1. 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

2. 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

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

4. 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

5. 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“?

6. 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

7. 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

8. Model of a committee

9. Voting power

10. 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

11. Penrose-Banzhaf power

12. Shapley-Shubik power index

13. Strictly proportional power

14. Index of efficiency

15. 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.

16. Quota interval of stable power

17. Quota interval of stable power

18. Quota interval of stable power

19. Number of intervals of stable power is finite

20. Number of intervals of stable power is finite, example

21. Number of intervals of stable power is finite, example

22. Mechanism of strictly proportional power

23. Mechanism of strictly proportional power

24. Mechanism of strictly proportional power - example

25. Mechanism of strictly proportional power - example

26. Mechanism of strictly proportional power - example

27. Deviation from proportionality

28. Deviation from proportionality

29. Index of fairness

30. Optimal quota

31. Optimal quota

32. Optimal quota

33. Optimal quota – example

34. Efficiency index

35. Fairness-efficiency matrix

36. Fairness-efficiency matrix

37. 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