1. 1 EFFICIENCY OF FAIRNESS IN VOTING SYSTEMS EPCS 2009 Athens, April 2-5, 2009 Frantisek Turnovec Charles University in Prague Institute of Economic Studies

2. 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. 3 Contents Fairness Efficiency Committee Voting power Quota intervals of stable power and marginal quotas Problem of optimal quota Fairness-efficiency matrix

4. 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. 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. 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. 7 Efficiency based on Coleman (1971) definition of „ability of a collectivity to act“ efficiency is an ability of the system to pass a proposal (to change status quo) how to achieve equality of voting power (at least approximately) to fair voting weights with a “reasonable” level of efficiency

8. 8 Model of a committee, basic concepts

9. 9 Voting power

10. 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. 11 Penrose-Banzhaf power

12. 12 Shapley-Shubik power index

13. 13 Strictly proportional power

14. 14 Index of efficiency

15. 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. 16 The same sets of winning configurations = the same voting power

17. 17 Blocking power = voting power

18. 18 Quota interval of stable power

19. 19 Quota interval of stable power

20. 20 Quota interval of stable power: example

21. 21 Weight increasing ordering

22. 22 Number of intervals of stable power is finite

23. 23 Number of intervals of stable power is finite, example

24. 24 Number of intervals of stable power is finite, example

25. 25 Mechanism of strictly proportional power

26. 26 Mechanism of strictly proportional power

27. 27 Mechanism of strictly proportional power - example

28. 28 Mechanism of strictly proportional power - example

29. 29 Mechanism of strictly proportional power - example

30. 30 Deviation from proportionality

31. 31 Deviation from proportionality

32. 32 Index of fairness

33. 33 Optimal quota

34. 34 Optimal quota

35. 35 Optimal quota

36. 36 Optimal quota – example

37. 37 Efficiency index

38. 38 Fairness-efficiency matrix

39. 39 Fairness-efficiency matrix

40. 40 Concluding remarks In the simple weighted committee with fixed number of members and voting weights there exists a finite number r of different quota intervals of stable power (r  2 n -1) generating finite number of power indices vectors Voting power is equal to blocking power what implies that number of different power indices vectors corresponding to majority quotas is equal at most to int(r/2) + 1 If the fair distribution of voting weights is defined, then fair distribution of voting power means to find a quota that minimizes distance between relative voting weights and relative voting power. Index of fairness is not a monotonic function of quota. Problem of optimal quota has an exact solution via finite number of majority marginal quotas

41. 41 Concluding remarks Index of efficiency defined as a probability to change status quo has also finite number of values corresponding to marginal quotas and is monotonic (decreasing) function of marginal quotas Problem „fairness versus efficiency“ can be represented by fairness-efficiency matrix and treated by methods of multi-objective decision making 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 and square root rule 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