Download A note on Minimal Unanimity and Ordinally Bayesian Incentive

Mathematical Social Sciences 53 (2007) 209 – 211
www.elsevier.com/locate/econbase
Short communication
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
210
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).
1
M. Núñez / Mathematical Social Sciences 53 (2007) 209–211
211
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.
2
We follow the representation of Majumdar and Sen (2004).