In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.

O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May's axioms, we can uniquely determine how to vote on three alternatives. In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, agrees with Minimax voting in all three-alternative elections, except perhaps in some improbable knife-edged elections in which ties may arise and be broken in different ways.

There is an extensive literature in social choice theory studying the consequences of weakening the assumptions of Arrow's Impossibility Theorem. Much of this literature suggests that there is no escape from Arrow-style impossibility theorems, while remaining in an ordinal preference setting, unless one drastically violates the Independence of Irrelevant Alternatives (IIA). In this paper, we present a more positive outlook. We propose a model of comparing candidates in elections, which we call the Advantage-Standard (AS) model. The requirement that a collective choice rule (CCR) be rationalizable by the AS model is in the spirit of but weaker than IIA; yet it is stronger than what is known in the literature as weak IIA (two profiles alike on $x, y$ cannot have opposite strict social preferences on $x$ and $y$). In addition to motivating violations of IIA, the AS model makes intelligible violations of another Arrovian assumption: the negative transitivity of the strict social preference relation $P$. While previous literature shows that only weakening IIA to weak IIA or only weakening negative transitivity of $P$ to acyclicity still leads to impossibility theorems, we show that jointly weakening IIA to AS rationalizability and weakening negative transitivity of $P$ leads to no such impossibility theorems. Indeed, we show that several appealing CCRs are AS rationalizable, including even transitive CCRs.