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2017 | OriginalPaper | Buchkapitel

Plurality, Borda Count, or Anti-plurality: Regress Convergence Phenomenon in the Procedural Choice

verfasst von : Takahiro Suzuki, Masahide Horita

Erschienen in: Group Decision and Negotiation. Theory, Empirical Evidence, and Application

Verlag: Springer International Publishing

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Abstract

We focus on voters’ preference profiles where at least two of the three selected voting rules (e.g. plurality, Borda count, and anti-plurality) produce different outcomes—thus, the voting body needs a procedural choice. While this situation evokes an infinite regress argument for the choice of rules to choose rules to choose rules to…and so on, we introduce a new concept named regress convergence, where every voting rule in the menu ultimately gives the same outcome within the finite steps of regress. We study the mechanism of this phenomenon in a large consequential society having a triplet of scoring rules. The results show that, in the menu of plurality, Borda count, and anti-plurality, the probability that the regress convergence happens is 98.2% under the Impartial Culture assumption and 98.8% under the Impartial Anonymous Culture assumption.

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Vorheriges Kapitel Multi-criteria Group Decision Making with Private and Shared Criteria: An Experiment

Nächstes Kapitel Estimating Computational Models of Dynamic Decision Making from Transactional Data

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If the alternatives are denoted A, B, and C, the seven rankings are \( A \succ B \succ C \), \( A \succ C \succ B \), \( C \succ A \succ B \), \( C \succ B \succ A \), \( A \succ B \,{ \sim }\,C \), \( A \sim C \succ B \), and \( C \succ A\,{ \sim }\,B \) (Saari and Tataru 1999).

Saari and Tataru [18] argue in their introduction that “Except in extreme cases such as where the voters are in total agreement, or where all procedures give a common outcome, it is debatable how to determine the ‘true wishes’ of the voters.” Clearly, the intuition of regress convergence lies in the latter “extreme cases,” though our results show that the phenomenon can occur relatively frequently in the choice of triplets of scoring rules.

Note that no rule chooses itself in the figure. Therefore, the weak convergence does not logically imply the stability of the menu of SCRs in either Houy [9]’s or Diss and Merlin [4]’s sense. We can also say that the existence of a self-selective rule does not imply regress convergence (See the trivial deadlock described in Definition 5 and Fig. 2).

Note that by definition we normalize the score assignment so that the top position gains one point and the worst position gains zero points.

In this article, we suppose the society uses the fixed set of SCRs, \( f_{1} , \ldots ,f_{m} \) for any level. The distinction between \( f_{j}^{1} \) and \( f_{j}^{2} \) by the superscripts is made based on the supposed agenda.

If we identify \( f \in F^{k} \) with its class \( C \subseteq X \), the consequentialism assumption is a way to introduce one’s preference on sets of alternatives. This is often called preference extension (Barbera et al. [1]). When seen in this way, our consequentialism assumption is the same as the Extension Rule. It is a natural requirement of most reasonable systems of preference extension to satisfy the Extension Rule (See also Endriss [6]).

In the present article, we adjust the number of alternatives and that of admissible SCRs. However, this is not essential. If the society has \( m^{\prime} \left( { \ne m} \right) \) alternatives and \( m \) scoring SCRs, we can modify the definition of trivial deadlock to be the case where every level-\( 2 \) scoring SCR chooses distinct singletons and \( f_{1}^{2} \left( {L^{1} } \right)\mathop \cup \nolimits \ldots \mathop \cup \nolimits f_{m}^{2} \left( {L^{1} } \right) = F^{1} \) for all \( L^{1} \in {\mathcal{L}}^{1} \left[ {L^{0} } \right] \). Then, our proofs for Lemma 1 and Proposition 1 also hold (though the specific value of \( p_{D} \) depends on \( m^{\prime} \)).

For a relatively small n, the probabilities of tied outcomes by famous scoring rules such as plurality and Borda count are studied by Gillet [7,8] and Marchant [12].

Note that these two properties (and thus, our Proposition 1) are also satisfied under IANC (impartial anonymous and neutral culture) model introduced by Eǧecioǧlu [5]. This is because each ANEC (anonymous and neutral equivalence class) has at most \( m\text{!} \) different AECs. Hence, the ratio of those ANECs including profiles such that ties happen or \( {\# }\left\{ {\left. {i \in N} \right | xL_{i} y} \right\} = n\alpha \) to the whole ANECs is at most \( \left( {m\text{!}} \right)^{2} \) times that of IAC , i.e. the ratio of AECs (anonymous equivalence class) causing ties or \( {\# }\left\{ {\left. {i \in N} \right | xL_{i} y} \right\} = n\alpha \) to the whole AECs. With our Lemma 3, we can confirm that the two asymptotic properties still hold under IANC. We thank an anonymous referee for encouraging us to consider about IANC also.

Technically speaking, we can find the similar use of a compatible linear ordering in Koray [10] and Koray and Slinko [11] . They define a Social Choice Function (SCF) \( f \) as self-selective at \( L^{0} \) relative to the menu of SCFs \( F^{1} \) if and only if there is a consequentially induced \( L^{1} \in \left( {{\mathcal{L}}\left( {F^{1} } \right)} \right)^{n} \) such that \( f^{2} \left( {L^{1} } \right) = f^{1} \). As Koray and Slinko stated (if we impose that the rule chooses itself for all compatible linear orderings), “it leads to a vacuous concept”. The same applies to regress convergence.

We assume the Hare system drops exactly one alternative with the least plurality score in each round (if two or more get the least score, it selects and drops one of them in neutral way).

As a step further in this direction, it is also shown that the asymptotic property of \( p_{WC} \approx 1 - p_{D} \) holds in a large consequential society having \( \left\{ {f_{P} ,f_{{E_{2} }} ,f_{A} ,f_{B} } \right\} \left( {m = 4} \right) \). The sketch of this calculation is found in the appendix.

Barbera, S., Bossert, W., Pattanaik, P.K.: Ranking sets of objects. In: Barbera, S., Hammond, P.J., Seidl, C. (eds.) Handbook of Utility Theory Volume II Extensions, pp. 893–977. Kluwer Academic Publishers, Dordrecht (2004)CrossRef

Barbera, S., Jackson, M.: Choosing how to choose: Self-stable majority rules and constitutions. Q. J. Econ. 119, 1011–1048 (2004)CrossRef

Diss, M., Louichi, A., Merlin, V., Smaoui, H.: An example of probability computations under the IAC assumption: the stability of scoring rules. Math. Soc. Sci. 64, 57–66 (2012)CrossRef

Diss, M., Merlin, V.: On the stability of a triplet of scoring rules. Theor. Decis. 69(2), 289–316 (2010)CrossRef

Eğecioğlu, Ö.: Uniform generation of anonymous and neutral preference profiles for social choice rules. Monte Carlo Methods Appl. 15(3), 241–255 (2009)

Endriss, U.: Sincerity and manipulation under approval voting. Theor. Decis. 74(3), 335–355 (2013)CrossRef

Gillet, R.: Collective indecision. Behav. Sci. 22(6), 383–390 (1977)CrossRef

Gillett, R.: The comparative likelihood of an equivocal outcome under the plurality, Condorcet, and Borda voting procedures. Public Choice 35(4), 483–491 (1980)CrossRef

Houy N.: A note on the impossibility of a set of constitutions stable at different levels. Mimeo (2004)

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Koray, S.: Self-selective social choice functions verify arrow and Gibbard- Satterthwaite theorems. Econometrica 68, 981–995 (2000)CrossRef

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Koray, S., Slinko, A.: Self-selective social choice functions. Soc. Choice Welf. 31(1), 129–149 (2008)CrossRef

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Marchant, T.: The probability of ties with scoring methods: some results. Soc. Choice Welf. 18(4), 709–735 (2001)CrossRef

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Merlin, V., Tataru, M., Valognes, F.: On the probability that all decision rules select the same winner. J. Math. Econ. 33(2), 183–207 (2000)CrossRef

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Suzuki, T., Horita, M.: How to order the alternatives, rules and the rules to choose rules: when the endogenous procedural choice regresses. In: Kamiński, B., Kersten, G.E., Shakun, M.F., Szapiro, T. (eds.) GDN2015. LNBIP, vol. 218, pp. 47–59. Springer, Heidelberg (2015)

Titel: Plurality, Borda Count, or Anti-plurality: Regress Convergence Phenomenon in the Procedural Choice
verfasst von: Takahiro Suzuki
Masahide Horita
Verlag: Springer International Publishing
Buch: Group Decision and Negotiation. Theory, Empirical Evidence, and Application
Print ISBN: 978-3-319-52623-2

Electronic ISBN: 978-3-319-52624-9

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-52624-9_4

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