The most popular voting rule, majority choice, may easily fa a unique social preference. Consider, for example, the simple three individuals A, B and C are asked to vote for three alternative states of the world, x, y, and z. Now suppose A prefers x to y and y to z, B also prefers y to z but instead prefers z to x, whereas C like B prefers z to x but like A prefers x to y. It is easily checked that x wins over y by the two vo C, y wins over z by the two votes of A and B, but, unfortunately x does not therefore win over z because z in turn wins over x by the two votes of B and C. In other words, in this simple case of three voters and three alternative options, the democratic method of majority choice leads to a stalemate.

What Arrow now demonstrated was that this stalemate can occur, not just under a constitution based on the principal of majority rule, but under every conceivable constitution except that of dictatorship: it is logically impossible to add up or otherwise combine the choices of individuals into an unambiguous social choice except by rigging the 'constitution'

....