Parallel Possibility Results of Preference
Aggregation and Strategy-proofness by using Prolog
Kenryo Indo
Department of Business Administration, Faculty of Economics, Kanto Gakuen University,
200 Fujiaku, Ota, Gunma 373-8515, Japan
Keywords: Arrow’s Impossibility Theorem, the Gibbard–satterthwaite Theorem, Restricted Domain, Super-arrovian
Domain, Profile Elimination, Prolog.
Abstract: Classical social choice theory provides axiomatic modeling for collective decision making in multi-agent
situations as functions of a set of profiles (i.e., tuples of transitive orderings). The celebrated Arrow’s
impossibility theorem (for unanimity-and-independence-obeying preference aggregation) and the Gibbard–
Satterthwaite theorem (for strategy-proof voting procedures) assume the unrestricted domain as well as the
transitivity of orderings. This paper presents a distribution map of all Arrow-type aggregation rules without
the unrestricted domain axiom for the two-individual three-alternative case in parallel with non-imposed
strategy-proof voting procedures by using a Prolog program that systematically removes profiles in the
super-Arrovian domains.
1 INTRODUCTION
Social choice theory studies axiomatic collective
decision making in multi-agent situations by
assuming the transitivity of individual preference
orderings (i.e., rankings), which is mapped into
certain collective decision outcomes.
Kenneth J. Arrow’s general impossibility
theorem is a classical result regarding the social
aggregation of a tuple of individual orderings (i.e., a
profile) into the ordering of society as a whole
(Arrow, 1951/63). A social welfare function (SWF)
is required to satisfy the following five axioms:
unrestricted individual orderings (U), the transitivity
of the ordering of society as a whole (T), unanimity,
namely the weak Pareto principle (P), independence
of irrelevant alternatives (I), and non-dictatorship
(D). Arrow proved that any aggregation rule that
satisfies the first four axioms should be dictatorial,
and therefore it is impossible to satisfy all five
axioms. Allan Gibbard and Mark Satterthwaite
independently proved that if there are three or more
candidates any voting procedure is non-imposed and
strategy-proof, namely every candidate has a
possibility to win and no individual can manipulate
the outcome of a vote by falsely reporting his or her
own preference, the procedure should be dictatorial
(Gibbard, 1973; Satterthwaite, 1975). These
classical results of social choice theory assume an
unrestricted domain.
In order to prove new possibility results as well
as classical impossibility theorems, this paper adopts
a computational step to axiomatically model social
choice under restricted domains instead of using
pure mathematics. Prolog language is useful to
program intelligent processing systems in the AI
research and industry. In addition, Prolog uses a
basic technology that stems from automated theorem
proving (Robinson, 1965) based on predicate logic.
Social choice theory has recently drawn the
attention of computer scientists, as it is the
foundation of mechanism design for multi-agent
systems as well as game theory. Tang and Lin
(2008) and Lin and Tang (2009) provided computer-
aided proofs of Arrow’s impossibility theorem and
the Gibbard–Satterthwaite theorem for two
individuals and three alternatives (which Lin and
Tang called the “base case”) by using a SAT solver,
and they proved the general case through
mathematical induction. Without exposure to the
source code, Tang and Lin also insisted that the base
case could be proven by using Prolog. Independent
of their work, Indo (2007) introduced a complete
Prolog program that proves Arrow’s theorem and
Wilson’s theorem for the “base case” in linear
ordering. Indo (2009) also extended this approach to
243
Indo K..
Parallel Possibility Results of Preference Aggregation and Strategy-proofness by using Prolog.
DOI: 10.5220/0004913302430248
In Proceedings of the 6th International Conference on Agents and Artificial Intelligence (ICAART-2014), pages 243-248
ISBN: 978-989-758-016-1
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
several classical results of social choice theory
including the Gibbard–Satterthwaite theorem and
domain restrictions. In order to generate a possible
domain and a social choice rule, Indo (2010)
developed a Prolog program that implemented the
systematic removal of a pair of profiles in the super-
Arrovian domains.
In this paper, a generalized version of the profile
elimination that removes arbitrary subsets in the
union of two super-Arrovian domains is applied to
prove the comprehensive distribution of Arrow-type
preference aggregation rules without the unrestricted
domain axiom for the base case, in parallel to non-
dictatorial strategy-proof social choice functions
(SCFs).
The remainder of this paper is organized as
follows. Section 2 describes the two classical
impossibility results and the domain conditions in
order to escape from impossibility. Section 3
introduces the alternative domain condition by
eliminating profiles in the paired super-Arrovian
domains. Section 4 explains a computational version
of social choice theory and the profile elimination
implemented in Prolog. Section 5 summarizes the
experimental results (i.e., automated proofs) of the
possibility of Arrow-type aggregation parallel to
non-imposed strategy-proof voting. Section 6
concludes.
2 CLASSICAL RESULTS
2.1 Preference Model
Given a set of individuals, N = {1, 2, …, n}, and a
set of a finite number of alternatives, A = {x, y, z,
…}. A is called agenda. A (weak) preference
ordering or ranking is defined as a complete and
transitive binary relation on A. R is complete if for
all x and y, either xRy, yRx, or both. R is transitive if
for all x, y, and z, if x R y and yRz, then xRz. A
binary relation {xRy, yRz, zRx} is intransitive.
The indifference relation w.r.t. a relation R,
which means xRy and yRx, is denoted by xIy. R is
anti-symmetric if for all x and y, if xIy, then x = y. P
stands for the strict part of R, namely x P y if xRy
and not yRx. A preference ordering R is called a
linear ordering if R = P(R
), i.e., R is strict.
Throughout this paper, we assume linear
ordering. In this case, we can consider any ordering
as a permutation of agenda A. Let profile R
N
= (R
1
,
R
2
, …, R
n
) be a combination of all individuals’
orderings. The set of all possible profiles U is called
the unrestricted domain (or universal domain).
Moreover, we assume that there exists a ranking of
society as a whole, R
S
, as well as individual rankings.
2.2 Preference Aggregation
Definition. An SWF is a function defined on a
subset of profiles D
U to the set of social
rankings. For any profile R
N
D, we say R
N
is
permissible. The ranking of society, R
S
= f(R
N
),
aggregated by an SWF f should satisfy the following
five conditions:
(U) The SWF is defined for every possible profile
(unrestricted domain).
(T) The social ranking R
S
should be transitive.
(P) For any pair of alternatives, x and y, if xR
i
y for
every individual i, then xR
S
y (unanimity).
(I) For any pair of alternatives, x and y, if every
individual has the same ranking regarding this pair
in two profiles R
N
and R
N
', then xR
S
y if xR
S
'y (the so-
called independence of irrelevant alternatives).
An individual i is called a dictator if for any pair
of alternatives, x and y, if xR
i
y, then xR
S
y.
(D) There is no dictator (i.e., non-dictatorship).
An SWF is termed resolute if the social ordering
is linear for every profile. We abuse the notion of
SWF when its domain is restricted to a subset of
profiles, thereby dropping the conditions U and D.
Theorem (Arrow’s Impossibility Theorem). If there
are one or more individuals and more than two
alternatives, then any SWF that satisfies U, T, P, and
I is dictatorial.
2.3 Voting Procedure
Definition. A (resolute) SCF is a function that
selects a single alternative from each non-empty
subset of alternatives (i.e., the agenda) for every
permissible profile.
An SCF is manipulable if an individual can
report a false ordering to establish a more preferable
outcome for herself/himself:
(A) An SCF is defined for every profile and the
agenda is restricted to A.
(S) An SCF is not manipulable (strategy-proofness
or non-manipulability).
(C) There is no alternative x as x is never selected
(as a single winner) for any profile or agenda (non-
imposition or a citizen’s sovereignty).
A single individual whose top-ranked alternative
is always selected as a winner is called a dictator. An
SCF is dictatorial if there is a dictator.
Theorem (The Gibbard–Satterthwaite Theorem). If
there are one or more individuals and more than two
alternatives, then any (resolute) SCF that satisfies A,
ICAART2014-InternationalConferenceonAgentsandArtificialIntelligence
244
S, and C is dictatorial.
2.4 Restricted Domain
Restricting permissible orderings for each individual
may help society escape from impossibilities. Many
classical domain conditions are known for a
pairwise-majority vote. Kalai and Muller (1977)
proposed decomposability into a relation that is
closed under a decisiveness implication as a
necessary and sufficient condition for the existence
of non-dictatorial SWFs, paralleling strategy-proof
voting procedures, assuming that every individual
has the same possible orderings (i.e., a common
admissible domain). Blair and Muller (1983)
modified this for an individual’s possible set of
orderings. Kalai, Muller, and Satterthwaite (1979)
extended decomposability to economic
environments (i.e., saturated domains). Arrow
proposed the “free triple” condition, while Kelly
(1994) and Ozdemir and Sanver (2007) elaborated
on this condition.
All the above literature argues that mathematical
conditions that restrict the domain of social choice
rule at most the possible set of orderings for each
individual. We depart from classical mathematical
approaches to social choice in two aspects. First, this
paper proposes finer-grained conditions that restrict
profiles as the permissible inputs of collective
decision making (see Section 5) and, second, to do
so we adopt computational proofs instead of
standard mathematical proofs, as we demonstrate in
the following sections.
3 ELIMINATION OF PROFILES
FROM THE SUPER-ARROVIAN
DOMAINS
This section introduces an alternative way in which
to find domains to avoid impossibility. By using the
backtrack mechanism, we can find all the possible
aggregation rules, at least in principle. Note that
even for two-individual three-alternative cases, a
naive backtrack is not computationally efficient for
generating all those functions. The cumulative
constraint in the recursive program plays a crucial
role in the negative proof for unrestricted domains.
However, this is not enough for analyzing restricted
domains. This section therefore introduces a profile
elimination procedure for removing a set of profiles
in order to generate versions of the SWF in restricted
domains more efficiently.
3.1 Super-Arrovian Domain
Let agenda A = {a, b, c}. If we take a subset of
profiles instead of the unrestricted domain, an
aggregation rule may satisfy all the axioms except
for U. Indeed, we can deliberately select a set of
profiles to be eliminated in order to escape from the
Arrow-type impossibility. These profiles have been
termed the super-Arrovian domain in the literature
(Fishburn and Kelly, 1997):
P
1
. (a P
4
c P
4
b, c P
4
b P
4
a) = ((a, c, b), (c, b, a)),
P
2
. (a P
5
b P
5
c, c P
5
a P
5
b) = ((a, b, c), (c, a, b)),
P
3
. (b P
6
a P
6
c, a P
6
c P
6
b) = ((b, a, c), (a, c, b)),
P
4
. (b P
1
c P
1
a, a P
1
b P
1
c) = ((b, c, a), (a, b, c)),
P
5
. (c P
2
b P
2
a, b P
2
a P
2
c) = ((c, b, a), (b, a, c)),
P
6
. (c P
3
a P
3
b, b P
3
c P
3
a) = ((c, a, b), (b, c, a)).
These six profiles propagate the decisiveness of
any subgroup for a pair of alternatives over all the
possible pairs of alternatives, and they are minimal
and sufficient for deducing a dictatorship under
Arrow’s axioms without U. There are also another
six profiles where Q
k
is (r
2
, r
1
), which corresponds to
P
k
= (r
1
, r
2
), k = 1, …, 6.
3.2 Profile Elimination
The profile elimination procedure implemented in
Prolog used first by Indo (2010) can provide
domains and rules finer than those conditions
introduced in Section 2.3. The next section explains
the computational proof for the impossibility results
and its modification to implement the profile
elimination.
In order to generate a possible domain and a
social choice rule more effectively, it is beneficial to
eliminate the profiles from M = {P
1
, …, P
6
}
{Q
1
,
…, Q
6
}. Indeed, Indo (2010) demonstrated that after
the removal of (P
k
, Q
j
) such that |(k – j) mod 6|
1,
the remaining domain that consists of 34 profiles has
18 Arrovian aggregation rules that can be reduced to
essentially six unanimous and constant rules (this
result can be verified by using
test1/0 in the Prolog
program). Those rules are maximally robust in the
sense of Dasgupta and Maskin (2008).
4 COMPUTATIONAL APPROACH
This section introduces the Prolog application for the
computational version of axiomatic social choice
theory. Because the basic technique adopted in this
paper is essentially the same as that used in previous
studies, we omit the detail. The source code is
ParallelPossibilityResultsofPreferenceAggregationandStrategy-proofnessbyusingProlog
245
available from http://www.xkindo.net/
sclp/pl/icaart2014.pl.
The program has been tested for version 6.4.1
running 64 bit Windows 7 (and it may run for any
PC that has installed SWI-Prolog version 5.6.52 or
later).
4.1 Social Choice Theory in Logic
Programming
A Prolog program consists of clauses. A generic
discrete functional form
f( Function, Domain,
Axiom )
is represented as follows:
f([ ], [ ], _).
f([ X - Y | F ], [ X | D ], Axiom):-
f(F, D, Axiom),
Goal =.. [ Axiom, X, Y, F],
Goal.
The first clause is the null function defined on an
empty domain. The second clause assigns
recursively a logically possible value without
violating any axiom given by the modeler under the
values
F that have already been assigned to the
subdomain
D. Note that, abusing the notation, we
write “
X-Y” to indicate an assignment of a value Y in
the region to a value
X in the domain. Predicate
=../2 in the second clause stands for a “term to list
conversion.”
The axioms of SWFs and SCFs can be written as
follows:
swf_axiom( X, Y, F):- rc( _, Y),
pareto( X - Y), iia( X - Y, F).
scf_axiom( X, Y, F):- x( Y),
\+ manipulable( _, X - Y, F).
The SWFs and SCFs defined on some domain D
can be written as follows:
swf( F, D):- f( F, D, swf_axiom),
\+ dictatorial_swf( _, F).
scf( F, D):- f( F, D, scf_axiom),
non_imposed(F),
\+ dictatorial_scf(_,F).
4.2 Profile Elimination
For the sake of later use, the rankings represented by
a predicate
rc/2 are as follows:
rc( 1, [a, c, b]).
rc( 2, [a, b, c]).
rc( 3, [b, a, c]).
rc( 4, [b, c, a]).
rc( 5, [c, b, a]).
rc( 6, [c, a, b]).
The possible profiles (pp/1) and unrestricted
domain (
all_pp/1) are written as follows:
pp( [P, Q]):- rc( _, P), rc(_, Q).
all_pp(U):- findall( O, pp(O), U)..
Note that these six rankings should be numbered
in the specified sequence (modulo 6). The super-
Arrovian domain P
k
and Q
j
(k, j = 1, …, 6) described
in Section 3.1 can be generated by pairing (and by
exchanging) indices such that (k, j) = (1, 5), (2, 6), (3,
1), (4, 2), (5, 3), and (6, 4).
The simple recursive program described in the
previous section is also useful for finding a way in
which to escape from the impossibility results. A
candidate domain consists of the remaining profiles
in the domain after a subset C
M has been
removed from the unrestricted domain U, where D =
U
C. In the Prolog program, select_n/3 (user-
defined) for generating subsets and
subtract/3
(builtin) for deleting list elements are used.
5 POSSIBILITY RESULTS
Tables 1–3 summarize the experimental results of
the profile elimination (these results are reproducible
by the automated proofs
test2/0 and test3/0 in
the Prolog program).
5.1 Possible SWFs
The top row in Table 1 (labeled 2) indicates the
cases that no Arrow-type aggregation rule exists
except for dictatorships. In particular, Cell (2, 12)
corresponds to the case of Arrow’s impossibility
theorem. Moreover, Cell (2, 6) valued 2 implies that
even when one super-Arrovian domain has been
completely eliminated, the other remaining super-
Arrovian domain is sufficient to prove a dictatorship.
Further, Cell (3, 10) and Cell (20, 0) both indicate
the 18 rules mentioned in Section 3.2. Note that
there are 4096 domains in total. The bottom row
coincides with binomial coefficients
12
C
column
.
Interestingly, Cell (k, 6) (k >2) suggests that any
exchange between P and Q can reverse impossibility
if the status quo domain is a complete removal of
one of the super-Arrovian domains. We term this the
exchange property.
5.2 Possible SCFs
With regard to Table 2, the total number of domains
is the same as shown in Table 1. Similar to the SWF
cases in Table 1, the top row labeled 2 also indicates
the impossibility results that no non-imposed
ICAART2014-InternationalConferenceonAgentsandArtificialIntelligence
246
Table 1: Arrow-type preference aggregation rules (SWFs)
generated by profile elimination in super-Arrovian
domains. The rows represent the counts of SWFs that
include two dictatorships, the columns represent the
remaining numbers of the super-Arrovian profiles, and the
cells contain the counts of restricted domains on which an
SWF exists.
Table 2: Non-imposed strategy-proof voting procedures
(SCFs) generated by profile elimination in super-Arrovian
domains. The rows represent the counts of SCFs that
include two dictatorships, the columns represent the
remaining numbers of the super-Arrovian profiles, and the
cells contain the counts of restricted domains on which an
SCF exists.
strategy-proof voting procedure exists except for the
dictatorship of each person. Cell (2, 12) valued 1
corresponds to the case of the Gibbard–Satterthwaite
theorem. Similar to Table 1, Cell (2, 6) valued 2
suggests that the super-Arrovian domain is sufficient
to prove a dictatorship. Cell (k, 6) also satisfies the
exchange property. Cell (3, 10) valued 18
corresponds to the 18 maximal domains.
Table 3: Parallel possibility results of SWFs and SCFs.
The rows represent the numbers of SCFs that include two
dictatorships, the columns represent the numbers of SWFs,
and the cells contain the counts of restricted domains.
5.3 Parallel Possibility Results
In Table 3, the two distributions seem to be
positively correlated, but the precise interrelation is
unclear. The top-left corner (2, 2) has a value of 169,
which indicates that the parallel impossibility results
have occurred in the 169 domains. Cell (3, 2)
implies that 24 domains have a strategy-proof voting
procedure but no non-dictatorial Arrow-type
preference aggregation. In Cell (196, 20), the two
super-Arrovian domains have been eliminated; the
remaining 24 profiles deduce 20 SWFs and 196
SCFs.
6 CONCLUSIONS
This paper presented a complete distribution map of
all Arrow-type aggregation rules without the
unrestricted domain axiom for the two-individual
three-alternative linear ordering case in parallel with
non-imposed strategy-proof voting procedures by
using a Prolog program that systematically removes
the arbitral subset of the super-Arrovian domains.
We can summarize the presented observations
into the following three parallel possibility results.
Result 1. The impossibility result no longer occurs if
more than half of the 12 profiles have been
swf 0123456789101112total
2 2 12 48 76 48 12 1 199
3 60 156 108 18 342
4 54 228 225 36 543
5 12 170 348 60 590
6 60 390 120 6 576
7 228 252 24 504
8 48 348 50 446
9 156 120 6 282
10 225 24 249
11 76 60 136
12 108 6 114
13 36 36
14 48 48
15 18 18
17 12 12
20 1 1
total 1 12 66 220 495 792 924 792 495 220 66 12 1 4096
sp 0123456789101112total
2 2 12 30 64 48 12 1 169
3 114 120 18 252
4 144 255 36 435
5 62 300 90 452
6 12 150 252 6 420
7 294 72 366
8 120 242 12 374
9 132 78 210
10 18 192 72 282
11 36 48 84
12 57 108 18 183
13 30 48 6 84
14 4 36 72 112
15 36 12 48
16 69 24 93
17 12 36 48
18 72 72
19 12 24 36
20 36 12 48
21 12 12
22 36 36 72
23 12 12
25 30 30
26 12 12
28 24 3 27
29 6 6
30 6 6
31 24 24
34 12 12
35 12 12
37 12 12
38 6 12 18
40 6 12 18
41 12 12
46 6 6
48 12 12
50 6 6
74 6 6
88 12 12
196 1 1
total 1 12 66 220 495 792 924 792 495 220 66 12 1 4096
swf
scf 2 3 4 5 6 7 8 91011121314151720total
2169 169
3 24 228 252
4 6 84 345 435
5 6 144 302 452
6 24 24 168 204 420
7 36 192 138 366
8 24 36 78 168 68 374
9 1212609630 210
10 36 36 18 120 60 12 282
11 24 12 24 24 84
12 30 36 30 24 48 12 3 183
13 6 2424 1218 84
14 24 48 30 10 112
15 12 24 12 48
16 12 18 48 12 3 93
17 12 24 12 48
18 60 12 72
19 12 12 12 36
20 6 6 24 12 48
21 12 12
22 12 24 24 12 72
23 12 12
25 24 6 30
26 12 12
28 3 24 27
29 6 6
30 6 6
31 24 24
34 12 12
35 12 12
37 12 12
38 12 6 18
40 12 6 18
41 12 12
46 6 6
48 12 12
50 6 6
74 6 6
88 12 12
196 11
total 199 342 543 590 576 504 446 282 249 136 114 36 48 18 12 1 4096
ParallelPossibilityResultsofPreferenceAggregationandStrategy-proofnessbyusingProlog
247
eliminated (see the top row in Table 1 and Table 2:
Cell (2, j) = 0 if j < 6).
In addition, the exchange property described in
the previous section is satisfied.
Result 2. The possibility may occur if more than two
of the 12 profiles are eliminated appropriately (see
the second and subsequent rows in Table 1 and
Table 2: If Cell (k, j) > 0 & k > 2, then j>2).
The minimal number of eliminations sufficient to
deduce a possibility is two. Indeed, these are the 18
profile pairs (see Cell (3, 10) in Table 1 and Table 2
as well as
test1 in the author’s Prolog program).
Result 3. (i) There are 169 domains where Arrow-
type aggregation (SWF) and non-dictatorial non-
imposed strategy-proof voting (SCF) are both empty
(i.e., Cell (2, 2) = 169 in Table 3). (ii) There are also
30 domains where SCF exists but SWF is empty (i.e.,
Cell (3, 2) = Cell (4, 2) = 0 in Table 3). (iii) There is
no domain where SWF exists but SCF is empty (i.e.,
Cell (2, j) = 0 if j > 2 in Table 3). (iv) In the other
domains, SWF and SCF are both non-empty.
Additionally, if we substitute Maskin
monotonicity and unanimity for strategy-proofness
and non-imposition, then Table 2 is the same as
shown in Table 1 (see
test4). Muller and
Satterthwaite (1977) proved the equivalence for the
unrestricted domain. Lastly, the n-person and m-
alternative case possibility result can be proven by
assuming that n – 2 individuals are dummy and
everyone is indifferent for m – 3 alternatives, but
further study in this regard is needed.
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