Mathematical Social Sciences 53 (2007) 209 – 211 www.elsevier.com/locate/econbase

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A note on Minimal Unanimity and Ordinally Bayesian Incentive Compatibility ☆ Matías Núñez Laboratoire d'Econométrie, Ecole Polytechnique, 1 Rue Descartes 75005, Paris, France Received 8 December 2005; received in revised form 15 May 2006; accepted 4 December 2006 Available online 24 January 2007

Abstract Majumdar and Sen (Majumdar, D., Sen, A., 2004. Ordinally Bayesian incentive compatible voting rules. Econometrica 72 (2), 523−540) extend the Gibbard–Satterthwaite theorem for Unanimous and Ordinally Bayesian Incentive Compatible (OBIC) social choice functions, assuming independent beliefs. We introduce a new weakening concept for unanimity: the Minimal Unanimity. Even under this weaker condition, we get a negative result: the minimally unanimous social choice functions that are OBIC with respect to independent beliefs are dictatorial. © 2006 Elsevier B.V. All rights reserved. Keywords: Gibbard–Satterthwaite theorem; Ordinally Bayesian Incentive Compatibility; Minimal Unanimity

JEL classification: D7; D70; D71

1. Introduction The Gibbard–Satterthwaite theorem states that all the voting systems that verify unanimity and strategy-proofness are dictatorial. Majumdar and Sen (2004) extend this negative conclusion for Unanimous and Ordinally Bayesian Incentive Compatible (OBIC) social choice functions, assuming independent beliefs. In this work, a weakening condition for the unanimity condition is given in the social choice function (SCF) context: minimal unanimity. It only requires that there ☆

I wish to thank Claude d'Aspremont and Jean-François Laslier for their help and useful comments during the project. I am also indebted to Efthymios Athanasiou, Luis Fontaine Campos, Clémence Christin, Hélène Latzer, Marc Leandri, Dipjyoti Majumdar, Jean François Mertens, Maia Stead, Isaac Tanguy, Giacomo Valletta and two anonymous referees for their help and valuable comments. E-mail address: [email protected]. 0165-4896/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2006.12.001

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M. Núñez / Mathematical Social Sciences 53 (2007) 209–211

exists at least one configuration such that if all the individuals have the same outcome as their best ranked alternative then this alternative must be chosen, and this for each outcome. As we will see, it leads us to the classic negative conclusion. 2. A negative result under Minimal Unanimity The framework used throughout is the one used in Majumdar and Sen (2004). Let us suppose that we have N agents with |N| ≥ 2. Each player has some strict preference over the set A = {a, b, c, …} of outcomes which we assume to be finite and such that |A| ≥ 3. The preference of voter i over the A outcomes will be denoted by the preference ordering Pi, where aPib means a is strictly preferred to b. Each preference ordering belongs to P, the set of strict and complete orderings of A. The preference profile P is a vector that describes the preference orderings of all the individuals in the society. Each P belongs to PN , the set of preference profiles. The social aggregation mechanism is a SCF, that is a mapping f: PN YA. For Pi ∈ P and k = 1, 2, 3, …, let rk(Pi) denote the k-th ranked alternative in Pi, i.e., rk(Pi) = a implies that |{b ∈ A|bPia| = k − 1. Definition 2.1. A SCF f is unanimous if f (P1, …, PN) = x whenever x = r1(Pi) for all individuals i ∈ N. Definition 2.2. A SCF f is minimally unanimous (MU) if for each different outcome x ∈ A, there exists a preference profile Px = (P1, …, PN) such that r1(Pi) = x for all i ∈ N and f (Px) = x. We denote by B(a, Pi ) = {b ∈ A|bPia} ∪ {a} the set of alternatives that are better than a under P . Property M, that is now presented, has been shown by Majumdar and Sen (2004) as necessary for OBIC SCFs under independent priors.1 i

Definition 2.3. (Property M) A SCF f satisfies Property M, if for all i ∈ N, for all k ∈ {1, …, |A|}, for all P−i and Pi , P⁎i such that B(rk(Pi ), Pi ) = B(rk(P⁎i ), P⁎i ), we have that: i ; P−i ÞaBðrk ðPi Þ; Pi Þ ½ f ðPi ; P−i ÞaBðrk ðPi Þ; Pi ÞZ ½ f ðP⁎

Property M can be interpreted as follows. Let f be a SCF and let P = (Pi, P−i) be a preference profile. Let P⁎i be a preference ordering that has the same first-ranked k elements as Pi. That is, we have the following set equality: B(rk(Pi ), Pi) = B(rk(P⁎i ), P⁎i ). Then Property M requires that if x is the elected social outcome under P and x is one of the top k elements of Pi, then the social outcome under (P⁎i , P−i) must be also one of these top k elements. Indeed, if P⁎i and Pi have the same best ranked element, Property M implies that f (P⁎i , P−i ) = f (Pi, P−i). Lemma 2.1. A SCF that is minimally unanimous and satisfies Property M is unanimous. Proof:. Let f be a SCF that satisfies Property M and that is minimally unanimous. For each x ∈ A, we denote by Px a profile where all voters rank x first and f (Px) = x. Let P = (P1, …, Pn) be an arbitrary profile where x is ranked first by all the voters. As r1 ðP1 Þ ¼ r1 ðPx1 Þ; then f ðP1 ; Px2 ; N ; PxN Þ ¼ f ðPx Þ ¼ x As r1 ðP2 Þ ¼ r1 ðPx2 Þ; then f ðP1 ; P2 ; Px3 ; N ; PxN Þ ¼ f ðP1 ; Px2 ; N ; PxN Þ ¼ f ðPx Þ ¼ x v As r1 ðPN Þ ¼ r1 ðPxN Þ; then f ðPÞ ¼ f ðP1 ; P2 ; N ; PN Þ ¼ f ðP1 ; P2 ; N ; PN −1 ; PxN Þ ¼ f ðPx Þ ¼ x The definitions of belief, OBIC with respect to μ, strategy-proofness, and the sets C and Δl have been omitted for simplicity. The reader can find them in Majumdar and Sen (2004).

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We can then change voters preferences from Px to any arbitrary P where all voters rank x first for each alternative x. The SCF f is unanimous. Example: This example shows graphically the implications of the lemma.2 Let A = {a, b, c}, N = {1, 2}. Individual 1's preferences appear along the rows and individual 2's along the columns. The SCF f on the left side of the implication is MU as we have that f (abc, acb) = a, f (bac, bac) = b and f (cab, cba) = c.

The SCF f on the right side of the implication is unanimous. As a consequence of the lemma, we can state the following theorem: Theorem 2.1. Let |A| ≥ 3. There exists a subset C of the set of independent beliefs Δl such that, if a SCF f is minimally unanimous and is OBIC with respect to μ ∈ C, then f is dictatorial. On the “minimality” of the minimal unanimous condition. Usually, unanimity is considered as an ethically founded and weakly demanding condition. However, assuming unanimity for a SCF implies that a considerable amount of social outcomes are fixed. In a game with N players and m outcomes where there exists (m!)N different social outcomes, we have m[(m − 1)!]N outcomes fixed. MU significantly weakens the unanimity condition. Indeed, we have m fixed values, instead of fixing m[(m − 1)!]N. However, one remark should be made about the “minimality” of the new condition. One standard weakening of unanimity condition is citizen sovereignty (CS). It states that for every alternative x there exists a preference profile P such that the social choice outcome is x. Under CS, we also have m fixed outcomes. MU is stronger than citizen sovereignty in a crucial way. Indeed, CS combined with Property M does not imply dictatorship. Whereas CS only requires every alternative to be socially attainable, MU requires it for some specific preference profiles. 3. Conclusion Unanimity has been relaxed to a new weaker concept, minimal unanimity. Unfortunately, even with the new condition, we still get a negative result under independent beliefs. The consequences of the interaction between minimal unanimity and OBIC under different hypotheses on the beliefs remain to be explored. It would be specially interesting to enlarge this approach assuming different forms of correlation between the beliefs of the voters. Reference Majumdar, D., Sen, A., 2004. Ordinally Bayesian incentive compatible voting rules. Econometrica 72 (2), 523–540.

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We follow the representation of Majumdar and Sen (2004).