# Lehrstuhl für Wirtschaftsmathematik

Mathematisches Institut, Fakultät für Mathematik, Physik und Informatik
Lehrstuhlinhaber: Prof. Dr. Jörg Rambau

# 23.06.2010, 18.15, H23 (RW)

Vortrag im Rahmen des neu gegründeten Forschungszentrums "Modellierung und Simulation sozioökonomischer Phänomene" (MODUS) am 23.06.2010 um 18:15 im H23 (RW II) von Prof. Dr. Josep Freixas zum Thema "Anonymous voting rules" (Ankündigung)

# Abstrakt

In this work we consider symmetric or anonymous (j,2) simple games, in which each voter chooses from among j ordered levels of approval and the outcome is â€˜yesâ€™ or â€˜no.â€™ Symmetric (j,2) simple games model some natural decision rules, such as pass-fail grading systems. The most conspicuous case arises for j = 3 which serves to model anonymous voting systems in which each voter may vote â€˜yes,â€™ abstain, or vote â€˜noâ€™.

Each symmetric (j,2) simple game is determined by the set of anonymous minimal winning profiles. This makes it possible to count the possible systems for small values of n and j, and the counts suggest some interesting patterns. The first concerns the number of anonymous voting rules with 3 levels of approval. The second exhibit a surprising symmetry certain for anonymous simple games.

In contrast to the situation for ordinary simple games, (2,2) simple games in our model, these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry. We consider several real-world examples, suggesting some attractive alternatives supplied by the general theory.

# 24.06.2010, 16.30, H19 (NW2)

Vortrag im Rahmen des mathematischen Kolloquiums am 24.06.2010 um 16:30 im H19 (NW II) von Prof. Dr. Josep Freixas zum Thema "Ordinal equivalence of power indices" (Ankündigung)

# Abstrakt

Different reasonable power indices provide different orderings of importance for voters, so that the evaluation of (a priori) power in (binary) voting systems is quite arbitrary since the rankings highly depend on the particular power index chosen. One is left with the hope that such discrepancies occur because the voting system at hand is rare enough. To better understand what happens, we analyze the ordinal equivalence of families of power indices.

Power indices based on symmetric probabilistic values (or semivalues) with positive coefficients (regular semivalues), as the Banzhaf index or the Shapley-Shubik index, share the same rankings of voters within the class of weakly complete games. Weakly complete games contain weighted and complete games and, therefore, the most common real voting systems. This partially solves the problem because power indices are mostly applied to non-complicated voting systems derived from real problems which always are weakly complete. However, the analytical problem of studying the ordinal equivalence, even for regular semivalues, is still not solved outside weakly complete games. We give an extension which implies the ordinal equivalence of the Banzhaf and Shapley-Shubik indices.

The Johnston index also shows a good behavior since it is ordinally equivalent to Banzhaf and Shapley-Shubik indices in a sufficiently large class of games containing complete games. Necessary and sufficient conditions are given to determine such a class. Oppositely, families of power indices based only on minimal winning coalitions, as Holler or Deegan-Packel indices, show an odd ranking behavior.

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