Graph-Theoretical Approaches to the Theory of Voting

The Condorcet set: Majority voting over interconnected propositions

“…This result can be viewed as a judgement aggregation counterpart to a well-known characterization of the top cycle as the outcome of non-strategic voting in simple binary tree agendas due to Miller (1977). Propositions 1.7 and 1.8 lend support to our argument for the sufficiency of Condorcet admissibility as a criterion of majoritarian decisions, since either result shows that every Condorcet admissible view is supportable by rules with unquestionable majoritarian credentials.…”

Optimizing group judgmental accuracy in the presence of interdependencies

“…(See Gelfand and Solomon, 1973;Gillispie, 1972;Black, 1958;Grainger, 1956;Barry, 1969;Baker, 1967Baker, , 1975. See also recent extensions by Gelfand and Solomon, 1974, 1977Grofman, 1978Grofman, , 1979Grofman, , 1980aKlevorick and Rothschild, 1978;Nagel and Neef, 1975;Grofman, Owen and Feld, 1982;Grofman, Feld and Owen, 1984 forthcoming;Grofman and Owen, 1984 forthcoming;Nitzan and Paroush, 1982;Miller, 1980;and Pinkham and Urken, 1981). 3.…”

“…If a group must choose from a set of alternatives (k >__ 2), then it may do so by using any one of a number of different binary choice procedures that decompose into sequences of pairwise (right fork or left fork) choices. The most common legislative procedure, the standard amendment procedure (see Farquharson, 1969;Roberts, 1973;and Miller, 1977), is a sequential binary choice process that pairs alternatives against one another, with the winner advancing into the next round. Other important voting procedures, for example, the Borda rule (Borda, 1781;Black, 1958;and Young, 1974), may also be represented in terms of the aggregation of information based entirely on pairwise choices.…”

“…These directions have been pursued by several authors. (See, for example, Feinberg, 1971;Walbert, 1971;Gelfand and Solomon, 1973, 1977Nagel and Neef, 1975;Klevorick and Rothschild, 1978;1980a. For detailed reviews of much of this work, see Penrod and Hastie, 1979;Grofman, 1980b.…”

The minimal covering set in large tournaments

“…1 These tournament solutions include the top cycle set (Schwartz, 1972;Miller, 1977), the uncovered set (Miller, 1980), the Banks set (Banks, 1985), the minimal covering set (Dutta, 1988), the bipartisan set (Laffond et al, 1993), the tournament equilibrium set (Schwartz, 1990), and others. A central principle in the search for new tournament solutions is that "smaller is better."…”