A Note on a Weak Dictator and the Veto Condition

Marija Srećković

doi:10.20944/preprints202506.1501.v1

Submitted:

17 June 2025

Posted:

18 June 2025

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Abstract

In this note we discuss the relationships between the weak dictatorship axiom, as introduced by A. Mas-Colel and H. Sonnenschein, and the vetoer axiom, as given by P. C. Fishburn, in a wider context of traditional Social Choice Theory (SCT). Namely, we prove the equivalence between these two axioms. This note can be a good formal reasoning exercise for students in the field of preference logic. This note is written as an outline for the 'second' lesson in SCT for non-mathematicians, which contains elementary formal logical argumentation, after the basics of preference logic, and before the extensive and complicated treatment of impossibility theorems.

Keywords:

dictatorship

;

weak dictator

;

vetoer

;

social choice theory

Subject:

Computer Science and Mathematics - Logic

Introduction

Social Choice Theory (SCT) has been the subject of the Nobel Prize in Economics on several occasions over the past decades (e.g., K. J. Arrow (1921–2017), Nobel Prize in Economics 1972; J. M. Buchanan (1919–2013) Nobel Prize in Economics 1986; A. K. Sen (1933-), Nobel Prize in Economics 1998). This is one of the additional reasons why SCT, with modifications and simplifications, is permanently interesting as teaching content at all levels of economics studies. On the other hand, contrary to this attractiveness, the problem is the difficulty in understanding the basic concepts of SCT.

The central themes of traditional Arrow-Sen SCT are impossibilities and interdeducibilities of sets of axioms appearing in the theory (see Arrow (1963), Sen (1970), Fishburn (1973), Bori\v ci\’c (2009)). This short note is devoted to deductive interdependences between dictatorship, as originally defined by Arrow (see Arrow (1963), Sen (1970)), the weak dictatorship, as given by Mas--Colel and Sonnenschein (1972), and the vetoer axiom, as introduced by Fishburn (1973). In previous papers, e.g., Sre\’ckovi\’c (2017), and Bori\v ci\’c and Sre\’ckovi\’c (2024), the authors deal primarily with the cardinal status of vetoing, but also include a discussion regarding some ordinal aspects of simplified versions of the well-known axioms of Arrow-Sen SCT. The simplification of traditional SCT is the central point in Boričić (2023, 2024), and Boričić and Srećković (2024).

The high mathematization of fragments of contemporary economic theories often presents a barrier to a wide range of economics students. Our goal is to make the formalism we use in this note, as part of the widespread contemporary symbolic logic, accessible to non-mathematician students in order to enable a better understanding and connection between form and meaning.

In this note, we focus on traditionally defined (but not simplified) conditions, as entities of a higher-order language, and the facts that the dictatorship implies weak dictatorship, that the dictatorship implies vetoing, and, finally, that the quantifier-free parts of the axioms of the weak dictatorship TWD and vetoing TV are equivalent. We follow the definitions and notations given in Boričić and Srećković (2024), but we prefer a more descriptive rather than a formal style of argumentation. It may seem that these relationships are anticipated and almost trivial, but we believe that such considerations contribute to the popularization of the theory and its better understanding by the wider community, primarily students. Finally, this note can be considered as an outline for the second lesson in SCT, after mastering the basic concepts of preference logic as given in Pefku (2025), but before the difficult lessons devoted to impossibility theorems.

Dictator, Weak Dictator, and Vetoer

The traditional definition is based on finite sets of individuals and alternatives V and X, respectively,

P_{i}

and

P

—individual and social strict preference relations on X,

R

—a social weak preference on X, and

I

—a social indifference relation on X. It is also supposed that

P

is asymmetric and transitive,

R

—linear and transitive, and

I

is reflexive, symmetric and transitive, as well as, that

R = P \cup I

, and

I = R \cup R^{- 1}

(v. Pefku (2025)).

Arrow’s dictatorship condition TD, the Mas-Colell and Sonnenschein weak dictatorship TWD, and Fishburn’s vetoer condition TV can be presented as follows:

TD: There exists an

i \in V

, such that for all n-tuples of individual preferences and each two alternatives

x, y \in X

,

x P_{i} y \to x P y .

TWD: There exists an

i \in V

, such that for all n-tuples of individual preferences and each two alternatives

x, y \in X

,

x P_{i} y \to x R y .

TV: There exists an

i \in V

, such that for all n-tuples of individual preferences and each two alternatives

x, y \in X

,

x P_{i} y \to \neg y P x .

The prefix “T” stands for the “traditional” versions of the axioms, as opposed to the “simplified” ones discussed in the papers Boričić (2023), Boričić and Srećković (2024), and Boričić (2024).

For negations NTD and NTV of TD and TV, respectively, Fishburn emphasised that NTV is stronger than NTD, “since a dictator is a vetoer, but not conversely” (see Fishburn (1973), p. 208). Here we provide an argument for contraposition, i.e., an equivalent statement for positive sentences.

Note that the quantification ‘for all n-tuples of individual preferences’ removes the above conditions from the first-order language.

Lemma 1.

(i) TD implies TV; (ii) TD implies TWD.

Proof.

(i) If we suppose TD, bearing in mind that

P

is asymmetric,

x P y \to \neg y P x

, for all

x, y \in X

, a fortiori, from

x P_{i} y \to x P y

, which is the quantifier-free part of TD we infer

x P_{i} y \to \neg y P x

, i.e., the quantifier-free part of TV.

(ii) Similarly, from

x P_{i} y \to x P y

, the quantifier-free part of TD, by weakening of the consequent, we infer

x P_{i} y \to x P y \lor x I y

, i.e.,

x P_{i} y \to x R y

, the quantifier-free part of TWD. □

Lemma 2.

Conditions TV and TWD are equivalent.

Proof.

In order to prove that TWD and TV are equivalent, firstly, let us suppose that TWD,

x P_{i} y \to x P y \lor x I y

, for all

x, y \in X

, is satisfied. The in-difference relation

I

is defined as

x I y

iff

\neg x P y \land \neg y P x

, and the strict prefe-rence relation

P

is asymmetric,

x P y \to \neg y P x

, for all

x, y \in X

, where, from TWD, a fortiori, we obtain TV:

x P_{i} y \to \neg y P x

. Conversely, we start with the fact that the weak preference

R

is linear:

x R y \lor y R x

, for all

x, y \in X

, i.e.,

x P y \lor x I y \lor y P x

, for all

x, y \in X

, which is logically equivalent to

\neg y P x \to x R y

, from which, through

x P_{i} y \to \neg y P x

, TV, we immediately have

x P_{i} y \to x R y

, for all

x, y \in X

, TDW. This means that a quantifier-free part of weak dictator axiom TWD is exactly a quantifier-free part of vetoer axiom TV, and vice versa. □

Conclusions

If we denote Arrow’s dictatorship, Mas-Colell’s and Sonnenschein’s weak dictatorship, and Fishburn’s vetoer axiom by TD, TWD and TV, respectively, then we can prove that, for the quantifier-free parts of these axioms, we have: TD implies both TWD and TV, and that TWD and TV are mutually equivalent. These facts can be a good formal reasoning exercise for students when dealing with logic of preference or social choice theory.

Acknowledgments

The author thanks Professor Branislav Boričić for suggesting to consider and prove the above facts.

References

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A Note on a Weak Dictator and the Veto Condition

Abstract

Keywords:

Subject:

Introduction

Dictator, Weak Dictator, and Vetoer

Conclusions

Acknowledgments

References

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