An empirical investigation of decision-making satisfaction in web-based decision support systems

Voting games with abstention: Linking completeness and weightedness

“…For S, T ∈ 3 N , we write S ⊆ 3 T to mean that either S = T or S may be transformed into T by shifting 1 or more voters to higher levels of approval. This is the same as saying Standard notions for coalitions in simple games naturally extend for tripartitions in (3,2) games: S is a losing tripartition whenever V (S) = 0, let L denote the set of losing tripartitions; S is a minimal winning tripartition provided that S is winning and for all T ∈ 3 N such that T ⊂ 3 S, T is losing, let W m denote the set of minimal winning tripartitions; S is a maximal losing tripartition provided that S is losing and for all T ∈ 3 N such that S ⊂ 3 T, T is winning, let L M denote the set of maximal losing tripartitions. It is clear that W and L form a bipartition of 3 N , and that each of the sets: W , L, W m , and L M uniquely determine the (3, 2) game.…”

“…Definition 1.3 A strongly weighted (3, 2) game is a weighted (3,2) game that admits a representation such that for every pair of voters p and r, either…”

“…The target system describes each situation in which partitions of voters are able to pass a new law or change the status quo. Making decisions in democratic organizations is regarded as a DSS and an investigation of the DSS literature reveals that research has mainly focused on the effects of design, implementation and use on decision outcomes (see e.g., [3,11]). …”

“…When absent voters are taken into account with a quorum (like in [4] or in [25]) the levels of input approval are not ordered and therefore, the results in this paper do not extend to that context. Some (3,2) voting systems are weighted (3, 2) systems which admit a representation by means of vector weights and a threshold for the system, and therefore their representations as weighted systems are useful to separate the two possible collective outcomes. A purpose of this paper is to link the completeness of some desirability relations, that determine the importance of voters in the system, with weighted systems with several ordered levels of approval for the input.…”

“…In Section 2 we introduce several notions of desirability for (3, 2) games and consider their completeness, their restrictions and extensions to simple games and the hierarchies they induce. In Section 3 we consistently link weighted (3,2) games with an appropriate class of complete (3, 2) games, and strongly weighted (3, 2) games with complete (3, 2) games with respect to the influence relation. In Section 4 different (3, 2) swap robustness properties, which restriction for the case of simple games constitutes a characterization of complete games, are established for the derived notions of completeness for (3,2) games.…”