Informational geometry of social choice

 Elementary geometry is used to understand, extend and resolve basic informational difficulties in choice theory. This includes axiomatic conclusions such as Arrow’s Theorem, Chichilnisky’s dictator, and the Gibbard–Satterthwaite result. In this manner new results about positional voting methods are outlined, and difficulties with axiomatic approach are discussed. A topological result about “dictatorial” behavior is offered. Received: 15 December 1993/Accepted: 22 April 1996     

Changes that cause changes

Beyond determining whether procedures can be manipulated, the real goal for any analysis of “strategic behavior” is to identify all settings where and when this can be done, who can do it, and what they should do. By applying the geometric approach of Saari [7, 8] to the Kemeny's Rule (KR), we demonstrate how surprisingly simple this analysis can be, we identify all three candidate KR strategic behavior, and we show how an almost identical analysis answers most other multiple profile concerns (e.g., the abstention paradox and when voters just make errors). We also introduce new measures, which can be used with any procedure, to compare strategic and other behavior involving “changes.” These measures help to identify settings where it may be more important to worry about honest mistakes than strategic voting. Received: 16 April 1999/Accepted: 29 September 1999     

Condorcet's jury theorem,dependency among jurors

With the aid of a simple discrete probability model, the Condorect Jury Theorem is extended to situations where there is dependency between judgement. It is shown that negative intra-voter correlation improves on jury competence in the sense that the probability of a correct decision increases. Positive correlation, on the other hand, tends to decrease jury competence. The results given here are obtained with the aid of simple recursion relations.     

The sensitivity of weight selection on the Condorcet efficiency of weighted scoring rules

A weighted scoring rule, Rule λ, on three alternative elections selects the winner by awarding 1 point to each voter's first ranked candidate, λ points to the second ranked candidate, and zero to the third ranked candidate. The Condorcet winner is the candidate that would defeat each other candidate in a series of pairwise elections by majority rule. The Condorcet efficiency of Rule λ is the conditional probability that Rule λ selects the Condorcet winner, given that a Condorcet winner exists. Borda rule (λ=1/2) is the weighted scoring rule that maximizes Condorcet efficiency. The current study considers the conditional probability that Borda rule selects the Rule λ winner, given that Rule λ elects the Condorcet winner with a large electorate. Received: 21 August 1996 / Accepted: 7 January 1997     

Single-Crossing, Strategic Voting and the Median Choice Rule

This paper studies the strategic foundation of the Representative Voter Theorem (Rothstein in: Pub Choice 72:193–212, 1991), also called the “second version” of the Median Voter Theorem. As a by-product, it also considers the existence of strategy-proof social choice functions over the domain of single-crossing preferences. The main result shows that single-crossing constitutes a domain restriction over the real line that allows not only majority voting equilibria, but also non-manipulable choice rules. In particular, this is true for the median rule, which is found to be group strategic-proof over the full set of alternatives and over every nonempty subset. In addition, the paper also examines the relation between single-crossing and order-restriction. And it uses this relation together with the strategy-proofness of the median rule to prove that the outcome predicted by the Representative Voter Theorem can be implemented in dominant strategies through a simple mechanism. This mechanism is a two-stage voting procedure in which, first, individuals select a representative among themselves, and then the winner chooses a policy to be implemented by the planner.     

THE EXTENDED COASE THEOREM AND LONG RUN EQUILIBRIUM: THE NONEQUIVALENCE OF LIABILITY RULES AND PROPERTY RIGHTS

Many authors have held that in a world with reasonable positive transactions costs, the assignment of liability for pollution damages affects profits, thus long run equilibrium. Therefore, the Coase Theorem fails for such cases. Others have defended the Theorem in such situations .
This paper shows that the disagreement has its roots in a confusion of liability rules and property rights. A simple general equilibrium model is used to show that the Theorem is correct for property rights assignments, but not for liability rules. Liability rules are shown to be, in effect, incomplete property rights, which leads to inefficiency .     

The relation between monotonicity and strategy-proofness

The Muller–Satterthwaite Theorem (J Econ Theory 14:412–418, 1977) establishes the equivalence between Maskin monotonicity and strategy-proofness, two cornerstone conditions for the decentralization of social choice rules. We consider a general model that covers public goods economies as in Muller–Satterthwaite (J Econ Theory 14:412–418, 1977) as well as private goods economies. For private goods economies, we use a weaker condition than Maskin monotonicity that we call unilateral monotonicity. We introduce two easy-to-check preference domain conditions which separately guarantee that (i) unilateral/Maskin monotonicity implies strategy-proofness (Theorem 1) and (ii) strategy-proofness implies unilateral/Maskin monotonicity (Theorem 2). We introduce and discuss various classical single-peaked preference domains and show which of the domain conditions they satisfy (see Propositions 1 and 2 and an overview in Table 1). As a by-product of our analysis, we obtain some extensions of the Muller–Satterthwaite Theorem as summarized in Theorem 3. We also discuss some new “Muller–Satterthwaite preference domains” (e.g., Proposition 3).     

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin in Rev Econ stud 66: 23–38, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackon and Palfrey in J Econ Theory 98: 1–25, 2001).I thank Toyo Sakai for his comments on a previous draft. I also thank two anonymous referees and an editor of this journal for helpful comments that improved this paper. A previous version circulated as “A note on Maskin monotonicity”. After the results presented here were obtained, I became aware of a new unpublished paper by Benoit and Ok (2004). The result of Theorem 2 and the discussion that follows is partially similar to their Theorem 1.     

A pedagogical proof of Arrow's Impossibility Theorem

In this note I consider a simple proof of Arrow's Impossibility Theorem (Arrow 1963). I start with the case of three individuals who have preferences on three alternatives. In this special case there are 13³=2197 possible combinations of the three individuals' rational preferences. However, by considering the subset of linear preferences, and employing the full strength of the IIA axiom, I reduce the number of cases necessary to completely describe the SWF to a small number, allowing an elementary proof suitable for most undergraduate students.  This special case conveys the nature of Arrow's result. It is well known that the restriction to three options is not really limiting (any larger set of alternatives can be broken down into triplets, and any inconsistency within a triplet implies an inconsistency on the larger set). However, the general case of n≥3 individuals can be easily considered in this framework, by building on the proof of the simpler case. I hope that a motivated student, having mastered the simple case of three individuals, will find this extension approachable and rewarding.  This approach can be compared with the traditional simple proofs of Barberà (1980); Blau (1972); Denicolò (1996); Fishburn (1970); Kelly (1988); Mueller (1989); Riker and Ordeshook (1973); Sen (1979, 1986); Suzumura (1988), and Taylor (1995). Received: 5 January 1999/Accepted: 10 December 1999     

Table of Contents

Proposed Rule on Benefits Raises Questions about Parity and Enforcement SAMHSA Releases Bupe‐OTP Final Rule Imprint of Founder Continues to Influence Father Martin's Proposed Rule's Language on Parity What Plans Must Include ONDCP's Botticelli: Prevention, Block Grant, and Recovery DSM‐5 Will Include SUDs Instead of Dependence and Abuse Improving Outcomes: Connecting Payment to Performance ABAM Continues to Moves to Make Addiction a Specialty Briefly Noted State News Coming up     

Certificates of optimality: the third way to biproportional apportionment

The biproportional apportionment problem (BAP) must be faced in many proportional electoral systems where seats must be allocated to parties within regions. BAP is a non-trivial optimization problem, and only sophisticated algorithms are currently available for solving it. The issue is: are they “writable” as an actual law? Citizens rightly demand simple, easy to understand, voting systems. The alternative, though, seems to have simple, but unsound electoral laws. We propose the following way out of this dilemma: leave to a mathematically sophisticated algorithm the task of producing an optimal apportionment, but attach to it a “certificate of optimality”, that is, describe a simple procedure whereby anybody can check, through some elementary operations, that the seat allocation output by the algorithm is indeed an optimal apportionment. We discuss one such certificate, based on the Max flow- min cut Theorem, relative to a parametric max flow method of ours for BAP.     

Annexations and alliances: When are blocs advantageous a priori?

For any simple voting game (SVG), we consider the question posed in the title from two different points of view as to what voting power means. We also distinguish between blocs imposed by annexation and those formed voluntarily, and present some general theoretical results concerning these notions. We illustrate our theoretical findings with examples using both toy SVGs and the Qualified Majority Voting rule of the Council of Ministers of the European Community (CMEC). We show that when voting power is understood as influence (I-power), forming a voluntary bloc may be advantageous even if its voting power is smaller than the sum of the original powers of its members; and it may be disadvantageous even if its voting power is greater than that sum. Received: 9 February 2000/Accepted: 19 October 2000     

More on independent decisiveness and Arrow's theorem

Denicolò [2, Theorem 1] strengthens Arrow's [1, p. 97] theorem by replacing the independence of irrelevant alternatives (IIA) condition by a strictly weaker one, relational independent decisiveness (RID). It is shown here that RID can be still substantially weakened. Yet, the new condition is equivalent to RID under the weak Pareto principle P and unrestricted domain U. In fact, any condition that can be put in place of IIA in Arrow's theorem must imply RID in the presence of P and U. Incidentally, it is argued that Denicolò's proof of his Theorem 1 contains an imprecision. Received: 7 March 2000/Accepted: 11 December 2000     

Assent-maximizing social choice

We take a decision theoretic approach to the classic social choice problem, using data on the frequency of choice problems to compute social choice functions. We define a family of social choice rules that depend on the population’s preferences and on the probability distribution over the sets of feasible alternatives that the society will face. Our methods generalize the well-known Kemeny Rule. In the Kemeny Rule, it is known a priori that the subset of feasible alternatives will be a pair. We define a distinct social choice function for each distribution over the feasible subsets. Our rules can be interpreted as distance minimization—selecting the order closest to the population’s preferences, using a metric on the orders that reflects the distribution over the possible feasible sets. The distance is the probability that two orders will disagree about the optimal choice from a randomly selected available set. We provide an algorithmic method to compute these metrics in the case where the probability of a given feasible set is a function only of its cardinality.     

THE DIVISION OF LABOR IN THE FIRM

In this paper, Adam Smith's theory of labor specialization is formulated and tested. The restrictions in the theory are explicitly derived, and Smith's Theorem — that specialization will rise with the output rate — is rigorously stated. A simple human capital model is then employed to derive testable implications of the theory. Wage equations are estimated for the U.S. shipbuilding industry during World War II. The peculiarities of the war data notwithstanding, the results are strongly supportive of the theory.     

When is Condorcet's Jury Theorem valid?

Existing proofs of Condorcet's Jury Theorem formulate only sufficient conditions for its validity. This paper provides necessary and sufficient conditions for Condorcet's Jury Theorem. The framework of the analysis is the case of heterogeneous decisional competence, but the independence assumption is maintained. Received: 23 January 1997/Accepted: 8 March 1997     

Social network research and human subjects protection: Towards more effective infectious disease control

The issue of human subject protection in relation to social network research on the spread and control of human pathogens is considered. As this area derives most of its concepts and methods from social network analysis more generally, the present discussion has wider relevance. One problem is that some Institutional Review Boards (IRBs) have assumed that if a participant (who gave informed consent) is to be asked to name network associates these too — automatically — would be human subjects from whom informed consent also must be obtained. Invariably, if this occurs proposed research — whatever its funding and potential contributions might be — is blocked. A conservative approach is taken here. The Common Rule is assumed to provide relevant guidance, the responsibility of IRBs is to make decisions based on the Common Rule, and in consequence the burden is on those proposing social network research to design — and defend — their planned work with this in mind. At the same time, it is argued that it is important not to stifle beneficial research by adding to one inherently conservative review process (of grant proposals) another (of IRBs) so that work is prevented simply because it is research at the frontiers rather than ‘safe’ research.     

Table of Contents

New Jersey Supreme Court to Rule on Methadone Treatment during Pregnancy Family and Community Linkages Key in Youth Treatment NSDUH: Daily Marijuana Use Increasing among Teens Blog Post Critical of IC&RC Training Full of Errors CSAT's Clark: SBIRT Trials Should Focus on Mild SUDs Briefly Noted Coming up     

Table of Contents

Opposing Voices on Essential Health Benefits: Insurers and Employers N.J. Agency Considers Sound Data Crucial in Tight Insurance Market Chamber of Commerce on HHS Delay in Proposed Rule FDA Likely to Move Hydrocodone to Schedule II Senator Harkin, Longtime Champion of Field, to Retire in 2014 Study: Smartphones Propel Recovery Briefly Noted Coming up     

Acyclic social choice from finite sets

This paper characterizes acyclic preference aggregation rules under various combinations of monotonicity, neutrality, decisiveness, and anonymity, in the spirit of Nakamura's (1979) Theorem on the core of simple games. Each of these characterizations can be seen to follow from Ferejohn and Fishburn's (1979) result on acyclic aggregation rules assuming only independence of irrelevant alternatives. We then state as corollaries to these characterizations extant results such as Sen's (1970) liberal paradox, Blau and Deb's (1977) theorem on the existence of vetoers, and Brown's (1975) theorem on the non-emptiness of the collegium.I thank David Austen-Smith, Michel Le Breton, Maurice Salles, and two anonymous referees for numerous enlightening conversations and comments. I also thank the National Science Foundation and Sloan Foundation for generous financial assistance.