A systematic approach to the construction of non-empty choice sets

Abstract

Suppose a strict preference relation fails to possess maximal elements, so that a choice is not clearly defined. I propose to delete particular instances of strict preferences until the resulting relation satisfies one of a number of known regularity properties (transitivity, acyclicity, or negative transitivity), and to unify the choices generated by different orders of deletion. Removal of strict preferences until the subrelation is transitive yields a new solution with close connections to the "uncovered set" from the political science literature and the literature on tournaments. Weakening transitivity to acyclicity yields a new solution nested between the strong and weak top cycle sets. When the original preference relation admits no indifferences, this solution coincides with the familiar top cycle set. The set of alternatives generated by the restriction of negative transitivity is equivalent to the weak top cycle set.

Dominant, Weakly Stable, Uncovered Sets: Properties and Extensions

“…MD=MU. Consequently, for tournaments the untrapped set coincides with the minimal dominant set (Duggan, 2007). lemma 3.…”

“…It is said that x traps y if x dominates y and is not reachable from y via µ, xµy and there is no path from y to x (Duggan, 2007). An untrapped set (Duggan, 2007) UT is comprised of all alternatives that are not trapped.…”

“…An untrapped set (Duggan, 2007) UT is comprised of all alternatives that are not trapped. UT is always nonempty and is nested between the strong and weak top cycles, MU⊆UT⊆MD (Duggan, 2007) It follows from the definitions that any dominant set is at the same time an undominated set. Thus Lemma 2 implies non-emptiness of MU, which implies non-emptiness of UT.…”

“…It turns out that there are five different definitions of covering: 1) y covers x if yµx and L(x)⊆L(y)∪H(y) (Duggan, 2007), then x is uncovered ⇔ ∀y: yµx ⇒ ∃z: xµz & zµy;…”

“…Therefore various sets of alternatives were proposed as solution concepts for majority voting games and/or as social choice rules. Below several such concepts are considered and compared: dominant set (Ward, 1961;Smith, 1973), minimal dominant set (Fishburn, 1977;Miller, 1977), undominated set (Ward, 1961), minimal undominated set (Schwartz, 1970(Schwartz, , 1972, weakly stable and minimal weakly stable sets (Aleskerov, Kurbanov, 1999), uncovered set (Fishburn, 1977;Miller, 1980), uncaptured and untrapped sets (Duggan, 2007). In addition we introduce a new solution concept -k-stable sets of alternativesand analyze its relations with other solution concepts listed above.…”