Cycles and Intractability in Social Choice Theory

We introduce the (j, k)-Kemeny rule – a generalization that aggregates weak orders. Special cases include approval voting, the mean rule and Borda mean rule, as well as the Borda count and plurality voting. Why, then, is the winner problem computationally simple for each of these other rules, but NP -hard for Kemeny? We show that winner intractability for the (j, k)-Kemeny rule first appears at the j = 3, k = 3 level. The proof reveals that computational complexity arises from the cyclic component in the fundamental decomposition −→w = −→w cycle + −→w cocycle of [16]. Thus the existence of majority cycles – the engine driving both Arrow’s impossibility theorem and the Gibbard-Satterthwaite theorem – also serves as a source of computational complexity in social choice.