Coalitional Manipulability of the Positional Social Choice Rules
Alexander Ivanov (HSE, ICS RAS),
Daniel Karabekyan (HSE),
Vyacheslav Yakuba (ICS RAS)
Abstract
We study a problem of coalitional manipulability in impartial culture for 8 positional
voting rules. Coalitional manipulability indices are estimated for 3 alternatives and the number
of agents between 3 and 8 inclusively.
1. Introduction
Gibbard (1973) and Satterthwaite (1975) showed that for at least three alternatives and
single-valued choice every non-dictatorial voting rule is individually manipulable. Later Duggan
and Schwartz (2000) had generalized this result for the case of multiple choice (when there can
be more than one alternative as the result of voting). Then interesting question arises: if we know
that every social choice rule is manipulable, how can we find the least manipulable one? This
question initiated a lot of papers studying to which extent known social choice rules are
manipulable. The non-exahustive list of papers is Chamberlin (1985), Nitzan (1985), Kelly
(1993), Aleskerov, Kurbanov (1999), Smith (1999), Favardin, Lepelley (2006), Pritchard,
Wilson (2007), Aleskerov et al. (2010a,b), Aleskerov et al. (2011).
All those papers consider individual manipulability, i.e. when only one agent may deviate
her preferences in order to obtain a better social choice. In this work we estimate coalitional
manipulability i.e. a situation when several agents deviate their preferences to obtain a better
social choice. We use the most popular and most native measure of manipulability: the share of
all manipulable profiles. This measure is used almost in all papers in this field.
The next important difference between papers is the way to deal with multiple choice.
Almost for all rules there are some profiles where these rules give tie as the result of voting.
Most of the papers use alphabetical tie-breaking rules: in the case of tie first alternative in
alphabetical order is chosen (for example, see Nitzan (1985), Aleskerov, Kurbanov (1999),
Favardin, Lepelley (2006)). The problem with this method is that it breaks symmetry between
the alternatives because first alternatives in alphabetical order have more chances to be selected
as final outcome. Pritchard and Wilson (2007) use random tie-breaking rule where in the case of
tie the final outcome is choosing randomly. In this case one can compare some sets of
alternatives using stochastic order. Voting rules in a more general framework of multiple choice
was studied in Aleskerov et al. (2010a,b). In our research we use the same model. Here we
estimate the degree of manipulability of nine voting rules in the case of multiple choice.
The structure of the paper is as follows. Section 2 introduces the basic notation and
concepts. Section 3 presents the indices to measure the degree of manipulability of social choice
rules and explains the computational scheme. Section 4 presents the social choice rules under
study. Section 5 states and discusses the results.
2. Framework
Here we use the same notations as in Aleskerov, Kurbanov (1999) and almost the same
model as in Aleskerov et al. (2010b). We consider a finite set A consisting of m alternatives,
m  3,4 . Let A = 2 A \  denote the set of all non-empty subsets of A . Each agent from a finite
set N = 1,..., n, n > 1 , is assumed to have a preference Pi  L over alternatives where L is the
set of linear orders on A .

An ordered n -tuple of preferences Pi is called a (preference) profile, P . A group

decision is made by a social choice rule based on P and is considered to be an element of A .
Thus we define a social choice rule as a mapping C : Ln  A .
Every agent i is assumed to have an extended preference EPi over A which is induced
by her preference Pi over A .
There is a lot of literature on axioms of preference extension, one can find them, for
example, in Barbera (1977), Gärdenfors (1976), Kelly (1977). The detailed overview can be
found in Barbera et al. (2004). In this paper we use the concept of strong manipulation with the
concept of leximin extension.
The concept of a leximin extension is similarly defined while it is based on the ordering
of two sets according to a lexicographic comparison of their worst elements. Again the elements
according to which the sets are compared will disagree at some step – except possibly when one
set is a subset of the other, in which case the larger set is preferred. 1 So, given any Pi  L and
any
distinct
X ,Y  A
where

X = x1 ,, x X

and

Y = y1 ,, y Y

are
such
that
j  1,, X  1 x j 1 Pi x j and j  1,, Y  1 y j 1 Pi y j , the leximin extended preference EPi
is defined as follows
1. If X = Y , then X EPi Y iff xh Pi yh for the greatest h  1,, k for which
xh  yh .
2. If X  Y and h  1,, min  X , Y  for which xh  yh , then X EPi Y iff
xh Pi yh for the smallest h  1,, min  X , Y  for which xh  yh .
3. If X  Y and xh = yh h  1,, min  X , Y  then X EPi Y iff X > Y .
In Leximin method if the preferences of
an agent are aPibPi c , then the extended
preferences will be aEPi a, bEPi bEPi a, cEPi a, b, cEPi b, cEPi c
1
For example, the leximin extension of the ordering a b c is {a}, {a, b}, {b}, {a, c}, {a, b, c}, {b, c}, {c}.
3. Manipulability Index
Number of alternatives being m , the total number of possible linear orders is equal to m!
, and total number of profiles with n agents is equal to (m!) n . Nitzan (1985) introduces the
following index, which was also used by Kelly (1993). We call this index as Nitzan-Kelly's
index and denote as NK , to measure the degree of manipulability of social choice rules:
NK =
d0
,
(m!) n
For coalitional manipulability this formula should be changed to
𝑁𝐾𝑐𝑜𝑎𝑙𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝑘=𝑙 =
𝑑0
(𝑚!)𝑛
where k is the maximum number of agents in one coalition and 𝑑0 is the number of profiles
where coalition of 𝑙 or less agents may manipulate.
4. Voting rules
The nine social choice rules we consider are defined as follows
1. Plurality Rule: Choose alternatives that have been admitted to be the best by the
maximum number of agents, i.e.



a  C ( P)  [x  A n  (a, P)  n  ( x, P)],

where n (a, P) = card{i  N | y  A aPi y}
2. q-Approval: Let us define

n  (a, P, q) = card{i  N | card{Di (a)}  q  1},

where Di (a)  y  A : yPi a is the upper contour set of a  A at Pi  L . That is n  ( a, P, q ) is
the number of agents for which a is ranked among the first q alternatives in their preference
ordering. The integer q can be called as the degree of procedure. We define q-Approval as
follows:



a  C ( P)  [x  A n  (a, P, q)  n  ( x, P, q)] ,
i.e., the alternatives which are admitted to be among the q best by the highest number of agents
are chosen. It can be easily seen that Plurality Rule is a particular case of q-Approval where
q = 1.

3. Borda's Rule: Let ri ( x, P) be the cardinality of the lower contour set of x  A in



Pi  P , i.e. ri ( x, P) = Li ( x) = b  A : xPi b . The sum of ri ( x, P) over all i  N is called the
Borda score of alternative x,
n

r (a, P) = ri (a, Pi ).
i =1
The alternatives with maximum Borda score are chosen., i.e.



a  C ( P)  [b  A, r (a, P)  r (b, P)].

4. Black's Procedure: Let us define the majority relation  for a given profile P :
xy  card{i  N | xPi y}  card{i  N | yPi x} .


Condorcet winner CW (P) in the profile P is an element undominated in the majority
relation  (constructed according to the profile), i.e.

CW ( P)  a | x  A, xa 
Pick the unique Condorcet winner if it exists and the Borda winner(s) otherwise.
5. Threshold rule (Aleskerov et al. 2010): Let v1 ( x) be the number of agents for which
the alternative x is the worst in their ordering, v2 ( x) – is the number of agents placing x the
second worst, and so on, vm (x) – the number of agents considering the alternative x the best.
Then we order the alternatives lexicographically. The alternative x is said to V -dominate the
alternative y if v1 ( x) < v1 ( y) or, if there exists k not more than m , s.t. vi ( x) = vi ( y) ,
i = 1,..., k  1 , and vk ( x) < vk ( y) . In other words, first, the number of worst places are compared,
if these numbers are equal then the number of second worst places are compared and so on. The
alternatives which are not dominated by other alternatives via V are chosen.
6. Hare's Procedure. First, if an alternative is chosen by simple majority of voters, then
this alternative is chosen, and the procedure stops. Otherwise, the alternative a with the
minimum number of votes is omitted. Then the procedure again applied to the set X = A \ {a}

and the profile P/X .
7. Antiplurality Rule. The alternative, which is regarded as the worst by the minimum
number of agents, is chosen, i.e.,



a  C ( P)  [x  An  (a, P)  n  ( x, P)],

where n  (a, P) = card {i  N | y  A yPi a} .
8. Nanson's Procedure. For each alternative Borda's count is calculated. Then average
 


count is calculated, r =  r (a, P)  / A , and alternatives c  A are omitted for which r (c, P ) < r
 aA


. Then the set X = {a  A : r (a, P)  r} is considered, and the procedure is applied to the profile

P/X . Such procedure is repeated until choice will not be empty.
5. Computation Scheme
To calculate NKcoalitional the brute force algorithm was used in order to get exact results. For
each combination of the number of agents and the number of alternatives we generated all
possible profiles. For every profile all possible preferences’ deviations were considered and
NKcoalitional index was calculated according to its definition.
6. Results
We are going to compare the results for coalitional manipulability (CM) indices with the
results for single-valued choice individual manipulability (SVIM) indices (Aleskerov, Kurbanov
(1999)) and with the results for multiple choice individual manipulability (MVIM) indices
(Aleskerov, Karabekyan, Sanver, Yakuba (2011)). The comparisons will be made for situations
with 3 alternatives and the number of agents between 3 and 8.
We take NKcoalitional with k = n as the index of coalitional manipulability for the
comparisons. It means that the index includes all the manipulations made by each possible
coalition which may contain from 1 agent to n-1 agent.
There are no results in Aleskerov, Kurbanov (1999) for Approval k=2 Rule and for
Threshold Rule, that is why for these two rules we compare only the results for the coalitional
manipulability and multiple choice manipulability.
Voting Rules
Plurality Rule
Approval q=2
Borda's Rule
Black's Procedure
Threshold Rule
Hare's Procedure
Antiplurality Rule
Nanson's
Procedure
Voting Rules
Plurality Rule
Approval q=2
Borda's Rule
3 agents
CM
SVIM
0,222 0,167
0,111
0,306 0,236
0,056 0,111
0,306
0,222 0,111
0,111 0,264
4 agents
MVIM CM
SVIM
0,222 0,333 0,185
0,111 0,296
0,306 0,361 0,310
0,056 0,292 0,144
0,306 0,403
0,222 0,333 0,093
0,111 0,296 0,276
0,056 0,111 0,056
6 agents
CM
SVIM
0,432 0,239
0,175
0,425 0,278
5 agents
MVIM CM
SVIM
0,333 0,370 0,232
0,296 0,375
0,361 0,417 0,286
0,236 0,139 0,170
0,403 0,282
0,333 0,093 0,093
0,296 0,375 0,286
0,292 0,144 0,236
7 agents
MVIM CM
SVIM
0,370 0,462 0,257
0,175 0,395
0,374 0,455 0,271
MVIM
0,370
0,375
0,370
0,116
0,258
0,093
0,375
0,093 0,139 0,069
8 agents
MVIM CM
SVIM
0,462 0,484 0,274
0,348 0,397
0,373 0,458 0,260
MVIM
0,384
0,374
0,368
Black's Procedure
Threshold Rule
Hare's Procedure
Antiplurality Rule
Nanson's
Procedure
0,372 0,150 0,250
0,426
0,349
0,201 0,097 0,170
0,175 0,288 0,175
0,204 0,183 0,138
0,475
0,385
0,150 0,112 0,126
0,395 0,288 0,348
0,405 0,151 0,249
0,342
0,274
0,202 0,122 0,178
0,397 0,284 0,374
0,372 0,145 0,252
0,146 0,137 0,072
0,399 0,141 0,243
Fig. 1 NK index
For Plurality Rule both coalitional manipulation results and multiple choice individual
manipulation results show higher degrees of manipulability than single-valued choice results. On
the other hand, multiple choice results and coalitional manipulation results have the same values
of NK index in situations with 3, 4, 5 and 7 agents. This regularity may take place because of the
similarities in definitions of these two indices.
For Approval Rule with q=2 coalitional manipulability results and multiple choice
individual manipulability results show the same degrees of manipulability for the number agents
from 3 to 6. After that, for 7 and 8 agents the NK index for coalitional manipulability is higher
than the other one.
In the graph with the results for Borda’s Rule an obvious separation of the values of the
three indices can be seen, because all the lines do not intersect each other. It means that
coalitional manipulability steadily demonstrates higher degrees of manipulability than both
single-valued choice and multiple choice individual manipulability. Additionally, multiple
choice manipulability reaches higher degrees of manipulability than single-valued choice
individual manipulability.
For Threshold Rule the NK index for n=3 and n=4 is the same for coalitional
manipulation and multiple choice individual manipulation, but after it coalitional manipulability
shows higher NK index than multiple choice one.
For Hare’s Procedure again the values of NK index are the same for coalitional
manipulability and for multiple choice individual manipulability for 3, 4 and 5 agents, and then
the degree of coalitional manipulability becomes higher. NK index for both coalitional
manipulability and for multiple choice individual manipulability is higher than NK index for
single-valued choice individual manipulability for all numbers of agents considered.
For Antiplurality Rule NK indices for coalitional manipulability and for multiple choice
individual manipulability differ only for the cases of 7 and 8 agents, where NK index for
coalitional manipulability is higher. Additionally, it seems that the results for single-valued
choice individual manipulability does not have anything common with them.
For Nanson’s Procedure NK indices for coalitional manipulability and for multiple choice
individual manipulability have the only equal result in the case of 3 agents. In other cases they
have different values.
Overall, we may notice several common characteristics of the graphs provided above:
1. NK index for coalitional manipulability is not less than NK index for multiple choice
individual manipulability (it can be proved by the definitions of these two
manipulability indices).
2. For small numbers of agents the values of NK indices for coalitional manipulation
and for multiple choice individual manipulation are equal. For larger numbers of
agents the degree of coalitional manipulability becomes higher than the degree of
multiple choice individual manipulability.
In this paper we have compared 8 different rules from their vulnerability to coalitional
manipulation point of. We have made comparisons with the previous research in the field of
individual manipulability. It is important to state that this result is not obligatory robust and if we
add additional rules to our analysis they can overcome these rules in manipulability sense. That
is why there are at least three ways to expand this research: study more rules, situations with
more agents and study more extended preferences concepts.
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