The ordinal egalitarian bargaining solution for finite choice sets

Rubinstein et al. (Econometrica 60:1171–1186, 1992) introduced the Ordinal Nash Bargaining Solution. They prove that Pareto optimality, ordinal invariance, ordinal symmetry, and IIA characterize this solution. A feature of their work is that attention is restricted to a domain of social choice problems with an infinite set of basic allocations. We introduce an alternative approach to solving finite social choice problems using a new notion called the Ordinal Egalitarian (OE) bargaining solution. This suggests the middle ranked allocation (or a lottery over the two middle ranked allocations) of the Pareto set as an outcome. We show that the OE solution is characterized by weak credible optimality, ordinal symmetry and independence of redundant alternatives. We conclude by arguing that what allows us to make progress on this problem is that with finite choice sets, the counting metric is a natural and fully ordinal way to measure gains and losses to agents seeking to solve bargaining problems.     

The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable

It is shown that the Majoritarian Compromise of Sertel (1986) is subgame-perfect implementable on the domain of strict preference profiles, although it fails to be Maskin-monotonic and is hence not implementable in Nash equilibrium. The Majoritarian Compromise is Pareto-optimal and obeys SNIP (strong no imposition power), i.e. never chooses a strict majority's worst candidate. In fact, it is “majoritarian approving” i.e. it always picks “what's good for a majority” (alternatives which some majority regards as among the better “effective” half of the available alternatives). Thus, being Pareto-optimal and majoritarian approving, it is majoritarian-optimal. Finally, the Majoritarian Compromise is measured against various criteria, such as consistency and Condorcet-consistency. Received: 31 January 1995/Accepted: 22 July 1998     

Cycling of simple rules in the spatial model

McKelvey [4] proved that for strong simple preference aggregation rules applied to multidimensional sets of alternatives, the typical situation is that either the core is nonempty or the top-cycle set includes all available alternatives. But the requirement that the rule be strong excludes, inter alia, all supermajority rules. In this note, we show that McKelvey's theorem further implies that the typical situation for any simple rule is that either the core is nonempty or the weak top-cycle set (equivalently, the core of the transitive closure of the rule) includes all available alternatives. Moreover, it is often the case that both of these statements obtain. Received: 13 October 1997/Accepted: 24 August 1998     

Limiting distributions for continuous state Markov voting models

This paper proves the existence of a stationary distribution for a class of Markov voting models. We assume that alternatives to replace the current status quo arise probabilistically, with the probability distribution at time t+1 having support set equal to the set of alternatives that defeat, according to some voting rule, the current status quo at time t. When preferences are based on Euclidean distance, it is shown that for a wide class of voting rules, a limiting distribution exists. For the special case of majority rule, not only does a limiting distribution always exist, but we obtain bounds for the concentration of the limiting distribution around a centrally located set. The implications are that under Markov voting models, small deviations from the conditions for a core point will still leave the limiting distribution quite concentrated around a generalized median point. Even though the majority relation is totally cyclic in such situations, our results show that such chaos is not probabilistically significant.We acknowledge the support of NSF Grants #SOC79-21588, SES-8106215 and SES-8106212.     

On the Average Minimum Size of a Manipulating Coalition

We study the asymptotic average minimum manipulating coalition size as a characteristic of quality of a voting rule and show its serious drawback. We suggest using the asymptotic average threshold coalition size instead. We prove that, in large electorates, the asymptotic average threshold coalition size is maximised among all scoring rules by the Borda rule when the number m of alternatives is 3 or 4, and by -approval voting when m ≥ 5.     

Aggregation of coarse preferences

We consider weak preference orderings over a set A _n of n alternatives. An individual preference is of refinement?≤n if it first partitions A _n into ? subsets of `tied' alternatives, and then ranks these subsets within a linear ordering. When ?<n, preferences are coarse. It is shown that, if the refinement of preferences does not exceed ?, a super majority rule (within non-abstaining voters) with rate 1− 1/? is necessary and sufficient to rule out Condorcet cycles of any length. It is argued moreover how the coarser the individual preferences, (1) the smaller the rate of super majority necessary to rule out cycles `in probability'; (2) the more probable the pairwise comparisons of alternatives, for any given super majority rule. Received: 29 June 1999/Accepted: 25 February 2000     

An axiomatization of the Kalai-Smorodinsky solution when the feasible sets can be finite

We axiomatize the Kalai-Smorodinsky solution (1975) in the Nash bargaining problems if the feasible sets can be finite. We show that the Kalai-Smorodinsky solution is the unique solution satisfying Continuity (in the Hausdorff topology endowed with payoffs space), Independence (which is weaker than Nash's one and essentially equivalent to Roth (1977)'s one), Symmetry, Invariance (both of which are the same as in Kalai and Smorodinsky), and Monotonicity (which reduces to a little bit weaker version of the original if the feasible sets are convex). Received: 4 November 1999/Accepted: 6 June 2001     

SYMMETRIC SCORING RULES AND A NEW CHARACTERIZATION OF THE BORDA COUNT

Young developed a classic axiomatization of the Borda rule almost 50 years ago. He proved it is the only voting rule satisfying the normative properties of decisiveness, neutrality, reinforcement, faithfulness and cancellation. Often overlooked is that the uniqueness of Borda applies only to variable populations. We present a different set of properties which only Borda satisfies when both the set of voters and the set of alternatives can vary. It is also shown Borda is the only scoring rule which will satisfy all of the new properties when the number of voters stays fixed. (JEL D71, D02, H00)     

Does majoritarian approval matter in selecting a social choice rule? An exploratory panel study

This study is an attempt to empirically detect the public opinion concerning majoritarian approval axiom. A social choice rule respects majoritarian approval iff it chooses only those alternatives which are regarded by a majority of “voters” to be among the “better half” of the candidates available. We focus on three social choice rules, the Majoritarian Compromise, Borda’s Rule and Condorcet’s Method, among which the Majoritarian Compromise is the only social choice rule always respecting majoritarian approval. We confronted each of our 288 subjects with four hypothetical preference profiles of a hypothetical electorate over some abstract set of four alternatives. At each hypothetical preference profile, two representing the preferences of five and two other of seven voters, the subject was asked to indicate, from an impartial viewpoint, which of the four alternatives should be chosen whose preference profile was presented, which if that is unavailable, then which if both of the above are unavailable, and finally which alternative should be avoided especially. In each of these profiles there is a Majoritarian Compromise-winner, a Borda-winner and a Condorcet-winner, and the Majoritarian Compromise-winner is always distinct from both the Borda-winner and the Condorcet-winner, while the Borda- and Condorcet-winners sometimes coincide. If the Borda- and Condorcet-winners coincide then there are two dummy candidates, otherwise only one, and dummies coincide with neither of the Majoritarian Compromise-, Borda- or Condorcet-winner. We presented our subjects with various types of hypothetical preference profiles, some where Borda respecting majoritarian approval, some where it failed to do so, then again for Condorcet, some profiles it respected majoritarian approval and some where it did not. The main thing we wanted to see was whether subjects’ support for Borda and Condorcet was higher when this social choice rule respected majoritarian approval than it did not. Our unambiguous overall empirical finding is that our subjects’ support for Borda and Condorcet was significantly stronger as they respect majoritarian approval.     

Minimal monotonic extensions of scoring rules

Noting the existence of social choice problems over which no scoring rule is Maskin monotonic, we characterize minimal monotonic extensions of scoring rules. We show that the minimal monotonic extension of any scoring rule has a lower and upper bound, which can be expressed in terms of alternatives with scores exceeding a certain critical score. In fact, the minimal monotonic extension of a scoring rule coincides with its lower bound if and only if the scoring rule satisfies a certain weak monotonicity condition (such as the Borda and antiplurality rule). On the other hand, the minimal monotonic extension of a scoring rule approaches its upper bound as its degree of violating weak monotonicity increases, an extreme case of which is the plurality rule with a minimal monotonic extension reaching its upper bound.

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How the size of a coalition affects its chances to influence an election

Since voting rules are prototypes for many aggregation procedures, they also illuminate problems faced by economics and decision sciences. In this paper we are trying to answer the question: How large should a coalition be to have a chance to influence an election? We answer this question for all scoring rules and multistage elimination rules, under the Impartial Anonymous Culture assumption. We show that, when the number of participating agents n tends to infinity, the ratio of voting situations that can be influenced by a coalition of k voters to all voting situations is no greater than $D_{m} \frac{k}{n}Since voting rules are prototypes for many aggregation procedures, they also illuminate problems faced by economics and decision sciences. In this paper we are trying to answer the question: How large should a coalition be to have a chance to influence an election? We answer this question for all scoring rules and multistage elimination rules, under the Impartial Anonymous Culture assumption. We show that, when the number of participating agents n tends to infinity, the ratio of voting situations that can be influenced by a coalition of k voters to all voting situations is no greater than , where D _m is a constant which depends only on the number m of alternatives but not on k and n. Recent results on individual manipulability in three alternative elections show that this estimate is exact for k=1 and m=3.

Arkadii SlinkoEmail:

     

Characterizing Paretian preferences

A characterization of a property of binary relations is of type M if it can be stated in terms of ordered M-tuples of alternatives. A characterization of finite type provides an easy test of whether preferences over a large set of alternatives possesses the property characterized. Unfortunately, there is no characterization of finite type for Pareto representability in . A partial result along the same lines is obtained for Pareto representability in , k>2.

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How to choose a non-controversial list with k names

Barberà and Coelho (WP 264, CREA-Barcelona Economics, 2007) documented six screening rules associated with the rule of k names that are used by diferent institutions around the world. Here, we study whether these screening rules satisfy stability. A set is said to be a weak Condorcet set à la Gehrlein (Math Soc Sci 10:199–209) if no candidate in this set can be defeated by any candidate from outside the set on the basis of simple majority rule. We say that a screening rule is stable if it always selects a weak Condorcet set whenever such set exists. We show that all of the six procedures which are used in reality do violate stability if the voters do not act strategically. We then show that there are screening rules which satisfy stability. Finally, we provide two results that can explain the widespread use of unstable screening rules.     

Partially monotonic bargaining solutions

The purpose of the paper is to study partially monotonic solutions for two-person bargaining problems. Partial monotonicity relates to the uncertainty a player has about the solution before bargaining. If the minimum utility a player can expect is greater in game T than in game S, and if T contains more alternatives than S, this may bring him to expect that his utility at the solution is greater in T than in S. Partially monotonic solutions reflect these expectations.One partially monotonic solution is axiomatized. The axioms of symmetry and independence of linear transformations are not explicitly assumed, although the solution has also these properties. The Kalai-Smorodinsky solution is shown to be the only continuous partially monotonic solution.This study was financed by the Yrjö Jahnsson Foundation, which is gratefully acknowledged. I like to thank an associate editor and a referee for their valuable suggestions, and the Yrjö Jahnsson Foundation Study Group on Public Economics for useful discussions     

On the likelihood of Condorcet's profiles

Consider a group of individuals who have to collectively choose an outcome from a finite set of feasible alternatives. A scoring or positional rule is an aggregation procedure where each voter awards a given number of points, w _j, to the alternative she ranks in j ^th position in her preference ordering; The outcome chosen is then the alternative that receives the highest number of points. A Condorcet or majority winner is a candidate who obtains more votes than her opponents in any pairwise comparison. Condorcet [4] showed that all positional rules fail to satisfy the majority criterion. Furthermore, he supplied a famous example where all the positional rules select simultaneously the same winner while the majority rule picks another one. Let P ^* be the probability of such events in three-candidate elections. We apply the techniques of Merlin et al. [17] to evaluate P ^* for a large population under the Impartial Culture condition. With these assumptions, such a paradox occurs in 1.808% of the cases. Received: 30 April 1999/Accepted: 14 September 2000     

Ranking opportunity sets: An approach based on the preference for flexibility

We describe a criterion to evaluate subsets of a finite set of alternatives which are considered as opportunity sets. The axioms for set comparison are motivated within the preference for flexibility framework. We assume the preference over the universal set of alternatives to be made of two disjoint binary relations. The result is the axiomatic characterization of a procedure which is formally similar to the leximax ordering, but in our case it incorporates the presence of some uncertainty about the decision-maker final tastes. Received: 20 January 1999/Accepted: 20 October 1999     

Reference functions and possibility theorems for cardinal social choice problems

 In this paper, we provide axiomatic foundations for social choice rules on a domain of convex and comprehensive social choice problems when agents have cardinal utility functions. We translate the axioms of three well known approaches in bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Kalai 1977) to the domain of social choice problems and provide an impossibility result for each. We then introduce the concept of a reference function which, for each social choice set, selects a point from which relative gains are measured. By restricting the invariance and comparison axioms so that they only apply to sets with the same reference point, we obtain characterizations of social choice rules that are natural analogues of the bargaining theory solutions. Received: 8 August 1994/Accepted: 12 February 1996     

Unequivocal majority and Maskin-monotonicity

The unequivocal majority of a social choice rule is a number of agents such that whenever at least this many agents agree on the top alternative, then this alternative (and only this) is chosen. The smaller the unequivocal majority is, the closer it is to the standard (and accepted) majority concept. The question is how small can the unequivocal majority be and still permit the Nash-implementability of the social choice rule; i.e., its Maskin-monotonicity. We show that the smallest unequivocal majority compatible with Maskin-monotonicity is n- ë \fracn-1m û{n-\left\lfloor \frac{n-1}{m} \right\rfloor} , where n ≥ 3 is the number of agents and m ≥ 3 is the number of alternatives. This value is equal to the minimal number required for a majority to ensure the non-existence of cycles in pairwise comparisons. Our result has a twofold implication: (1) there is no Condorcet consistent social choice rule satisfying Maskin-monotonicity and (2) a social choice rule satisfies k-Condorcet consistency and Maskin-monotonicity if and only if k 3 n- ë \fracn-1m û{k\geq n-\left\lfloor \frac{n-1}{m}\right\rfloor}.     

Strategy-proof resolute social choice correspondences

We qualify a social choice correspondence as resolute when its set valued outcomes are interpreted as mutually compatible alternatives which are altogether chosen. We refer to such sets as “committees” and analyze the manipulability of resolute social choice correspondences which pick fixed size committees. When the domain of preferences over committees is unrestricted, the Gibbard–Satterthwaite theorem—naturally—applies. We show that in case we wish to “reasonably” relate preferences over committees to preferences over committee members, there is no domain restriction which allows escaping Gibbard–Satterthwaite type of impossibilities. We also consider a more general model where the range of the social choice rule is determined by imposing a lower and an upper bound on the cardinalities of the committees. The results are again of the Gibbard–Satterthwaite taste, though under more restrictive extension axioms.     

Opportunity sets and individual well-being

 An opportunity set ranking rule assigns an ordering of opportunity sets to each individual utility function (defined on the universal set of alternatives) within the domain of this rule. Using an axiomatic approach, this paper characterizes a general class of opportunity set ranking rules which are based on the utilities associated with the elements of an opportunity set. It is argued that the addition of an alternative to a given opportunity set is not necessarily desirable in terms of overall well-being, and this position is reflected in replacing a commonly used monotonicity axiom with an alternative condition. Received: 15 May 1995/Accepted: 14 December 1995