By David J. Mayston (auth.)
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Additional resources for The Idea of Social Choice
It will no longer always be the case (see p. 69 below) that a pairwise relation XI PX 2 , defined by a SCR, over a set containing Xl' x 2 and other alternatives, will imply that if S were to contain just Xl and X 2 we would still have XI PX 2 and therefore a social decision for Xl against X 2• Thus one cannot necessarily associate the same direct decision-making implication to a single pairwise statement of 'social preference' taken in isolation. Rather for the general case these statements take upon a significance when viewed as steps towards a social decision over the whole of S, with the interpretation of a single pairwise vote dependent upon the future agenda which follows from taking one route rather than another at this particular stage of the decision process.
R n ), being a rule, or functional relation, such that one and only one weak social preference relation R is implied for any set of individual R1,relations. 2 In a similar way to the individual case, we define a Social Choice Set, C(S), as the set of all points Xl in S such that xIRx for every X in S. C(S) may be empty as in Example 1. If C(S) contains two or more alternatives, 'social indifference' between all of those in C(S) is implied. The behavioural association of social indifference with an action of social choice in the name of the social body is usually left undefined in the literature, and in the absence of such an operational interpretation, social indifference, like the other social preference relations, remains purely a statement.
Under transitivity this should imply X3PX I, but in fact we have X3Ixl under a direct comparison. Black, however, suggests that when the number of individuals is even, in the event of a tie one of the voters should be made chairman with a casting vote. The principal difficulty with tlus procedure is that any vote by all members of the electorate on who should become the chairman is open to disagreement, possibly itself resulting in a tie and also intransitivities. A prior specification of which individual is to become chairman in the event of a tie breaks certain of the properties of MDR, by placing additional relative weight on that individual in such cases, when, as here, one assumes sincere voting.