Game-theoretic analysis of voting in committees

In this chapter we adopt the axiomatic approach in order to find (new) voting procedures to committees that are immune against deviations by coalitions of voters. We shall now describe our approach. Let G be a committee and let A be a finite set of m alternatives, m [greater-or-equal, slanted] 2. Our problem is to find a social choice function F that will enable the members of G to choose one alternative out of A. We insist that F will have the following properties.(i) F should be Paretian, monotonic, and preserve the symmetries of G;(ii) the power structure induced by F should coincide with G;(iii) for each profile RN of (true) preferences of N (i.e., the set of members of G), F(RN) should be the outcome of a strong Nash equilibrium (in the strategic game specified by F and RN).Let, again, G be a committee and let A be a set of m alternatives, m [greater-or-equal, slanted] 2. The pair (G, A) is called a choice problem. A social choice function that satisfies the foregoing three criteria (i)-(iii), is called a strong representation of (G, A). If G is weak, that is, G has a vetoer, then (G, A) has a strong representation for every value of m. If G does not contain a vetoer, then there exists a natural number [mu](G) [greater-or-equal, slanted] 2 (the capacity of G), such that (G, A) has a strong representation if and only if 2 [less-than-or-equals, slant] m [less-than-or-equals, slant] [mu](G). A family of algorithms, called feasible elimination procedures, produces a strong representation to any choice problem (G, A) whenever such a representation exists. Feasible eliminations procedures produce all the strong representations of symmetric committees.