An Introduction to Decision Theory

Megan, Joe and Nick have decided to spend the summer holiday together. Unfortunately, they disagree on where to go. The set of alternatives includes Acapulco (a), Belize (b) and Cape Cod (c). Everyone in the group agrees that it would be fair to make the decision by voting, and their preferences are as follows.

It seems plausible to maintain that, conceived of as a group, the friends prefer a to b, because in a pairwise choice a will get two votes (from Megan and Nick) whereas b will get just one (from Joe). Furthermore, the group prefers b to c, because b will get two votes (from Megan and Joe) and c one (from Nick). Finally, the group also prefers c to a, because c will get two votes (from Joe and Nick) and a one (from Megan). However, by now the voting rule has produced a cyclic preference ordering: a is preferred to b, and b to c, and c to a. This holds true although none of the individual preference orderings is cyclic.

The observation that the majority rule can give rise to cyclic preference orderings is not a novel discovery. This point was extensively analyzed already by the French nobleman Marquis de Condorcet in the eighteenth century. Nowadays this result is known as the voting paradox. It serves as a classic illustration of the difficulties that arise if a group wishes to aggregate the preferences of its individual members into a joint preference ordering. Evidently, if group preferences are cyclic, they cannot be choice guiding. (But why not roll a die and let chance make the decision? Answer: This does not explain why one option is better than another. It can hardly be better to spend the summer in Acapulco because a die landed in a certain way on a particular occasion.)