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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.
REFERENCES
Arrow, K. J. 1951/1963. Social Choice and Individual
Values. Wiley. New York, 2
nd
Edition.
Blair, D., Muller, E. 1983. Essential aggregation
procedure on restricted domains of preferences,
Journal of Economic Theory, 30, 34–53.
Dasgupta, P., Maskin, E. 2008. On the robustness of
majority rule. Journal of the European Economic
Association, 6(5), 949–973.
Fishburn, P. C., Kelly, J. S. 1997. Super-Arrovian domains
with strict preferences. SIAM Journal on Discrete
Mathematics, 10(1), 83–95.
Gibbard, A. 1973. Manipulation of voting schemes: a
general result. Econometrica, 41(4), 587–601.
Indo, K. 2007. Proving Arrow's theorem by Prolog,
Computational Economics, 30(1), 57–63.
Indo, K. 2009. Modeling a small agent society based on
the social choice logic programming. In T. Terano et
al. (Eds.), Agent-based Approaches in Economic and
Social Complex Systems V. Springer Verlag, Tokyo pp.
93–104.
Indo, K. 2010. Generating social welfare functions over
restricted domains for two individuals and three
alternatives using Prolog, In A. Tavidze (Ed.),
Progress in Economics Research, 18, Nova Science,
New York, pp. 163–187.
Kalai, E., Muller, E. 1977. Characterization of domains
admitting nondictatorial social welfare functions and
nonmanipulable voting procedures, Journal of
Economic Theory, 16, 457–469.
Kalai, E., Muller, E., Satterthwaite, M.A. 1979. Social
welfare functions when preferences are convex,
strictly monotonic and continuous, Public Choice, 34,
87–97.
Kelly, J. S. 1994. The free triple assumption, Social
Choice and Welfare, 11(2), 97–101.
Lin, F., Tang, P. 2009. Computer-aided proofs of Arrow's
and other impossibility theorems, Artificial
Intelligence, 173(11), 1041–1053.
Ozdemir, U., Sanver, M. R. 2007. Dictatorial domains in
preference aggregation, Social Choice and Welfare,
28(1), 61–76.
Muller, E., Satterthwaite, M. A. 1977. The equivalence of
strong positive association and strategy-proofness.
Journal of Economic Theory, 14(2), 412–418.
Robinson, A. 1965. A machine-oriented logic based on the
resolution principle, JACM 12(1): 23–41.
Satterthwaite, M. A. 1975. Strategy-proofness and
Arrow’s conditions: Existence and correspondence
theorems for voting procedures and social welfare
functions. Journal of Economic Theory, 10(2), 187–
217.
Tang, P., Lin, F. 2008. Computer-aided proofs of Arrow's
and other impossibility theorems, In Proceedings of
the Twenty-Third AAAI Conference on Artificial
Intelligence (AAAI-08), 114–118.
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