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Social Choice and Welfare

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01-09-2015

Dynamic stable set as a tournament solution

Author: Hannu Vartiainen

Published in: Social Choice and Welfare | Issue 2/2015

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Abstract

We define a notion of dynamic (vNM) stable set for a tournament relation. Dynamic stable set satisfies (i) an external direct stability property, and (ii) an internal indirect stability property. Importantly, stability criteria are conditioned on the histories of past play, i.e. the dominance system has memory. Due to the asymmetry of the defining stability criteria, a dynamic stable set is stable both in the direct and in the indirect sense. We characterize a dynamic stable set directly in terms of the underlying tournament relation. A connection to the covering set of Dutta (J Econ Theory 44:63–80, 1988) is established. Using this observation, a dynamic stable set exists. We also show that a maximal implementable outcome set is a version of the ultimate uncovered set.

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Recent contributions include Anesi Anesi (2006, (2010, (2012), Anesi and Seidmann (2013), Diamantoudi and Xue (2007), Ehlers (2007), Kultti and Vartiainen (2007), Penn (2008), Mauleon et al. (2011). Greenberg (1990) provides a useful taxonomy of the stable set-approaches.

For discussions on tournament solutions, see Moulin (1986), Dutta (1988), and Laslier (1997).

Harsanyi (1974) points out that the standard stable set fails to take farsightedness appropriately into account since after a blocking the play may revert back to the stable set in a way that is beneficial to the blocking party. Xue (1998) argues that indirect dominance, where the agents block an alternative keeping in mind the outcome that is eventually implemented, is not a satisfactory basis for a stable set either. Nothing prevents an intermediate mover along the dominance sequence from deviating from the plan.

Extending the solution to a more general framework à la Chwe (1994) should, however, be doable along the avenues of Vartiainen (2011).

Vartiainen (2013) establishes further connections of the concept of a consistent choice set to models of dynamic decision making.

The uncovered set is due to Fishburn (1977), and Miller (1980).

See Duggan (2012) and Duggan and Kalandrakis (2012) for the existence and other results in the general class of stochastic games that also contains the legislative bargaining models discussed here.

The result may be sensitive to the details of the underlying model, see Kalandrakis (2004).

Mariotti (1997) is a precursor in this literature, analysing coalition formation in normal form games. To the best of our knowledge, his paper is the first one to formally study history dependence in coalition formation.

For example, if \(R\) reflects the majority rule, then \(x_{k}\) may be challenged with \(y\) by any majority coalition of individuals.

In fact, Harsanyi (1974) seems to advocate strict stable set, comparable to history independent version of Definition 1.

We adopt the weak version of indirect dominance-relation (also used by Konishi and Ray 2003; Ray 2007). The original definition due to Chwe (1994) assumes strict relations.

Following Harsanyi (1974), the solution might be called dynamic strict stable set.

For different versions of covering, see Duggan (2013).

Chwe (1994) gives an example where the consistent set is not contained by the uncovered set and, hence, by the ultimate uncovered set.

Due to history dependence, indirect and direct dominance need not coincide.

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Anesi V (2006) Committees with farsighted voters: a new interpretation of stable sets. Soc Choice Welf 27:595–610CrossRef

Anesi V (2010) Noncooperative foundations of stable sets in voting games. Games Econ Behav 70:488–493CrossRef

Anesi V (2012) A new old solution for weak tournaments. Soc Choice Welf 39:919–930CrossRef

Anesi V, Seidmann D (2013) Bargaining over an endogenous agenda. Theor Econ 9:445–492CrossRef

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Title: Dynamic stable set as a tournament solution
Author: Hannu Vartiainen
Publication date: 01-09-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0897-0

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Representing a democratic constituency in negotiations: delegation versus ratification

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