voting paradox the possibility that if there are three candidates, A, B, and C, for democratic choice, with at least three choosers, and the choosers are asked to make sequential choices among pairs of candidates, A could defeat B by a majority vote, B could defeat C, and C could defeat A. (This would be the outcome if the choosers’ preferences were ABC, BCA, and CAB.) Hence, although each individual voter may have a clear preference ordering over the candidates, the collective may have cyclic preferences, so that individual and majoritarian collective preference orderings are not analogous. While this fact is not a logical paradox, it is perplexing to many analysts of social choice. It may also be morally perplexing in that it suggests majority rule can be quite capricious. For example, suppose we vote sequentially over various pairs of candidates, with the winner at each step facing a new candidate. If the candidates are favored by cyclic majorities, the last candidate to enter the fray will win the final vote. Hence, control over the sequence of votes may determine the outcome. It is easy to find cyclic preferences over such candidates as movies and other matters of taste. Hence, the problem of the voting paradox is clearly real and not merely a logical contrivance. But is it important? Institutions may block the generation of evidence for cyclic majorities by making choices pairwise and sequentially, as above. And some issues over which we vote provoke preference patterns that cannot produce cycles. For example, if our issue is one of unidimensional liberalism versus conservatism on some major political issue such as welfare programs, there may be no one who would prefer to spend both more and less money than what is spent in the status quo. Hence, everyone may display single-peaked preferences with preferences falling as we move in either direction (toward more money or toward less) from the peak. If all important issues and combinations of issues had this preference structure, the voting paradox would be unimportant. It is widely supposed by many public choice scholars that collective preferences are not single-peaked for many issues or, therefore, for combinations of issues. Hence, collective choices may be quite chaotic. What order they display may result from institutional manipulation. If this is correct, we may wonder whether democracy in the sense of the sovereignty of the electorate is a coherent notion. See also ARROW’S PARADOX, DECISION THEORY, SOCIAL CHOICE THEORY. R.Ha. wang, pa, Chinese political titles meaning ‘king’ and ‘hegemon’, respectively. A true wang has the Mandate of Heaven and rules by te rather than by force. The institution of the pa developed during a period in which the kings of China lacked any real power. In order to bring an end to political chaos, the most powerful of the nobles was appointed pa, and effectively ruled while the wang reigned. During the Warring States period in China (403–221 B.C.), rulers began to assume the title of wang regardless of whether they had either the power of a pa or the right to rule of a wang. After this period, the title of Emperor (ti or huang-ti) replaced wang. B.W.V.N.