Now consider how the vote would go among the three possible pairs of flavors. In a vote between two flavors it is assumed that people vote for the one of the two which is highest in their preferences, even though their number one choice may be different from the two being considered.

In a choice between vanilla and chocolate, the X groups would vote for vanilla, the Y group would also vote for vanilla and the Z group would vote for chocolate. So vanilla would win two-thirds of the votes and we could say that vanilla is socially preferred to chocolate.

In a choice between chocolate and strawberry the X group would vote for chocolate, the Y group would vote for strawberry and the Z group would vote for chocolate so chocolate would win. So chocolate is preferred to strawberry. Rationally we would expect that this would imply that vanilla would be preferred to strawberry. But consider a social choice by majority voting between vanilla and strawberry. The X group would vote for vanilla, the Y group would vote for strawberry and the Z group would vote for strawberry. So strawberry is socially preferred to vanilla.

Thus we have the irrational result that socially vanilla is preferred to chocolate and chocolate is preferred to strawberry but strawberry is preferred to vanilla.

Kenneth Arrow's Impossibility Theorem

Kenneth Arrow examined the problem rigorously by specifying a set of requirements that should be satisfied by an acceptable rule for constructing socially preferences from individual preferences; i.e.,

• Social preferences should be complete in that given a choice between alternatives A and B it should say whether A is preferred to B, or B is preferred to A or that their is a social indifference between A and B.
• Social preferences should be transitive; i.e., if A is preferred to B and B is preferred to C then A is also preferred to C.
• If every individual prefers A to B then socially A should be preferred to B.
• Socially preferences should not depend only upon the preferences of one individual; i.e., the dictator.
• Social preferences should be independent of irrelevant alternatives; i.e., the social preference of A compared to B should be independent of preferences for other alternatives.

What Kenneth Arrow was able to prove mathematically is that there is no method for constructing social preferences from arbitrary individual preferences. In other words, there is no rule, majority voting or otherwise, for establishing social preferences from arbitrary individual preferences.

This was a major result and for it and other work Kenneth Arrow received the Nobel prize in economics.

There is one way out of this impasse for making social decisions through the political process. If the individual preferences have some commonality then social preferences can be constructed. If the alternatives can be represented as being elements of a spectrum and the preferences of the individuals exhibit single peakedness then social preferences can be constructed.