A voting method is Condorcet consistent if in any election in which one candidate is preferred by majorities to each of the other candidates, this candidate--the Condorcet winner--is the unique winner of the election. Condorcet consistent voting methods form an important class of methods in the theory of voting (see, e.g., Fishburn 1977; Brams and Fishburn 2002, 8; Zwicker 2016, 2.4; Pacuit 2019, 3.1.1). Although Condorcet methods are not currently used in government elections, they have been used by several private organizations (see Wikipedia contributors 2020b) and in over 30,000 polls through the Condorcet Internet Voting Service (https://civs.cs.cornell.edu). Recent initiatives in the U.S. to make available Instant Runoff Voting (Kambhampaty 2019), which uses the same ranked ballots needed for Condorcet methods, bring Condorcet methods closer to political application. Indeed, Eric Maskin and Amartya Sen have recently proposed the use of Condorcet methods in U.S. presidential primaries (Maskin and Sen 2016, 2017a,b). In the meantime, Condorcet methods continue to be used by committees, clubs, etc.

We propose six axioms concerning when one candidate should defeat another in a democratic election involving two or more candidates. Five of the axioms are widely satisfied by known voting procedures. The sixth axiom is a weakening of Kenneth Arrow's famous condition of the Independence of Irrelevant Alternatives (IIA). We call this weakening Coherent IIA. We prove that the five axioms plus Coherent IIA single out a method of determining defeats studied in our recent work: Split Cycle. In particular, Split Cycle provides the most resolute definition of defeat among any satisfying the six axioms for democratic defeat. In addition, we analyze how Split Cycle escapes Arrow's Impossibility Theorem and related impossibility results.

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.