Short Communication
War and peace in veto voting

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Abstract

Let I={i1,,in} be a set of voters (players) and A={a1,,ap} be a set of candidates (outcomes). Each voter iI has a preference Pi over the candidates. We assume that Pi is a complete order on A. The preference profile P={Pi,iI} is called a situation. A situation is called war if the set of all voters I is partitioned in two coalitions K1 and K2 such that all voters of Ki have the same preference, i=1,2, and these two preferences are opposite. For a simple class of veto voting schemes we prove that the results of elections in all war situations uniquely define the results for all other (peace) situations. In other words, the results depend only on the veto (or effectivity) function. We give several other examples from game (and from graph) theory with the same property.

Section snippets

Main theorem

We follow standard concepts and notation of veto voting theory; see, e.g., [5], [6], and [8, Section 8.4] . Let I={i1,,in} be a set of voters (players) and A={a1,,ap} be a set of candidates (outcomes). Each voter iI has a preference (a complete order) Pi over all candidates. The set of all preferences P={Pi,iI} is called a preference profile or a situation. A situation is called war if the set of voters I is partitioned in two coalitions K1 and K2 such that all voters of Ki have the same

An equivalent statement

The theorem can be equivalently reformulated as follows.

The veto function is defined as a mapping V:2I×2A{0,1}; that is, V has two arguments: a coalition of voters KI and a block of candidates BA. The equalities V(K,B)=1 and V(K,B)=0 mean that K can, and respectively cannot, veto B. The complementary function E(K,B)=V(K,AB) is called the effectivity function; see [6] Section 7.2 and [9] Chapter 6.

Each pair of distributions μ:IZ+ and λ:IZ+, generates a veto function V=Vμ,λV(K,B)=1iffiKa

Proof of Theorem 2

In this section we will consider only simple veto orders. Then, without any loss of generality, we can assume that permutation τ is identical; that is first i1 distributes all veto cards, then i2, etc. In this case argument τ becomes irrelevant and we will omit it in all formulas. In particular, pair (λ,μ) already defines a voting scheme.

We will make use of the terminology from the popular game Sea Battle; see, for example, http:/www.karpolan.com/sea-battle. Given a scheme (λ,μ), a voter iI,

On properties of game structures and graphs that depend only on the corresponding veto functions

The main result of this paper states that the SCC of a simple veto voting scheme is uniquely defined by its effectivity (or equivalently, veto) function. The following two results are similar:

  • (i)

    Nash solvability of a two-person game form g depends only on its effectivity function Eg.

  • (ii)

    The core of a normal form game (g,u) depends only on its utility function u and effectivity function Eg.

The definitions follow. Let standardly I and A be a set of voters (players) and candidates (outcomes)

Acknowledgements

I am thankful to Endre Boros and Leonid Khachiyan for helpful remarks.

References (9)

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The research was supported by the National Science Foundation, Grant IIS-0118635, and by DIMACS, the NSF Center for Discrete Mathematics and Theoretical Computer Science.

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