1. Social Choice Rules and their Manipulability
Fuad Aleskerov
National Research University Higher School of Economics
and Institute of Control Sciences of Russian Academy of Sciences
COST Meeting, Istanbul
November 2-4, 2015 1

2. Authors Fuad Aleskerov (NRU HSE and Institute of Control Sciences)
Daniel Karabekyan (NRU HSE)
Remzi Sanver (Bilgi University)
Vyacheslav Yakuba (NRU HSE and Institute of Control Sciences)
Alexander Ivanov (Skoltech and NRU HSE)
Website manip.hse.ru

3. 1. Introduction Manipulability problem
How can we evaluate a degree of manipulability of social choice procedures? Such procedures aggregate individual preferences, defined as an ordering of alternatives, into social choice, which may contain one or several alternatives.
Our approach to the problem.
In construction to build a house engineers check soil mechanic conditions and based on the number of floors of the building just look to the handbook and define how deep should be the base.
Our idea is to construct such a handbook for colleagues working in the voting models. The tradition in this field when we deal with the manipulation phenomenon is to state the manipulability of the rules under different assumptions either to find domain or other type restriction preventing manipulability
Main trends: single or multi-valued choices, individual or coalitional deviation of sincere preferences.

4. Background
Gibbard (1973) and Satterthwaite (1975): every aggregation procedure is manipulable on unrestricted domain (if some rather weak conditions hold).
Duggan, Swartz (2000), Ching, Zhou (2002) and Benoit (2002): the similar result for the case of multi-valued choice.
Chamberlin (1985) and Nitzan (1985): first time studied the degree of manipulability of aggregation procedures.
Kelly (1993) and Aleskerov, Kurbanov (1999) studied the degree of manipulability with the assumption of Impartial Culture (all profiles are equally likely) and alphabetical tie-breaking rule.
Lepelley, Valognes (2003), Favardin, Lepelley (2006) and Pritchard, Wilson (2006): the problem for Impartial Anonymous Culture assumption (all voting situations are equally likely) and alphabetical tie-breaking rule.
Aleskerov et al. (2011, 2012) estimated the degree of manipulability for the case of multi-valued choice (without using any tie-breaking rule) and for Impartial Culture (IC).

5. 2. Aggregation procedures
The research has been conducted on about 30 known aggregation procedures.
The list of rules is as follows:
Plurality rule
Approval rule q=2,3
Inverse Plurality rule
Borda’s rule
Black’s procedure
Minimal Dominant Set
Minimal Undominated Set
Minimal Weekly Stable Set
Fishburn Rule
Uncovered Set I,II
Richelson rule
Copeland rule I,II,III
Simpson’s rule
MinMax rule
Strong q-Paretian simple majority
Strong q-Paretian plurality
Strongest q-Paretian simple majority
Condorcet practical
Threshold rule
Nanson’s rule
Inverse Borda’s rule
Hare’s procedure
1,2,3-stable set
Coombs rule
We will remind definitions for several most important rules

6. Aggregation procedures

7. Aggregation procedures

8. 3. General Notions The Model’s Parameters Overview
Number of alternatives m = 3, 4, 5
Number of agents n = , each agent has a preference (linear order) over the set of alternatives
30 aggregation procedures with multi-valued choice
Extended preferences (4 EP for 3 alternatives; 10 EP for 4 alternatives, 12 EP for 5 alternatives) to compare all possible social choices
Impartial Culture and Impartial Anonymous Culture
Individual and Coalitional manipulability

9. Impartial Culture and Impartial Anonymous Culture
While in Impartial Culture (IC) model it is assumed that all profiles are equally likely, in Impartial Anonymous Culture (IAC) model the voting situations are equally likely, i.e. all profiles obtained by permutation of agents correspond to one voting situation.

10. Multi-valued choice
While the rules are expected to give a single alternative as a choice result, in many cases the choice obtained is of set of several alternatives. Some rules yield single alternatives with high probability, but some do not.
In the example of 5 alternatives and 91 agents, the Black and Borda rules produce single-valued choice in more than 95% of profiles, the Richelson and Copeland rules have only around 75% of such choice. And there are rules (such as Condorcet practical) which frequently return the full set of alternatives.

11. Strong and Weak Manipulability
Strong manipulability: a voter can compare all possible social choices
Strong manipulability extensions: Leximin, Leximax, Risk-averse, Risk-lover (for 3 alternatives)
Weak manipulability: a voter can compare not all possible social choices
Weak manipulability extensions: Kelly, Gardenfors-Zwicker, Expected Utility

12. Extended Preferences for Weak Manipulability
For instance, Kelly extension: Social Choice 1 (SC1) is better than Social Choice 2 (SC2) if and only if all alternatives in SC1 are not worse than all alternatives in SC and there is at least one alternative in SC2 which is worse than one alternative in SC1 a b ab c ac bc
abc 1 -1

13. Extended Preferences for Strong Manipulability
The simple illustration of extended preferences in multiple choice model is the following. For three alternative let for some agent the preferences are a>b>c.
How to compare two possible choice sets {a,c} and {b}?
It depends on the additional assumptions about the agent, and it requires to define the extensions of the single-valued preferences to preferences on the sets. There are several extensions.
Let us explain the Leximin extension. Consider two collective choices X,Y, and the agent's preferences P.
If X and Y consist of equal number of elements, we sort elements of each set in the descending order according to the preferences P, and find the first position form top, on which the elements of the two sets differ. The set with better element in this position dominates the other set.
If X and Y consist of different number of elements, we sort elements of each set in the ascending order, and look from the first elements in the ordering (from the worst element). Then we compare elements on the first position, on which elements differ. The set with better element in that position dominates. In elements of two seta are equal on each position in the ordering, than the wider set dominates.

14. Extended Preferences for Strong Manipulability
Leximin extension for 3 alternatives looks as follows:
The Leximax extension for 3 alternatives yields the following ordering:

15. Manipulability indices: NK
Nitzan-Kelly index calculates the share of manipulable voting situations among all voting situations.
- number of manipulating profiles
- number of all profiles in IC model
- number of all profiles in IAC model

16. Index of efficiency of manipulation, I2
- average benefit of manipulation in terms of ranks
Two presented formulae show the indices normalized for the IC and IAC models

17. Index of efficiency of manipulation, I3
- maximum benefit of manipulation in terms of places
Two formulae show the indices normalized for the IC and IAC models

18. Computational scheme
The following computer modelling scheme to calculate manipulability indices is used:
1. Generate 1,000,000 profiles, separately for IAC and IC
2. For each profile we determine whether it is manipulable or not. A profile is considered manipulable if there is at least one agent who can manipulate,
3. If the profile is manipulable, we increase d0 index (number of manipulable profiles) by 1,
4. Calculate NK index by dividing d0 by the total number of generated profiles, i.e. 1,000,000.
5. Calculate I2 and I3 indices simultaneously with NK

19. Coalitional manipulability model
The coalitional manipulation model assumes that
the coalitions should contain agents with the same preferences, and
insincere preferences submitted by agents of the coalition in the process of manipulation should be the same.
The method for performing coalitional calculations is as follows. Since coalitions contain agents with the same preferences, it is enough to try to manipulate by one of the coalitions of the particular size, and then count the number of such coalitions, which can be constructed form the agents with the same preferences in a profile.

20. Results: manipulability
The complete set of the results contains thousands of tables and obviously cannot be reprinted in any text.
The complete set of the results can be found on the website
manip.hse.ru

21. Results: manipulability
All NK indices for IAC are decreasing with the growing number of agents. This is the same as in Impartial Culture. The explanation is pretty simple: as soon as we consider individual manipulability, the ‘weight’ of one’s preferences and its impact are decreasing with the growing number of agents.
For most of aggregation procedures we can notice ‘periodical behavior’ of either 2 or 3 agents in their values. The same is true for NK index in Impartial Culture.
Periods for aggregation procedures by the numbers of agents exist and seem to be the same as in the case of Leximin preferences extension method.
2-Approval rule is again the worst in terms of manipulability. Its values start even from 0.64, which stands for more than half manipulable profiles.
There is no rule which is the least manipulable. Hare’s procedure shows best results at the beginning (small number of agents), but then it is surpassed by Nanson’s rule with the growing number of agents.

22. Results: manipulability
There is no least manipulable procedure, but the following aggregation procedures show minimal manipulability in substantial number of cases:
Hare’s procedure
Nanson’s procedure
Minimal Dominant Set
Minimal Undominated Set
The Third Copeland’s rule
Fishburn’s rule

23. Individual manipulability, NK

24. Individual manipulability, leximin, m=3
Kelly’s index
Individual manipulability, leximin, m=3

25. Individual manipulability, leximin, m=4

26. Individual manipulability, leximax, m=5
Kelly’s index
Individual manipulability, leximax, m=5

27. Individual manipulability, leximax, m=3

28. Individual manipulability, leximin, m=4

29. Individual manipulability, leximin, m=5

30. Individual manipulability, leximax, m=4

31. Individual manipulability, leximax, m=5

32. Individual Manipulability, NK, IC

33. Individual Manipulability, NK, IC

34. Individual Manipulability, NK, IC

35. Individual Manipulability, NK, IC

36. Individual Manipulability, NK, IC

37. Individual Manipulability, NK, IAC

38. Individual Manipulability, NK, IAC

39. Individual Manipulability, NK, IAC

40. Individual Manipulability, NK, IAC

41. Individual Manipulability, NK, IAC

42. Individual manipulability, I - indices

43. Individual,leximin, m=3, n=3

44. Individual,leximin, m=3, n=5

46. Individual I1+, leximin, m=3

47. Individual I1=, leximin, m=3

48. Individual I1-, leximin, m=3

50. Coalitional manipulability

51. Coalitional manipulability, weak, IC
NK index for m=5, calculated for IC model.
Weak manipulability: Kelly extension.

52. Coalitional manipulability, strong, IC
NK index for rules for m=5, calculated for IC model.
Leximax extension.

53. Coalitional manipulability, weak, I1, IC
I1+ index for m=5, calculated for IC model.
Weak manipulability: Kelly extension.

54. Coalitional manipulability, strong, I1, IC
I1+ index for rules for m=5, calculated for IC model.
Leximax extension.

55. Coalitional manipulability, I1 index, IC

56. Coalitional manipulability, strong, I2, IC
Average effectiveness of manipulability (I2) index for rules for m=5, calculated for IC model. Leximax extension.

57. Coalitional manipulability, strong, I3, IC
Maximum effectiveness of manipulability (I3) index for rules for m=5, calculated for IC model. Leximax extension.

58. Coalitional manipulability, weak, IAC
NK index for rules for m=5, calculated for IAC model.
Weak manipulability: Kelly extension.

59. Coalitional manipulability, weak, I1, IAC
I1+ index for rules for m=5, calculated for IAC model.
Weak manipulability: Kelly extension.

60. Coalitional manipulability, strong, IAC
NK index for rules for m=5, calculated for IAC model.
Leximax extension.

61. Coalitional manipulability, strong, I1 IAC
I1+ index for rules for m=5, calculated for IAC model.
Leximax extension.

62. Coalitional manipulability, I1 index, IAC,

63. Coalitional manipulability, strong, I2 IAC
Average effectiveness of manipulability (I2) index for rules for m=5, calculated for IAC model. Leximax extension.

64. Coalitional manipulability, strong, I3, IAC
Maximum effectiveness of manipulability (I3) index for rules for m=5, calculated for IAC model. Leximax extension.

65. 8. Manipulability and Decisiveness
Some social choice rules show low manipulability, because they yield the whole set of alternatives for most profiles. It makes them not only almost non-manipulable under multi-valued choice, but also useless, because they do not actually make any choice at all. Decisiveness is the criterion which can be combined with manipulability indices to exclude such procedures.

66. Example Frequencies of choice sets m=5, n=25
One can see Condorcet practical with high frequency for "abc" choice set and Strong q-paretian simple majority with second high value.
abc
Condorcet practical
0.93
Strong q-Paretian simple majority
0.19

67. D- index
We construct the D-index for the case of m=3. Consider the frequencies of all choice sets. Let M1 be the sum of frequencies of all choices, which consist of only one alternative. Let M2 be the sum of frequencies of 2-alternative choices. And let M3 be the sum of frequencies of 3-alternative choices
D-index is defined as the following sum: D= 1*M1+2*M2+3*M3,
i.e. It is the sum of the number of alternatives in choice set, multiplied by cumulative frequencies of these sets.

68. Values of D-index
There are 3 types of rules. Low values of D index: Plurality, Approval q=2, Inverse Plurality, Borda, Black, Simpson, MinMax, Strong q-Paretian plurality, Threshold, Nanson, Inverse Borda, Hare, Coombs, Approval q=3 for 4 alts.
Average values: Minimal dominant set, Minimal undominated set, Minimal weekly stable set, Fishburn, Uncovered set I, Uncovered set II, Richelson, Copeland I, II, II, Strongest q-Paretian simple majority, 1,2, 3-stable set.
Condorcet practical and Strong q-paretian simple majority rules have high values of D index

69. Nitzan-Kelly – D-index space

70. Decisiveness, m=4, n=100, IC

71. 9. Conclusion Many aggregation procedures
3, 4, 5 alternatives, agents (not all agents)
Multi-valued choice, preference extensions: strong and weak
Impartial Culture and Impartial Anonymous Culture
Coalitional manipulability
Manipulability indices: Nitzan-Kelly and effectiveness I2, I3
Decisiveness

72. Thank you!
manip.hse.ru

73. Appendix 1. Definitions of aggregation rules

74. Appendix 1. Definitions of aggregation rules

75. Appendix 1. Definitions of aggregation rules

76. Appendix 1. Definitions of aggregation rules

77. Appendix 1. Definitions of aggregation rules

78. Appendix 1. Definitions of aggregation rules

79. Appendix 1. Definitions of aggregation rules

80. Appendix 1. Definitions of aggregation rules

81. Appendix 1. Definitions of aggregation rules

82. Extended preferences
Axiomatic definition of Leximin

83. Extended preferences