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Methods of Election
... The Borda-Condorcet debate can be envisioned in a broader picture: what are the relationships between the rankings on a set of candidates A and the rankings on the subset B included in A for a given preference profile when we use a scoring rule? Apart from classical studies analyzing the relationships between pairwise voting and scoring rules (see Dodgson, 1876;Nanson, 1882;Smith, 1973;Fishburn and Gehrlein, 1976), the first extension of these contributions is due to Fishburn (1981) who showed that there always exist a preference profile for which removing any candidate from A leads to the reversed ranking on the remaining set of candidates. In a seminal paper, Saari (1988) generalized this result by studying simultaneously the rankings on all the subsets of A for a given profile. ...
... , w B 2 m −(m+1) ) is the collection of scoring vectors used for each proper subset B of A. 14 However, this is not the case if we seek for generalized voting rules in other families; see for example the Copeland rule (Saari and Merlin, 1996). 15 An early version of this result can be tracked back to Nanson (1882). Modern proofs are proposed by Smith (1973) and Fishburn and Gehrlein (1976). ...
We know since the works of Gehrlein and Fishburn (1980, 1981), Fishburn (1981) and Saari (1987, 1988, 1990) that, the collective rankings of scoring rules are not stable when some alternatives are dropped from the set of alternatives. However, in the literature, attention has been mainly devoted to the relationship between pairwise majority vote and scoring rules rankings. In this paper, we focus on the relationships between four-candidate and three-candidate rankings. More precisely, given a collective ranking over a set of four candidates, we determine under the impartial culture condition, the probability of each of the six possible rankings to occur when one candidate is dropped. As a consequence, we derive from our computations, the likelihood of two paradoxes of committee elections, the Leaving Member Paradox (Staring, 1986) and the Prior Successor Paradox which occur when an elected candidate steps down from a two-member committee.
... 1 the relationships between the rankings on a set A of candidates and the rankings on the subset B included in A for a given preference prole? Apart from classical studies analyzing the relationships between pairwise voting and scoring rules (see Dodgson (1876); Nanson (1882); Smith (1973); Fishburn and Gehrlein (1976)), the rst extension of these is due to Fishburn (1981) who showed that there always exist a preference prole for which removing any candidate from A leads to the reversed ranking on the remaining candidates. In a seminal paper Saari (1988) generalized this result by studying simultaneously the ranking on all the subsets of A for a given prole. ...
... Theorems 1 to 3 establish the superiority of the Borda count in the class of scoring rule when one wishes to minimizes the types of inconsistencies we can observe across subsets. However, they do not tell us whether the likelihood of paradoxes are rare oddities or not, and whether the Borda count also minimizes the probability of 11 This results can be tracked back to Nanson (1882). Modern proofs are proposed by Smith (1973) and Fishburn and Gehrlein (1976). ...
We know since the works of Gehrlein and Fishburn (1980, 1981), Fishburn
(1981) and Saari (1987, 1988, 1990) that, the collective rankings of scoring rules are not stable when some alternatives are dropped from the set of alternatives. However, in the literature, attention has been mainly devoted to the relationship between pairwise majority vote and scoring rules rankings. In this paper, we focus on the relationships between four-candidate and three-candidate rankings. More precisely, given a collective ranking over a set of four candidates, we determine under the impartial culture condition, the probability of each of the six possible rankings to occur when one candidate is dropped. As a consequence, we derive from our computations, the likelihood of two paradoxes of committee elections, the Leaving Member paradox (Staring, 1986) and of the Prior Successor Paradox which occur when an elected candidate steps down from a two-member committee.
... An election system is said to be Condorcet-consistent if it elects the Condorcet winner when one exists. Many interesting election systems, such as that of Borda, are not Condorcet-consistent [Nan82]. ...
... An election system with this property is said to be Condorcet (or weak Condorcet) consistent. Many election systems, such as that of Borda, are not Condorcet consistent [Nan82]. ...
... Still, they showed that SHIFT-BRIBERY for Borda is NP-complete, yet can be efficiently approximated to within a factor of 2 (which was generalized by Elkind and Faliszewski (2010) for all scoring protocols). Recently, Maushagen et al. (2018) studied the complexity of SHIFT-BRIBERY for iterative voting rules such as those by Baldwin (1926) and Nanson (1882), and showed that they are NP-complete as well. These two voting rules proceed in rounds and eliminate in each round the candidates performing worst (namely, the candidates with lowest Borda score in Baldwin and those with scores lower than the average Borda score in Nanson), and the remaining candidates win. ...
Borda Count is one of the earliest and most important voting rules. Going far beyond voting, we summarize recent advances related to Borda in computational social choice and, more generally, in collective decision making. We first present a variety of well known attacks modeling strategic behavior in voting—including manipulation, control, and bribery—and discuss how resistant Borda is to them in terms of computational complexity. We then describe how Borda can be used to maximize social welfare when indivisible goods are to be allocated to agents with ordinal preferences. Finally, we illustrate the use of Borda in forming coalitions of players in a certain type of hedonic game. All these approaches are central to applications in artificial intelligence.
... In the weighted version of approval-based multiwinner voting (Thiele, 1895;Janson, 2018;Brill et al., 2018;Lackner and Skowron, 2019), if we will apply independence of unanimously approved candidates (analogous to our independence of unanimous winners), we will obtain geometric sequences of scores which include the top-k rule and a refinement of the Chamberlin-Courant rule as particular cases. Nor are these questions relevant for scoring rules only -one might think that no reasonable rule would violate something as weak as independence of unanimous losers, but Nanson's rule (Nanson, 1882;McLean and Urken, 1995;Felsenthal and Nurmi, 2018) does exactly that. ...
Scoring rules are widely used to rank athletes in sports and candidates in elections. Each position in each individual ranking is worth a certain number of points; the total sum of points determines the aggregate ranking. The question is how to choose a scoring rule for a specific application. First, we derive a one-parameter family with geometric scores which satisfies two principles of independence: once an extremely strong or weak candidate is removed, the aggregate ranking ought to remain intact. This family includes Borda count, generalised plurality (medal count), and generalised antiplurality (threshold rule) as edge cases, and we find which additional axioms characterise these rules. Second, we introduce a one-parameter family with optimal scores: the athletes should be ranked according to their expected overall quality. Finally, using historical data from biathlon, golf, and running we demonstrate how the geometric and optimal scores can simplify the selection of suitable scoring rules, show that these scores closely resemble the actual scores used by the organisers, and provide an explanation for empirical phenomena observed in golf tournaments. We see that geometric scores approximate the optimal scores well in events where the distribution of athletes’ performances is roughly uniform.
... The iterative scoring rules eliminate all the candidates who obtain strictly less than the average score at each stage of the elimination process. Given the notation of Section 2.2, if the iterative scoring rule is associated with λ = 1, this defines the Kim-Roush voting rule (Kim and Roush, 1996); we get the Nanson rule (Nanson, 1883) if the iterative scoring rule is associated with λ = 1 2 . For our framework with three candidates, when we eliminate all the candidates who obtain strictly less than the average score, the following scenarios are conceivable: ...
The Borda Effect, first introduced by Colman and Poutney (Behav Sci 23:15–20, 1978), occurs in a preference aggregation process using the Plurality rule if given the (unique) winner there is at least one loser that is preferred to the winner by a majority of the electorate. Colman and Poutney (1978) distinguished two forms of the Borda Effect: the Weak Borda Effect, describing a situation under which the unique winner of the Plurality rule is majority dominated by only one loser; and the Strong Borda Effect, under which the Plurality winner is majority dominated by each of the losers. The Strong Borda Effect is well documented in the literature as the Strong Borda Paradox. Colman and Poutney (1978) showed that the probability of the Weak Borda Effect is not negligible; but they only focused on the Plurality rule. In this note, we extend the work of Colman and Poutney (1978) by providing, for three-candidate elections, representations of the limiting probabilities of the (Weak) Borda Effect for the whole family of scoring rules and scoring runoff rules. Our analysis leads us to highlight that there is a relation between the (Weak) Borda Effect and Condorcet efficiency. We perform our analysis under the assumptions of Impartial Culture and Impartial Anonymous Culture, which are two well-known assumptions often used for such a study.
... For completeness of results, we should mention well-studied voting rules that satisfy the mutual majority criterion:Nanson's (1882), see alsoMcLean and Urken (1995),Baldwin's (1926), single transferable vote(Hare, 1859), Coombs' (1964, maximal likelihood(Kemeny, 1959), ranked pairs(Tideman, 1987),Schulze's (2011), successive elimination, and Bucklin's (see e.g.Felsenthal and Nurmi, 2018), median voting rule (Bassett and Persky, 1999), majoritarian compromise(Sertel and Yılmaz, 1999), q-approval fallback bargaining(Brams and Kilgour, 2001), and those tournament solutions which are refinements of the top cycle(Good, 1971;Schwartz, 1972). For their formal definitions and properties, we also adviseBrandt et al. (2016),Felsenthal and Nurmi (2018), Fischer et al. (2016),Taylor (2005), Tideman (2006), and Zwicker (2016.18 ...
We study voting rules with respect to how they allow or limit a majority from dominating minorities: whether a voting rule makes a majority powerful, and whether minorities can veto the candidates they do not prefer. For a given voting rule, the minimal share of voters that guarantees a victory to one of their most preferred candidates is the measure of majority power, and the minimal share of voters that allows them to veto each of their least preferred candidates is the measure of veto power. We find tight bounds on these minimal shares for voting rules that are popular in the literature and in real elections. We order these rules according to majority power and veto power. The instant-runoff voting has both the highest majority power and the highest veto power and the plurality rule has the lowest. In general, the higher the majority power of a voting rule is, the higher its veto power. The three exceptions are: voting with proportional veto power, Black’s rule, and Borda rule, which have a relatively low level of majority power and a high level of veto power and thus provide minority protection. Our results can shed light on how voting rules provide different incentives for voter participation and candidate nomination.
... 8 (q, k)-MM is even more general than the concept q-PSC formalized in [1] if the latter is applied to single-winner elections. The weak mutual majority criterion defined in [26] turns as a particular case of q = k/(k + 1). 9 Also one can see that unanimity criterion is equivalent to (1 − ε, 1)-MM with infinitely small ε > 0. 10 For completeness of results, we should mention well-studied voting rules that satisfy the mutual majority criterion: Nanson's [29,30], Baldwin's [4], single transferable vote [23], Coombs [13], sequential majority comparison [45], maximal likelihood [25], ranked pairs [41], beat paths [33], median voting rule [6], Bucklin's [40], majoritarian compromise [35], q-approval fallback bargaining [9], and those tournament solutions which are refinements of the top cycle [21,34]. For their formal definitions and properties, we also advise [10,18,38,40,45]. ...
We study voting rules with respect to how they allow or limit a majority to dominate minorities. For this purpose we propose a novel quantitative criterion for voting rules: the qualified mutual majority criterion (q, k)-MM. For a fixed total number of m candidates, a voting rule satisfies (q, k)-MM if whenever some k candidates receive top k ranks in an arbitrary order from a majority that consists of more than q ∈ (0, 1) of voters, the voting rule selects one of these k candidates. The standard majority criterion is equivalent to (1/2, 1)-MM. The standard mutual majority criterion (MM) is equivalent to (1/2, k)-MM, where k is arbitrary. We find the bounds on the size of the majority q for several important voting rules, including the plurality rule, the plurality with runoff rule, Black's rule, Condorcet least reversal rule, Dodgson's rule, Simpson's rule, Young's rule and monotonic scoring rules; for most of these rules we show that the bound is tight.
... Several systems were associated with a scoring system by awarding different scores to different rank orders, and pick the candidate with the maximum score as the winner. Some popular voting systems with this procedure are: Nauru (Benjamin Reilly, 2002) [11], Borda Count (1781) [3], Condorcet Least-reversal System (1785) [12], and Nanson-Baldwin Elimination (Edward J. Nanson, 1882) [13]. ...
... The Borda scores are then revised, taking only the remaining candidates into account. The procedure repeats until a Borda winner can be determined (Nanson 1883). ...
Providing a high quality of service in public transportation is essential to reduce dissatisfactions stemming from traffic congestion and noise. Public transport providers need to find ways to dilute the effects of immoderate use of private cars in big cities while maintaining a sufficient level of customer satisfaction. This study aimed to identify the key service quality (SQ) factors that drive passenger satisfaction in Istanbul’s rail transit (RT) system using data obtained from an extensive survey conducted by the Istanbul Public Transportation Co. A total of 11,116 passengers who used rail transport from May 15–June 3, 2012, and June 17–July 3, 2013, were interviewed in person. The relative importance of the SQ factors was assessed so that service provision could be prioritized and the enhancement of passenger satisfaction can be achieved employing several social choice techniques. The results indicate that, from an overall perspective, waiting time, crowdedness in cars, and fare are the SQ factors that best reflect the public good.
... The large number of Condorcet's writings on mathematics, philosophy, politics and economics present a problem of interpretation (for a brief history of some reactions, see Faccarello 1989). Commentaries generally referred to a vague theory of evolution and progress associated with the 1795 Esquisse, and, after the Second World War, to his ideas on elections to which Georges-Théodule Guilbaud (1952), Gilles-Gaston Granger (1956), Duncan Black (1958) and Kenneth Arrow (1963) drew attention -they had been almost forgotten for some 150 years, with the exceptions of Edward John Nanson (1882Nanson ( [1907) and Charles Lutwidge Dodgson (alias Lewis Carroll 1876). While much is still to be done, recent research has made it possible to identify a quite different intellectual stature of Condorcet. ...
... In fact, two of the most widely known voting rules are explicitly divided on the issue of "should a pairwise majority winner automatically be the winner of an election?": Condorcet's practical method [6] and the Borda count [4] answer this question in diametrically opposed ways. ...
We propose a new point of view in the long-standing problem where several voters have expressed a linear order relation (or ranking) over a set of candidates. For a ranking a > b > c to represent a group's opinion, it would be logical that the strength with which a > c is supported should not be less than the strength with which either a > b or b > c is supported. This intuitive property can be considered a monotonicity constraint, and has been addressed before. We extend previous approaches in the following way: as the voters are expressing linear orders, we can take the number of candidates between two candidates to be a measure of the degree to which one candidate is preferred to the other. In this way, intensity of support is both counted as the number of voters who indicate a > c is true, as well as the distance between a and c in these voters' rankings. The resulting distributions serve as input for a natural ranking rule that is based on stochastic monotonicity and stochastic dominance. Adapting the previous methodology turns out to be non-trivial once we add some natural feasibility constraints.
... These conditions guarantee that there is a pairwise winner and pairwise loser. However, paradoxes can still occur where the Borda winner(loser) is not the pairwise winner(loser), even though the Borda count must rank the pairwise winner above the pairwise loser [19] (see Fact 1.2). ...
Pardoxes in voting has been an interest of voting theorists since the 1800's when Condorcet demonstrated the key example of a voting paradox: voters with individually transitive rankings produce an election outcome which is not transitive. With Arrow's Impossibility Theorem, the hope of finding a fair voting method which accurately reflected society's preferences seemed unworkable. Recent results, however, have shown that paradoxes are unlikely under certain assumptions. In this paper, we corroborate results found by Gehrelin for the probabilities of paradoxes, but also give results which indicate paradoxes are extremely likely under the right conditions. We use simulations to show there can be many situations where paradoxes can arise, dependent upon the variability of voters' preferences, which echo Saari's statements on the topic.
... The conclusion now follows from the result (Saari [16]) that the Condorcet winner is BC strictly ranked above the Condorcet loser and that no other restrictions exist among the rankings. (Special cases of this result were known by Nanson [14] and maybe even Borda.) Instead of finding all q n 's which define the indicated KR ranking, we use the larger set consisting of all convex combinations of the vertices of the a 1 a 2 · · · a n transitive ranking region plus all (n − 1)! vectors defined by C n r for any strict ranking r of the n candidates. ...
By using geometry, a fairly complete analysis of Kemeny's rule (KR) is obtained. It is shown that the Borda Count (BC) always ranks the KR winner above the KR loser, and, conversely, KR always ranks the BC winner above the BC loser. Such KR relationships fail to hold for other positional methods. The geometric reasons why KR enjoys remarkably consistent election rankings as candidates are added or dropped are explained. The power of this KR consistency is demonstrated by comparing KR and BC outcomes. But KR's consistency carries a heavy cost; it requires KR to partially dismiss the crucial "individual rationality of voters" assumption.
... Black, however, proposed that the Borda winner be chosen when no Condorcet winner existed. Nanson (1882) proposed a complex scheme based on the Borda rule that has the property that it always yields the Condorcet winner when one exists (see McLean and Urken 1995: 57-60;Tangian 2013: 203). In total, a and b have the same number of both first and second preferences: with sincere voting, b's third preference from Voter 4 trumps a's fourth preference from Voter 3, but that advantage vanishes if Voter 1 votes strategically. ...
This article examines strategic elements of voter behaviour in parliamentary elections where the voting method is a scoring rule other than plurality: the Borda Count, which is used for the election of ethnic minorities in Slovenia, and the Dowdall rule, which is used in the Pacific island state of Nauru in multi-seat districts. After first examining the general properties of scoring rules, and generating theoretical differences between the two rules, we look at empirical evidence from Nauru and Slovenia. This casts a doubt on predictions based simply on a voting rule's mathematical properties and on the accuracy of assumptions of sincere rank ordering.
... If a Condorcet winner exists then the Nanson method elects him or her (cf. Nanson, 1883;McLean an Urken, 1995, ch. 14). ...
In iterative voting systems, candidates are eliminated in consecutive rounds until either a fixed number of rounds is reached or the set of remaining candidates does not change anymore. We focus on iterative voting systems based on the positional scoring rules plurality, veto, and Borda and study their resistance against shift bribery attacks introduced by Elkind et al. [1] and Kaczmarczyk and Faliszewski [2]. In constructive shift bribery (Elkind et al. [1]), an attacker seeks to make a designated candidate win the election by bribing voters to shift this candidate in their preferences; in destructive shift bribery (Kaczmarczyk and Faliszewski [2]), the briber’s goal is to prevent this candidate’s victory. We show that many iterative voting systems are resistant to these types of attack, i.e., the corresponding decision problems are NP-hard. These iterative voting systems include iterated plurality as well as the voting rules due to Hare, Coombs, Baldwin, and Nanson; variants of Hare voting are also known as single transferable vote, instant-runoff voting, and alternative vote.
A voting rule that permits some voters to favor a candidate by revealing only the initial segment of their sincere rankings is said to be vulnerable to the truncation paradox. In this paper, we consider four models for counting truncated ballots, optimistic, pessimistic (the most common), averaged, and round-down. Under the impartial anonymous culture assumption, the choice of model generally has a real impact on truncation-paradox vulnerability, but exceptions exist. When the election is decided by a one-shot scoring rule, the optimistic model is invulnerable to the truncation paradox, but all other models are vulnerable. We identify new voting rules immune to the truncation paradox, such as the Modified Borda Count. To obtain a more complete picture of the impact of processing model, we assess the likelihood of the truncation paradox in three-candidate elections with large electorates, focusing not only on one-shot scoring rules but also scoring rules with one-by-one or below-average elimination. Our assessment confirms that the processing model for truncated ballots may really matter.
Borda Count is one of the earliest and most important voting rules and has been central to many applications in artificial intelligence. We study the problem of control in Borda elections where an election chair seeks to either make a designated candidate win (constructive case), or prevent her from winning (destructive case), via actions such as adding, deleting, or partitioning either candidates or voters. These scenarios have been studied for many voting rules and the related control problems have been classified in terms of their computational complexity. However, for one of the most prominent natural voting rules, the Borda Count, complexity results have been known for only half of these cases until recently. We settle the complexity for all missing cases, focusing on the unique-winner model. We also exhibit two of the very rare cases where the complexity of control problems differs depending on the winner model chosen: For destructive control by partition and by run-off partition of candidates when ties promote, Borda is resistant in the unique-winner model (i.e., these two control problems are NP-hard), yet is vulnerable in the nonunique-winner model (i.e., one can decide in polynomial time whether control is possible). Finally, we turn to the model of online control in sequential elections that was recently proposed by Hemaspaandra et al. [62], [61]. We show that sequential Borda elections are vulnerable to constructive and destructive online control by adding or deleting candidates, whereas we obtain coNP-hardness results for all types of online voter control in sequential Borda elections.
An important and surprising phenomenon in voting theory is the No-Show Paradox (NSP), which occurs if a voter is better off by abstaining from an election. While it is known that certain voting rules suffer from this paradox in principle, the extent to which it is of practical concern is not well understood. We aim at filling this gap by analyzing the likelihood of the NSP for six Condorcet extensions (Black’s rule, Baldwin’s rule, Nanson’s rule, Max-Min, Tideman’s rule, and Copeland’s rule) under various preference models using Ehrhart theory as well as extensive computer simulations. We find that, for few alternatives, the probability of the NSP is rather small (less than 4% for four alternatives and all considered preference models, except for Copeland’s rule). As the number of alternatives increases, the NSP becomes much more likely and which rule is most susceptible to abstention strongly depends on the underlying distribution of preferences.
We study voting rules with respect to how they allow or limit a majority from dominating minorities: whether a voting rule makes a majority powerful and whether minorities can veto the candidates they do not prefer. For a given voting rule, the minimal share of voters that guarantees a victory to one of the majority’s most preferred candidates is the measure of majority power; and the minimal share of voters that allows the minority to veto each of their least preferred candidates is the measure of veto power. We find tight bounds on such minimal shares for voting rules that are popular in the literature and used in real elections. We order the rules according to majority power and veto power. Instant-runoff voting has both the highest majority power and the highest veto power; plurality rule has the lowest. In general, the greater is the majority power of a voting rule, the greater its veto power. The three exceptions are: voting with proportional veto power, Black’s rule and Borda’s rule, which have relatively weak majority power and strong veto power, thus providing minority protection. Our results can shed light on how voting rules provide different incentives for voter participation and candidate nomination.
Although two-round voting procedures are common, the theoretical voting literature rarely discusses any such rules beyond the traditional plurality runoff rule. Therefore, using four criteria in conjunction with two data-generating processes, we define and evaluate nine “runoff pair selection rules” that comprise two rounds of voting, two candidates in the second round, and a single final winner. The four criteria are: social utility from the expected runoff winner (UEW), social utility from the expected runoff loser (UEL), representativeness of the runoff pair (Rep), and resistance to strategy (RS). We examine three rules from each of three categories: plurality rules, utilitarian rules and Condorcet rules. We find that the utilitarian rules provide relatively high UEW and UEL, but low Rep and RS. Conversely, the plurality rules provide high Rep and RS, but low UEW and UEL. Finally, the Condorcet rules provide high UEW, high RS, and a combination of UEL and Rep that depends which Condorcet rule is used.
We compare in this paper two classes of sequential scoring rules: the first class eliminates at each step the candidate with the lowest score whereas the second one eliminates the candidates whose scores are equal to or lower than the average score of the candidates remaining in contention. We show that, in three-candidate elections, the second method is susceptible to improve the ability of the sequential scoring rules to avoid monotonicity paradoxes, but this benefit is offset by a decrease in the Condorcet efficiency of these rules.
A social dichotomy function maps a collection of weak orders to a set of dichotomous weak orders. Every dichotomous weak order partitions the set of alternatives into approved alternatives and disapproved alternatives. The Borda mean rule returns all dichotomous weak orders that approve all alternatives with above-average Borda score and disapprove alternatives with below-average Borda score. We show that the Borda mean rule is the unique social dichotomy function satisfying neutrality, reinforcement, faithfulness, and the quasi-Condorcet property. Our result holds for all domains of weak orders that are sufficiently rich, including the domain of all linear orders and the domain of all weak orders.
During the 20th century, impossibility theorems have become an important part of mathematics. Arrow's impossibility theorem (1950) stands out as one of the first impossibility theorems outside of pure mathematics. It states that it is impossible to design a welfare function (or a voting method) that satisfies some rather innocent looking requirements. Arrow's theorem became the starting point of social choice theory that has had a great impact on welfare economics. This paper will analyze the history of Arrow's impossibility theorem in its mathematical and economic contexts. It will be argued that Arrow made a radical change of the mathematical model of welfare economics by connecting it to the theory of voting and that this change was preconditioned by his deep knowledge of the modern axiomatic approach to mathematics and logic.
The no-show paradox occurs whenever a group of identically-minded voters is better off abstaining than by voting according to its preferences. Moulin’s (J Econ Theory 45:53–64, 1988) result states that if one wants to exclude the possibility of the no-show paradox, one has to resort to procedures that do not necessarily elect the Condorcet winner when one exists. This paper examines ten Condorcet-consistent and six Condorcet-non-consistent procedures in a restricted domain, viz., one where there exists a Condorcet winner who is elected in the original profile and the profile is subsequently modified by removing a group of voters with identical preferences. The question asked is whether the no-show paradox can occur in these settings. It is found that only two of the ten Condorcet-consistent procedures investigated (Maximin and Schwartz’s procedure) are not vulnerable to the no-show paradox, whereas only two of the six non-Condorcet-consistent ranked procedures investigated (Coombs’ and the Negative Plurality Elimination Rule procedures) are vulnerable to this paradox in the restricted domain. In other words, for a no-show paradox to occur when using Condorcet-consistent procedures it is not, in general, necessary that a top Condorcet cycle exists in the original profile, while for this paradox to occur when using (ranked) non-Condorcet-consistent procedures it is, almost always, necessary that the original profile has a top cycle.
This paper compares the vulnerability of Borda Elimination Rule (BER) and of Nanson Elimination Rule (NER) to monotonicity paradoxes under both fixed and variable electorates. It is shown that while NER is totally immune and BER is vulnerable to monotonicity failure in 3-candidate elections, neither of these two rules dominates the other in n-candidate elections (n > 3) when no Condorcet Winner exists. When the number of competing alternatives is larger than three and no Condorcet Winner exists, we find profiles where NER violates monotonicity while BER does not, profiles where BER violates monotonicity while NER does not, as well as profiles where both NER and BER violate monotonicity. These findings extend to both fixed and variable electorates, as well as to situations where the initial winners under both rules are the same, as well as to situations where the initial winners under both rules are different. So, which of the two rules should be preferred in terms of monotonicity in n-candidate elections (n > 3) where no Condorcet Winner exists, depends on the kind of profiles one can expect to encounter in practice most often. Nevertheless, in view of the results of 3-candidate elections under other scoring elimination rules, we conjecture that inasmuch as BER and NER exhibit monotonicity failures, it is more likely to occur in closely contested elections.
These reflections, written in honor of Kenneth Arrow, sketch out how one political scientist thinks about Arrow’s theorem and its implications for voting rules. The basic claim is that Arrow’s theorem means that all real-world voting rules are problematic in two quite specific ways—namely, they can be neither ‘strategyproof’ nor ‘spoilerproof’. However, Condorcet’s pairwise version of majority rule, while not a fully specified voting rule because of the cyclical majorities problem, is itself both strategyproof and spoilerproof. Moreover, the cycling problem seems to occur only rarely in practice.
18 voting procedures for electing a single candidate are introduced and briefly commented upon. The procedures fall into three classes in terms of the type of voter input and Condorcet consistency: non–ranked procedures, ranked procedures that are not Condorcet–consistent and ranked ones that are Condorcet–consistent. The first class consists of four procedures, the second consists of six procedures and the third class consists of eight procedures.
The common reaction to social choice anomalies is that while these may be theoretically feasible, their role in practice is likely to be very limited since the profiles giving rise to the paradoxes are exceptional rather than common. We argue that while the significance and observation of the general no-show paradoxes and monotonicity failures may be limited, the same is not true of P-TOP and P-BOT paradoxes. These pertain to the voters’ best and worst alternatives. Hence, the occurrences associated with them are most likely to be observed and talked about. Vulnerability to these paradoxes creates bizarre incentives for voters. Failures on monotonicity, on the other hand, have more to do with the legitimacy of the voting outcomes. It is important to find out the structural properties of settings giving rise to monotonicity-related paradoxes. Thus far very little is known about these properties. We have related the paradoxes to the existence of majority cycles in the electorate. Our general finding is that the presence of a core or Condorcet winner does not, in general, make it harder to construct profiles that lead to paradoxes. Sometimes the opposite is true: the presence of cycles makes it harder, not easier, to construct paradoxical profiles. Finally, we speculate about reasons for continued use of non-monotonic rules.
In practice, selecting an energy project for development requires balancing criteria and competing stakeholder priorities to identify the best alternative. Energy source selection can be modeled as multi-criteria decision-maker problems to provide quantitative support to reconcile technical, economic, environmental, social, and political factors with respect to the stakeholders' interests. Decision making among these complex interactions should also account for the uncertainty present in the input data. In response, this work develops a stochastic decision analysis framework to evaluate alternatives by involving stakeholders to identify both quantitative and qualitative selection criteria and performance metrics which carry uncertainties. The developed framework is illustrated using a case study from Fairbanks, Alaska, where decision makers and residents must decide on a new source of energy for heating and electricity. We approach this problem in a five step methodology: (1) engaging experts (role players) to develop criteria of project performance; (2) collecting a range of quantitative and qualitative input information to determine the performance of each proposed solution according to the selected criteria; (3) performing a Monte-Carlo analysis to capture uncertainties given in the inputs; (4) applying multi-criteria decision-making, social choice (voting), and fallback bargaining methods to account for three different levels of cooperation among the stakeholders; and (5) computing an aggregate performance index (API) score for each alternative based on its performance across criteria and cooperation levels. API scores communicate relative performance between alternatives. In this way, our methodology maps uncertainty from the input data to reflect risk in the decision and incorporates varying degrees of cooperation into the analysis to identify an optimal and practical alternative.
An important issue in river water quality management is taking into account the role played by wastewater dischargers in the decision-making process and in the implementation of any proposed waste load allocation program in a given region. In this study, a new decision-making methodology, called 'stochastic social choice rules' (SSCR), was developed for modeling the bargaining process among different wastewater dischargers into shared environments. For this purpose, the costs associated with each treatment strategy were initially calculated as the sum of treatment cost and the fines incurred due to violation of water quality standards. The qualitative simulation model (QUAL2Kw) was then used to determine the penalty function. The uncertainty associated with the implementation of strategies under the economic costs (i.e., the sum of treatment and penalty costs) was dealt with by a Monte-Carlo selection method. This method was coupled with different social choice methods to identify the best solution for the waste load allocation problem. Finally, using the extended trading-ratio system (ETRS), the most preferred treatment strategy was exchanged among dischargers as the initial set of discharge permits aimed at reducing the costs and encouraging dischargers to participate in the river water quality protection scheme. The proposed model was finally applied to the Zarjoub River in Gilan Province, northern Iran, as a case study. Results showed the efficiency of the proposed model in developing waste load allocation strategies for rivers.
In situations when a group of people has to make a decision based on the set of individual preferences, they use a certain aggregation method, in particular, voting. One of the main problems for any non-dictatorial social choice rule is the possibility for the voters to achieve a more preferable outcome of the voting by misrepresenting their preferences. Such actions on behalf of the voters are called manipulation, or strategic voting. One approach used to compare social choice rules in terms of how hard they are to manipulate is to find the complexity classes of manipulation problems for a given aggregation method. In this work, we present a survey of the studies of complexity classes of manipulation problems under various model assumptions and constraints.
Three factors motivated me to write this chapter: The recent passage (25 February 2010) by the British House of Commons of the Constitutional Reform and Governance Bill, clause #29 of which states that a referendum will be held by 31 October 2011 on changing the current single member plurality (aka first-past-the-post, briefly FPTP) electoral procedure for electing the British House of Commons to the (highly paradoxical) alternative vote (AV) procedure (aka Instant Runoff ).1 Similar calls for adopting the alternative vote procedure are voiced also in the US. My assessment that both the UK and the US will continue to elect their legislatures from single-member constituencies, but that there exist, from the point of view of social-choice theory, considerably more desirable voting procedures for electing a single candidate than the FPTP and AV procedures. A recent report by Hix et al. (2010) – commissioned by the British Academy and entitled Choosing an Electoral System – that makes no mention of standard social-choice criteria for assessing electoral procedures designed to elect one out of two or more candidates.
Preferential voting where the voters rank candidates in order of preference plays an important role in many decision making problems and have been studied intensively. Yet there are too many variations and many popular methods are promulgated differently in different regions. Hence, some iconic conventional methods are reviewed for syntactic patterns and categorized. A nomenclature for these voting methods is suggested to reveal their syntactic patterns. Over a thousand of voting methods are devised from the conventional procedural patterns. Over 60 representative voting methods are used to reveal their semantic relationship in the form of hierarchical clustering tree. All preferential voting methods perform significantly different from the simplest plurality method.
The publication of the first book by Kenneth Arrow and Hervé Raynaud, in 1986, led to an important wave of research in the field of axiomatic approach applied to managerial logic. Managerial Logic summarizes the prospective results of this research and offers consultants, researchers, and decision makers a unified framework for handling the difficult decisions they face. Based on confirmed results of experimental psychology, this book places the problem in a phenomenological framework and shows how the influence of traditional methods has slowed the effective resolution of these problems. It provides a panorama of principal concepts and theorems demonstrated on axiomatized methods to guide readers in choosing the best alternatives and rejecting the worst ones. Finally, it describes the obtained extensions, often paradoxical, reached when these results are extended to classification problems. The objective of this book is also to allow the decision maker to find his way through the plethora of "multicriterion methods" promoted by council organizations. The meta-method it proposes will allow him to distinguish the wheat from the chaff. The collaboration with Kenneth Arrow comes essentially from the fact that his work influenced all subsequent works quoted in this book. His famous impossibility theorem, his gem of a PhD thesis, and his various other works resulted in him receiving the Nobel Prize for economy just before meeting Hervé Raynaud who was at that time a visiting professor at Berkeley University in California. Their mutual publications serve as the basis for the axiomatic approach in multicriterion decision-making.
Das Vergleichen und Aggregieren von Informationen ist ein zentraler Bereich in der Analyse von Wahlsystemen. In diesen müssen die verschiedenen Meinungen von Wählern über eine Menge von Kandidaten zu einem möglichst gerechten Wahlergebnis aggregiert werden. In den meisten politischen Wahlen entscheidet sich jeder Wähler durch Ankreuzen für einen einzigen Kandidaten. Daneben werden aber auch Rangordnungsprobleme als eine Variante von Wahlsystemen untersucht. Bei diesen bringt jeder Wähler seine Meinung in Form einer totalen Ordnung über der Menge der Kandidaten zum Ausdruck, wodurch seine oftmals komplexe Meinung exakter repräsentiert werden kann als durch die Auswahl eines einzigen, favorisierten Kandidaten. Das Wahlergebnis eines Rangordnungsproblems ist dann eine ebenfalls totale Ordnung der Kandidaten, welche die geringste Distanz zu den Meinungen der Wähler aufweist. Als Distanzmaße zwischen zwei totalen Ordnungen haben sich neben anderen Kendalls Tau-Distanz und Spearmans Footrule-Distanz etabliert. Durch moderne Anwendungsmöglichkeiten von Rangordnungsproblemen im maschinellen Lernen, in der künstlichen Intelligenz, in der Bioinformatik und vor allem in verschiedenen Bereichen des World Wide Web rücken bereits bekannte, jedoch bislang eher wenig studierte Aspekte in den Fokus der Forschung. Zum einen gewinnt die algorithmische Komplexität von Rangordnungsproblemen an Bedeutung. Zum anderen existieren in vielen dieser Anwendungen unvollständige „Wählermeinungen“ mit unentschiedenen oder unvergleichbaren Kandidaten, so dass totale Ordnungen zu deren Repräsentation nicht länger geeignet sind. Die vorliegende Arbeit greift diese beiden Aspekte auf und betrachtet die algorithmische Komplexität von Rangordnungsproblemen, in denen Wählermeinungen anstatt durch totale Ordnungen durch schwache oder partielle Ordnungen repräsentiert werden. Dazu werden Kendalls Tau-Distanz und Spearmans Footrule-Distanz auf verschiedene nahe liegende Arten verallgemeinert. Es zeigt sich dabei, dass nun bereits die Distanzberechnung zwischen zwei Ordnungen ein algorithmisch komplexes Problem darstellt. So ist die Berechnung der verallgemeinerten Versionen von Kendalls Tau-Distanz oder Spearmans Footrule-Distanz für schwache Ordnungen noch effizient möglich. Sobald jedoch partielle Ordnungen betrachtet werden, sind die Probleme NP-vollständig, also vermutlich nicht mehr effizient lösbar. In diesem Fall werden Resultate zur Approximierbarkeit und zur parametrisierten Komplexität der Probleme vorgestellt. Auch die Komplexität der Rangordnungsprobleme selbst erhöht sich. Für totale Ordnungen effizient lösbare Varianten werden für schwache Ordnungen NP-vollständig, für totale Ordnungen NP-vollständige Varianten hingegen liegen für partielle Ordnungen teilweise außerhalb der Komplexitätsklasse NP. Die Arbeit schließt mit einem Ausblick auf offene Problemstellungen.
Originally published in 1951, Social Choice and Individual Values introduced "Arrow's Impossibility Theorem" and founded the field of social choice theory in economics and political science. This new edition, including a new foreword by Nobel laureate Eric Maskin, reintroduces Arrow's seminal book to a new generation of students and researchers. "Far beyond a classic, this small book unleashed the ongoing explosion of interest in social choice and voting theory. A half-century later, the book remains full of profound insight: its central message, 'Arrow's Theorem,' has changed the way we think."-Donald G. Saari, author of Decisions and Elections: Explaining the Unexpected. © 1951, 1963,2012 by Cowles Foundation for Research in Economics at Yale University. All rights reserved.
Although no electoral system can be perfect, some of those that are in use are very much better than others. A number of methods are reviewed and reasons are given for believing that Condorcet's method of paired comparisons is to be preferred where a single seat is to be contested, but the single transferable vote (STV) method where more than one seat is involved. A tentative suggestion is made of a means of combining these two methods into a single system.
It is shown how simple geometry can be used to analyze and discover new properties about pairwise and positional voting rules as well as for those rules (e.g., runoffs and Approval Voting) that rely on these methods. The description starts by providing a geometric way to depict profiles, which simplifies the computation of the election outcomes. This geometry is then used to motivate the development of a "profile coordinate system," which evolves into a tool to analyze voting rules. This tool, for instance, completely explains various longstanding "paradoxes," such as why a Condorcet winner need not be elected with certain voting rules. A different geometry is developed to indicate whether certain voting "oddities" can be dismissed or must be taken seriously, and to explain why other mysteries, such as strategic voting and the no-show paradox (where a voter is rewarded by not voting), arise. Still another use of geometry extends McGarvey's Theorem about possible pairwise election rankings to identify the actual tallies that can arise (a result that is needed to analyze supermajority voting). Geometry is also developed to identify all possible positional and Approval Voting election outcomes that are admitted by a given profile; the converse becomes a geometric tool that can be used to discover new election relationships. Finally, it is shown how lessons learned in social choice, such as the seminal Arrow's and Sen's Theorems and the expanding literature about the properties of positional rules, provide insights into difficulties that are experienced by other disciplines.
Multiperson decision making is reviewed from the perspective of social choice theory and the theory of elections. The review interweaves abstract theory and practical concerns that deserve consideration in evaluating alternative election methods and in designing good election systems.
We investigate manipulation of the Borda voting rule, as well as two elimination style voting rules, Nanson's and Baldwin's voting rules, which are based on Borda voting. We argue that these rules have a number of desirable computational properties. For unweighted Borda voting, we prove that it is NP-hard for a coalition of two manipulators to compute a manipulation. This resolves a long-standing open problem in the computational complexity of manipulating common voting rules. We prove that manipulation of Baldwin's and Nanson's rules is computationally more difficult than manipulation of Borda, as it is NP-hard for a single manipulator to compute a manipulation. In addition, for Baldwin's and Nanson's rules with weighted votes, we prove that it is NP-hard for a coalition of manipulators to compute a manipulation with a small number of candidates.
Because of these NP-hardness results, we compute manipulations using heuristic algorithms that attempt to minimise the number of manipulators. We propose several new heuristic methods. Experiments show that these methods significantly outperform the previously best known heuristic method for the Borda rule. Our results suggest that, whilst computing a manipulation of the Borda rule is NP-hard, computational complexity may provide only a weak barrier against manipulation in practice. In contrast to the Borda rule, our experiments with Baldwin's and Nanson's rules demonstrate that both of them are often more difficult to manipulate in practice. These results suggest that elimination style voting rules deserve further study.
A strong Condorcet winner (SCW) is an alternative, x, that a majority of voters rank higher than z, for every other alternative, z. A weak Condorcet winner (WCW) is an alternative, y, that no majority of voters rank below any other alternative, z, but is not a SCW. There has been some confusion in the voting/social choice literature as to whether particular voting rules that are SCW-consistent are also WCW-consistent. The purpose of this paper is to revisit this issue, clear up the confusion that has developed, and determine whether three additional SCW-consistent voting rules—that as far as we know have not been investigated to date regarding their possible WCW consistency—are indeed WCW-consistent.
The present notes are an improved version of the notes that served as teaching materials for the course Introduction to Judgment Aggregation given at the 23rd European Summer School on Logic, Language and Information (ESSLLI'11, Ljubljana). The notes are structured as follows: Section 1 introduces the field of judgment aggregation, its relations to preference aggregation and some formal preliminaries. Section 2 shows that the paradox that originated judgment aggregation is not a problem limited to propositionwise majority voting but a more general issue, illustrated by an impossibility theorem of judgment aggregation that is here proven. The relaxation of some conditions used in impossibility results in judgment aggregation may lead to escape routes from the impossibility theorems. These escape routes are explored in Section 3. Section 4 presents the issue of manipulation that arises when voters strategically misrepresent their true vote in order to force a different outcome in the aggregation process. Finally, we conclude by sketching a list of on-going research in the field of judgment aggregation (Section 5).
Voting systems aggregate preferences efficiently and are often used for deciding conservation priorities. Desirable characteristics of voting systems include transitivity, completeness, and Pareto optimality, among others. Voting systems that are common and potentially useful for environmental decision making include simple majority, approval, and preferential voting. Unfortunately, no voting system can guarantee an outcome, while also satisfying a range of very reasonable performance criteria. Furthermore, voting methods may be manipulated by decision makers and strategic voters if they have knowledge of the voting patterns and alliances of others in the voting populations. The difficult properties of voting systems arise in routine decision making when there are multiple criteria and management alternatives. Because each method has flaws, we do not endorse one method. Instead, we urge organizers to be transparent about the properties of proposed voting systems and to offer participants the opportunity to approve the voting system as part of the ground rules for operation of a group.
Sistemas de Votación para Decisiones Ambientales
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