Simple voting games (SVGs) are mathematical idealisations of decision-making by a council or board for example the EU Council of Ministers or the UN Security Council. -- The theory of SVGs includes structure-preserving mappings. Until now, these have not been organised in a category. -- We start in the most natural way, with the objects of the category being SVGs conceived as sets of sets of voters and arrows being isomorphisms, bloc formation and inclusion maps. -- An alternative category, or sequence of categories, of SVGs, is simple to define. Co is the category with two objects and a single non-identity arrow between them. Cn+i is the arrows category of Cn. Encouragingly, the category-theoretic duals, products and coproducts correspond to duals, meets (products) and joins (coproducts) in SVGs. This category is a partial order. We develop a new notation for SVGs and have almost immediate proof of substantial results, for example the construction of the constant-sum extension of a game and the fact that the Banzaf-Penrose measure for a bloc of two voters is equal to the sum of the Banzaf-Penrose measures for the two voters. -- The Cn are also lattices. We can find the bipartitions as a sublat-tice which is also Boolean algebra. -- Lattice homomorphisms between the Cn correspond to structure-preserving mappings of SVGs. We can build another category with these. -- We also have a bijective mapping between SVGs and ordered pairs of a simplicial complex and its Alexander dual. This connects the theory of SVGs with topology.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:628065 |
| Date | January 2012 |
| Creators | Terrington, Simon |
| Publisher | King's College London (University of London) |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://kclpure.kcl.ac.uk/portal/en/theses/developing-a-representation-of-simple-voting-games-within-category-theory(221e7401-8dae-4f6c-9b36-4f4e23835aeb).html |
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