Short CommunicationWar and peace in veto voting☆
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 be a set of voters (players) and be a set of candidates (outcomes). Each voter has a preference (a complete order) Pi over all candidates. The set of all preferences 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 ; that is, V has two arguments: a coalition of voters and a block of candidates . The equalities and mean that K can, and respectively cannot, veto B. The complementary function is called the effectivity function; see [6] Section 7.2 and [9] Chapter 6.
Each pair of distributions and , generates a veto function
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 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 , a voter ,
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 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.
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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.