Impartial culture

Impartial culture (IC) or the culture of indifference^[1] is a probabilistic model used in social choice theory for analyzing ranked voting method rules.^[2][3]

The model is understood to be unrealistic, and not a good representation of real-world voting behavior, however, it is useful for mathematical comparisons of voting methods under reproducible, worst-case scenarios.^{[4][5][6][1][7]}

The model assumes that each voter provides a complete strict ranking of all the candidates (with no equal rankings or blanks), which is drawn from a set of all possible rankings. For ${\displaystyle m}$ candidates, there are ${\displaystyle m!}$ possible strict rankings (permutations).^[2]

There are three variations of the model that use different subsets of the full set of possible rankings, so that different election permutations are drawn with different probabilities:

Impartial culture

This model assumes that each voter's ranking is randomly selected from a uniform distribution. If these are chosen by ${\displaystyle n}$ voters, there are thus ${\displaystyle m!^{n}}$ possible elections ("preference profiles".)^[2]

Impartial anonymous culture

This reduces the set of possible elections by eliminating those that are equivalent if the voter identities are unknown.^[2][8] For example, the two-candidate, three-voter election {A>B, A>B, B>A} is equivalent to the election where the second and third voters swap votes: {A>B, B>A, A>B}, and so all variations on this set of votes are only included once. The set of all such elections is called the anonymous equivalence class (AEC), and if the strict rankings are being chosen by ${\displaystyle n}$ voters, there are ${\displaystyle \left({\frac {n+m!-1}{m!-1}}\right)}$ possible elections.^[2]

This is also referred to as the "Dirichlet" or "simplex" model.^[9][10][11]

Impartial, anonymous, and neutral culture

This reduces the set of possible elections further, by eliminating those that are equivalent if the candidate identities are unknown. For example, the two-candidate, three-voter election {A>B, A>B, B>A} is equivalent to the election where the two candidates are swapped: {B>A, B>A, A>B}.^[2]

References

1. Van Deemen, Adrian (March 2014). "On the empirical relevance of Condorcet's paradox". Public Choice. 158 (3–4): 311–330. doi:10.1007/s11127-013-0133-3. S2CID 154862595. The impartial culture assumption has been criticized extensively as being implausible and empirically irrelevant
2. Eğecioğlu, Ömer; Giritligil, Ayça E. (October 2013). "The Impartial, Anonymous, and Neutral Culture Model: A Probability Model for Sampling Public Preference Structures". The Journal of Mathematical Sociology. 37 (4): 203–222. doi:10.1080/0022250X.2011.597012. S2CID 17266150.
3. Guilbaud, Georges-Théodule (2012). "Les théories de l'intérêt général et le problème logique de l'agrégation" [Theories of general interest and the logical problem of aggregation]. Revue économique (in French). 63 (4): 659. doi:10.3917/reco.634.0659.
4. Lehtinen, Aki; Kuorikoski, Jaakko (June 2007). "Unrealistic Assumptions in Rational Choice Theory". Philosophy of the Social Sciences. 37 (2): 115–138. doi:10.1177/0048393107299684. S2CID 145169622. …Using unrealistic assumptions may thus have a reasonable methodological function even if we know how to describe reality in a more realistic way…
5. Tsetlin, Ilia; Regenwetter, Michel; Grofman, Bernard (1 December 2003). "The impartial culture maximizes the probability of majority cycles". Social Choice and Welfare. 21 (3): 387–398. doi:10.1007/s00355-003-0269-z. S2CID 15488300. it is widely acknowledged that the impartial culture is unrealistic … the impartial culture is the worst case scenario
6. Tideman, T. Nicolaus; Plassmann, Florenz (2008). "The Source of Election Results: An Empirical Analysis of Statistical Models of Voter Behavior". CiteSeerX 10.1.1.504.3181. Voting theorists generally acknowledge that they consider this model to be unrealistic
7. Gehrlein, William V.; Lepelley, Dominique (2011). "Voting Paradoxes and Their Probabilities". Voting Paradoxes and Group Coherence. Studies in Choice and Welfare. pp. 1–47. doi:10.1007/978-3-642-03107-6_1. ISBN 978-3-642-03106-9. if we use conditions that tend to exaggerate the likelihood of observing paradoxes and find that the probability is small with such calculations, the paradox is assuredly very unlikely to be observed in reality.
8. Veselova, Yuliya A. (2014). "Comparing Probabilistic Models: IC, IAC, IANC". Editorial Express. S2CID 33200881.
9. Quas, Anthony (March 2004). "Anomalous outcomes in preferential voting". Stochastics and Dynamics. 04 (1): 95–105. doi:10.1142/S0219493704000912. In the simplex model, an assumption is made that vote allocations are uniformly distributed on the simplex.
10. Tideman, T. Nicolaus; Plassmann, Florenz (2010). "The Structure of the Election-Generating Universe" (PDF). London School of Economics and Political Science. p. 5.
11. de Mouzon, Olivier; Laurent, Thibault; Le Breton, Michel; Lepelley, Dominique (March 2020). "The theoretical Shapley–Shubik probability of an election inversion in a toy symmetric version of the US presidential electoral system" (PDF). Social Choice and Welfare. 54 (2–3): 363–395. doi:10.1007/s00355-018-1162-0. S2CID 148626981. It can be shown that IAC is equivalent to the assumption that the preferences of the voters are independent and identically distributed according to a multinomial distribution conditional on a prior uniform draw … This prior is a special case of a Dirichlet distribution.