Modeling group preferences: IC, IAC, and IANC
Veselova Y.A.1
One of the basic problems in computational social choice is measuring probability of
some paradox or manipulation to occur in a randomly chosen voters’ preference profile.
Computation of these probabilities bases on the assumptions of which preference profiles should
be considered as the same representation of group preference. There are two basic models:
impartial culture model (IC) and impartial anonymous culture model (IAC). The IC model was
introduced in (Guilbaud 1952); it assumes that a set of all preference profiles is used for
generating voters’ preferences. The second model, IAC, is based on anonymity of voters. It was
introduced in (Kuga and Nagatani 1974) and (Gehrlein and Fishburn 1976). The book (Gehrlein
and Lepelley 2011) summarizes numerous studies of probabilities of voting paradoxes both in IC
and IAC. The question of manipulability of social choice rules in the IAC model was thoroughly
investigated by Pritchard and Wilson (2007), Lepelley and Valognes (2002), Favardin and
Lepelley (2006), and Slinko (2006).
Recently another model was introduced – impartial anonymous and neutral culture model
(IANC), which adds neutrality to the assumption of anonymity. In this model, such preference
profiles which differ in permutation of voters or permutation of alternatives are regarded as the
same representation of voters’ preferences, or equivalent. A set of equivalent preference profiles
form an anonymous and neutral equivalence classes (ANECs). This model was presented in
(Egecioglu 2005) and (Egecioglu and Giritligil 2009). They introduced a way of calculating the
number of anonymous and neutral equivalence classes and an algorithm for their uniform
random generation.
However, the position of IANC model relative to IC and IAC has not been defined.
Before conducting some experiments in IANC model it would be useful to understand whether
computational results in IANC differ from those in IC and IAC model. If they differ, then to
what extent do they differ?
We also discuss when these three models are more appropriate to use. There are some
advantages for each of them. For example, the strongest argument for the IAC model is the
computational usability. In the IANC model computations include uniform generation of the
1
International Laboratory of Decision Choice and Analysis, National Research University Higher
School of Economics, Moscow, Russia.
E-mail: yul-r@mail.ru.
preference profiles representing each ANEC. The algorithm was introduced in (Egecioglu 2005).
Complexity of the ANECs generation is higher than the complexity of uniform preference
profiles generation. However, there is a significant reduction of the number of objects to be
considered (preference profile in IC and ANECs in IANC) shown in Table 1. It means that
results of computational experiments with Monte-Carlo generation in IANC should be more
accurate than in IC. Even comparing IANC model with IAC this reduction is the more
considerable the higher the number of alternatives. The problem considering IANC is about its
intuitive interpretation. What are the equivalence classes? What is their application? We
considering manipulability index. In the IC model, the Nitzan-Kelly’s index is the proportion of
manipulable profiles in the set of all preference profiles. In the IANC model we consider not
profiles, but equivalence classes, and the Nitzan-Kelly’s index in IANC is a proportion of
manipulated equivalence classes. It is obvious that every fair voting rule should satisfy at least
anonymity and neutrality. Thus, ANEC could be considered as separate type of group
preferences from the perspective of anonymous and neutral social choice rules. In other words,
IANC model provides a universal domain for such rules, the set of ANECs.
Using combinatorial methods and elements of group theory, we estimate the difference of
indexes in IANC and IC models. We study properties of equivalence classes with maximal and
minimal number of elements and evaluate the maximal difference of probabilistic measures for
the number of voters and alternatives up to 10. After that, we compare IANC model with IAC
and conclude that the difference between them is insignificant when the number of voters or
alternatives exceeds 10. This allows us to obtain the asymptotic difference between IC and IANC
models.
Finally, we illustrate the theoretical study with examples of computation of NitzanKelly’s manipulability index in IC and IANC model for four social choice rules with 3
alternatives and up to 30 voters.
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