Inducing Cooperation through Weighted Voting and Veto Power

We study the design of voting rules for committees representing heterogeneous groups (countries, states, districts) when cooperation among groups is voluntary. While efficiency recommends weighting groups proportionally to their stakes, we show that accounting for participation constraints entails overweighting some groups, those for which the incentive to cooperate is the lowest. When collective decisions are not enforceable, cooperation induces more stringent constraints that may require granting veto power to certain groups. In the benchmark case where groups differ only in their population size (i.e., the apportionment problem), the model provides a rationale for setting a minimum representation for smaller groups. (JEL C73, D71, D72)

A Mathematical Analysis of an Election System Proposed by Gottlob Frege

In 1998 a long-lost proposal for an election law by Gottlob Frege (1848–1925) was rediscovered in the Thüringer Universitäts- und Landesbibliothek in Jena, Germany. The method that Frege proposed for the election of representatives of a constituency features a remarkable concern for the representation of minorities. Its core idea is that votes cast for unelected candidates are carried over to the next election, while elected candidates incur a cost of winning. We prove that this sensitivity to past elections guarantees a proportional representation of political opinions in the long run. We find that through a slight modification of Frege’s original method even stronger proportionality guarantees can be achieved. This modified version of Frege’s method also provides a novel solution to the apportionment problem, which is distinct from all of the best-known apportionment methods, while still possessing noteworthy proportionality properties.

Approval-Based Apportionment

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we are able to provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity.

A New Quota Approach to Electoral Disproportionality

In this paper electoral disproportionality is split into two types: (1) Forced or unavoidable, due to the very nature of the apportionment problem; and (2) non-forced. While disproportionality indexes proposed in the literature do not distinguish between such components, we design an index, called “quota index”, just measuring avoidable disproportionality. Unlike the previous indexes, the new one can be zero in real situations. Furthermore, this index presents an interesting interpretation concerning transfers of seats. Properties of the quota index and relationships with some usual disproportionality indexes are analyzed. Finally, an empirical approach is undertaken for different countries and elections.

A Comparative Study on Objective Functions of Product Rate Variation and Discrete Apportionment Problems

Setting the proper objective functions to optimize the decision making situations is prevalent in most of the mathematical programming problems. In this paper, we formulate the mathematical models of product rate variation and discrete apportionment problems. Furthermore, a brief comparative study of the objective functions to both the problems is reported in terms of inequality measures, precisely indicating the equitably efficient frontier for production rate variation problem via discrete apportionment. The largest reminder algorithm and rank-index algorithm for the apportionment problem are discussed briefly.Journal of Institute of Science and Technology, 2014, 19(1): 69-74

Reliability Optimization for Multi-State Series-Parallel System Design Using Ant Colony Algorithm

In this paper, an algorithm based ant colony approach (ACA) is designed to find the optimal configuration for redundancy apportionment problem (RAP) of series-parallel multi-state system (MSS). In order to achieve maximum system reliability under cost and performance constraints, the algorithm takes advantage of ACA’s combinatorial optimization, uses heuristic information about reliability to arrange components from available choice. Different from traditional nominal performance and reliability system, universal generating function (UGF) is introduced and used to estimate the multi-state system reliability. Finally the experiment has shown that the presented algorithm can provide optimal and near optimal solutions, and have efficient and convenient calculation performance.