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Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 284/731 4. Decision Making (2) 1. Voting Theorem 4.11 (Arrow (1951)) If the social welfare function f ∗ is (1) pareto efficient and (2) independent from irrelevant alternatives, then there always exists a dictator: for all U ⊆ ∏ |A| Ordpar ∃i ∈ A : ∀o, o ′ ∈ O : o≺ i o ′ To be more precise: for all U ⊆ ∏ |A| i=1 Ord iff o≺ ≺ ∗ o ′ . Ord i=1 Ordpar ∃i ∈ A : ∀〈≺ 1 , . . . , ≺ |A| 〉 ∈ U : ∀o, o ′ ∈ O, o≺ i o ′ iff o f ∗ (〈≺ 1 , . . . , ≺ |A| 〉) o ′ .
Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 285/731 4. Decision Making (2) 1. Voting Proof (of Arrows theorem). The proof is the third proof given by John Geanakoplos (1996) and based on the following Lemma 4.12 (Strict Neutrality) Consider two pairs of alternatives a, b and α, β. Suppose each voter strictly prefers a to b or b to a, i.e. for all i: a≺ ≺ i b or b≺ ≺ i a Suppose further that each voter has the same preference for α, β as she has for a, b. Then either a≺ ≺ ∗ b and α ≺ ∗ β or b≺ ≺ ∗ a and β ≺ ∗ α.
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9. Agents based on FOL 1. First Ord
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9. Agents based on FOL 7. Reference
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