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Dependence and Independence in Social Choice: Arrow’s Theorem

Abstract

One of the goals of social choice theory is to study the group decision methods that satisfy two types of desiderata. The first type ensures that the group decision depends in the right way on the voters’ opinions. The second type ensures that the voters are free to express any opinion, as long as it is an admissible input to the group decision method. Impossibility theorems, such as Arrow’s Theorem, point to an interesting tension between these two desiderata. In this paper, we argue that dependence and independence logic offer an interesting new perspective on this aspect of social choice theory. To that end, we develop a version of independence logic that can express Arrow’s properties of preference aggregation functions. We then prove that Arrow’s Theorem is derivable in a natural deduction system for the first-order consequences of our logic.

Keywords

  • Social Choice
  • Group Decision
  • Social Ranking
  • Social Choice Function
  • Social Choice Theory

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    In formal work on social choice theory, it is common to identify a voter’s preference over a set of alternatives X with her ranking over the set of alternatives. In general, a ranking of the alternatives is only one way in which a voter may express her preference over the set of alternatives. Consult [17] for a discussion of the main philosophical issues here.

  2. 2.

    For simplicity, we restrict attention to a finite set of alternatives. This restriction is not necessary for what follows, though it does have some implications on the design of the formal language used to describe a social choice model.

  3. 3.

    In this article, we set aside any game-theoretic issues around whether voters have an incentive to report their true preferences.

  4. 4.

    Also, R maj P may not be complete if there is an even number of voters. There are a variety of ways to modify the definition of the majority ordering to ensure completeness when there are an even number of voters.

  5. 5.

    One can also explore alternative definitions of Unanimity of varying strengths. For example, if all voters weakly rank candidate a above candidate b, then society does so as well.

  6. 6.

    A full discussion of this result is beyond the scope of this article. See [46] for a precise statement of the Müller-Satterthwaite Theorem (including the additional assumptions needed to prove the equivalence) and a discussion of the relevant literature.

  7. 7.

    Properties of group decision methods are often called “axioms” in the social choice literature. However, the principles studied in the social choice literature do not have the same status as the axioms of, for example, Peano arithmetic or the axioms defining a group. As should be clear from the discussion in this section, many of the so-called axioms of social choice are certainly not “self-evident,” and may require extensive justification.

  8. 8.

    The interested reader can consult [16] and [23] for the details of the natural deduction system. We do not include the system here since we are only proving the existence of a derivation of Arrow’s Theorem rather than providing a derivation. We will take up this challenge in the extended version of this paper.

  9. 9.

    Note that a slightly different connective \(\sim '\) with the semantics \(M\models _{S} \sim '\varphi\) iff \(M\nvDash _{S}\varphi\) is known as classical negation in the dependence logic literature.

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Pacuit, E., Yang, F. (2016). Dependence and Independence in Social Choice: Arrow’s Theorem. In: Abramsky, S., Kontinen, J., Väänänen, J., Vollmer, H. (eds) Dependence Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-31803-5_11

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