FAIRNESS AND EFFICIENCY IN VOTING SYSTEMS

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FAIRNESS AND EFFICIENCY IN VOTING SYSTEMS GACR 402/09/1066 Research Seminar 01-09, February 16, 2009 FAIRNESS AND EFFICIENCY IN VOTING SYSTEMS Author: F. Turnovec UK FSV, IES, Prague February 16, 2009

Abstract Fair representation of voters in a committee representing different voters’ groups is being broadly discussed during last few years. Assuming we know what the fair representation is, there exists a problem of optimal quota: given a “fair” distribution of voting weights, how to set up voting rule (quota) in such a way that distribution of a priori voting power is as close as possible to distribution of voting weights. Together with optimal quota problem a problem of trade-off between fairness and efficiency (ability of a voting body to make decisions) is formalized by a fairness-efficiency matrix

Contents Fairness Efficiency Committee Voting power Quota interval of stable power Problem of optimal quota Fairness-efficiency matrix

Related literature Penrose, 1946, indirect voting power Felsenthal and Machover, 1998, 2007, mathematical properties of theory of power indices Laruelle and Widgrén, 1998, proof of referendum type voting power Leech and Aziz, 2008, empirical evaluation of indirect voting power in EU Słomczyński W. and K. Życzkowski 2006, optimal quota for square root Słomczyński W. and K. Życzkowski 2007, Jagiellonian compromise Turnovec, Mercik, Mazurkiewicz 2008, decisiveness, pivots and swings Baldwin and Widgrén, 2004, comparative analysis of different voting rules Berg and Holler, 1986, model of strictly proportional power

Fairness n units (e.g. regions, political parties) with different size of population (voters), represented in a super-unit committee that decides different agendas relevant for the whole entity each unit representation in the committee has some voting weight (number of votes) by voting system we mean an allocation of voting weights in elections and committees, the form of the ballot and rules for counting the votes to determine outcome of voting What allocation of weights is „fair“?

Fairness: voting weights = voting power voting weight is not the same thing as voting power understood as an ability to influence outcome of voting voting power indices are used to evaluate a probability that a particular voter is “decisive in voting” in the sense that if her vote is YES, then the outcome of voting in committee is YES, and if she votes NO, then the outcome is NO assuming, that a principle of fairness is selected for a distribution of voting weights, we are addressing the question how to achieve equality of voting power (at least approximately) to fair voting weights; fairness = proportionality of voting power to voting weights

Efficiency efficiency is an ability of the system to reach decisions, probability that a proposal will be passed how to achieve equality of voting power (at least approximately) to fair voting weights with a “reasonable” level of efficiency

Model of a committee

Voting power

Swings and pivots two basic concepts of decisiveness are used: swing position - ability of individual voter to change by unilateral switch from YES to NO outcome of voting, pivotal position - such position of individual voter in a permutation of voters expressing ranking of attitudes of members to voted issue (from most preferable to least preferable) and corresponding order of forming of winning configuration, in which her vote YES means YES outcome of voting and her vote NO means NO outcome of voting Penrose-Banzhaf power index = probability of swing Shapley-Shubik power index = probability of pivot

Penrose-Banzhaf power

Shapley-Shubik power index

Strictly proportional power

Index of efficiency

Fairness and quota Fair voting system is usually formulated in terms of voting weights. Let w be a fair distribution of voting weights (whatever principle is used to justify it), then the voting system used is fair if the committee [N, q, w] has the property of strictly proportional power For given N and w the only variable we can vary to design fair voting system is quota q.

Quota interval of stable power

Quota interval of stable power

Quota interval of stable power

Number of intervals of stable power is finite

Number of intervals of stable power is finite, example

Number of intervals of stable power is finite, example

Mechanism of strictly proportional power

Mechanism of strictly proportional power

Mechanism of strictly proportional power - example

Mechanism of strictly proportional power - example

Mechanism of strictly proportional power - example

Deviation from proportionality

Deviation from proportionality

Index of fairness

Optimal quota

Optimal quota

Optimal quota

Optimal quota – example

Efficiency index

Fairness-efficiency matrix

Fairness-efficiency matrix

Concluding remarks Słomczyński and Życzkowski introduced optimal quota concept within the framework of so called Penrose voting system as a principle of fairness in the EU Council of Ministers voting and related it exclusively to Penrose-Banzhaf power index In this paper it is treated in a more general setting as a property of any simple weighted committee and any well defined power measure Problem of optimal quota has an exact solution wia marginal quotas and intervals of stable power Proble „fairness versus efficiency“ can be represented by fairness-efficiency matrix and treated by methods of multi-critera decision making