the Hobbit, who has the weakest desire to take the Ring but knows that someone must throw it into the fires of Mordor. In modeling this scenario as a game, assume there are four players: Boromir, Frodo, Gimli, and Legolas. (There were more, of course, including Aragorn and Elrond, but let’s keep it simple.) Each of them has a preference ordering, shown in the following table, as to who should take on the task of carrying the One Ring.
Of the three non-Hobbits, each prefers to take on the task himself. Each would prefer that other than themselves and Frodo, no one should take the Ring. As for Frodo, he doesn’t really want to do it and prefers to do so only if no one else will. The game is one in which all players simultaneously make a choice among the four people. Only if they all agree—a unanimity voting rule is put in place—is someone selected; otherwise, no one takes on this epic task. Find all symmetric Nash equilibria.
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