The voters' paradox, spin, and the Borda count

Abstract

Electrical engineers employ some methods of linear algebra, derived from Homology Theory, to decompose the flow of current in a complex circuit into two components. The same decomposition can be applied to a ‘circuit’ containing nodes representing the candidates in a multicandidate election, connected by ‘wires’ carrying flows of net voter preference. In this case, the cyclic component measures the tendency towards a voters' paradox, while the cocyclic component measures the spreads in the Borda counts. When the cocyclic component is stronger, it masks the cycles in the cyclic component, and a voters' paradox is avoided; we call this ‘Borda Dominance’. Methods based on this decomposition provide a host of necessary and sufficient conditions for various degrees of transitivity of majority preference. Sen's well-known sufficiency theorem, together with some stronger theorems, are shown to depend upon a strong ‘double’ form of the masking phenomenon. This mathematically natural generalization of Sen's key hypothesis is revealed to be equivalent to a new, quantitative form of transitivity. Because the approach provides fresh insight into the underlying source of the voters' paradox, it appears to represent a promising new tool in social choice theory, with applications beyond those in the current paper.