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Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 278/731 4. Decision Making (2) 1. Voting We are now relaxing our assumptions considerably (and thereby strengthening Arrow’s theorem). Voting based on partial orders The voting of agent i is described by a binary relation ≺ i ⊆ O × O, which we assume to be irreflexive, transitive and antisymmetric: Ties are now allowed! We denote by Ord par the set of all such binary relations.
Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 279/731 4. Decision Making (2) 1. Voting Definition 4.9 (Social welfare function) A social welfare function is any function ≺ ∗ f ∗ : U → Ord par ; (≺ 1 , . . . , ≺ |A| ) ↦→ ≺ ∗ For each V ⊆ 2 O \ {∅} the function f ∗ w.r.t. U induces a choice function C 〈≺1 ,...,≺ |A| 〉 as follows: { V −→ V C 〈≺1 ,...,≺ |A| 〉 = def v ↦→ C 〈≺1 ,...,≺ |A| 〉(v) = max ≺ ∗ max ≺ ∗ | V to ≺ ∗ | V . ≺ ∗ | V v is the set of all maximal elements in v according Each tuple u = (≺ 1 , . . . , ≺ |A| ) determines the election for all possible v ∈ V . v
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9. Agents based on FOL 7. Reference
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