The Family of Least Square Values for Transferable Utility Games

Ruiz, Luis M.; Valenciano, Federico; Zarzuelo, José Manuel

doi:10.1006/game.1997.0622

Cited by 103 publications

(103 citation statements)

References 8 publications

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“…Lemma 1 (Ruiz et al 1998) Every linear and efficient value ψ having the equal treatment property is of the form: (Weber 1988) if in every monotone game v (i.e., satisfying…”

Section: And Every Constant C ∈ R -Equal Treatment Property: If Playmentioning

confidence: 99%

“…van den Brink et al (2011)) in order to distinguish between this property and more fundamental Young's monotonicity. Coalitional monotonicity appears under the same name in Young (1985) and Ruiz et al (1998); local monotonicity is a well-known generalization of a property of power indices, also known as desirability. 3 In fact, Young assumes symmetry, but it is clear from his proof that equal treatment is sufficient.…”

Section: S ⊃ T ⇒ V(s) ≥ V(t )) the Individual Values Of All Players Amentioning

confidence: 99%

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“Procedural” values for cooperative games

Malawski

2012

Int J Game Theory

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This paper introduces a new notion of a "procedural" value for cooperative TU games. A procedural value is determined by an underlying procedure of sharing marginal contributions to coalitions formed by players joining in random order. We consider procedures under which players can only share their marginal contributions with their predecessors in the ordering, and study the set of all resulting values. The most prominent procedural value is, of course, the Shapley value obtaining under the simplest procedure of every player just retaining his entire marginal contribution. But different sharing rules lead to other interesting values, including the "egalitarian solution" and the Nowak and Radzik "solidarity value". All procedural values are efficient, symmetric and linear. Moreover, it is shown that these properties together with two very natural monotonicity postulates characterize the class of procedural values. Some possible modifications and generalizations are also discussed. In particular, it is shown that dropping one of monotonicity axioms is equivalent to allowing for sharing marginal contributions with both predecessors and successors in the ordering.

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“…Lemma 1 (Ruiz et al 1998) Every linear and efficient value ψ having the equal treatment property is of the form: (Weber 1988) if in every monotone game v (i.e., satisfying…”

Section: And Every Constant C ∈ R -Equal Treatment Property: If Playmentioning

confidence: 99%

Section: S ⊃ T ⇒ V(s) ≥ V(t )) the Individual Values Of All Players Amentioning

confidence: 99%

“Procedural” values for cooperative games

Malawski

2012

Int J Game Theory

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“…In addition, we suggest generalizations of the axioms proposed for characteristic function form games (such as the nullifying player, the neutral dummy player, or the coalitional monotonicity axioms), to adapt them to situations with externalities. Our method allows us to extend the equal division value (Van den Brink, 2007), the equal surplus value (Driessen and Funaki, 1991), the consensus value (Ju, Borm, and Ruys, 2007), the -egalitarian Shapley value (Joosten, 1996), and the least-square family (Ruiz, Valenciano, and Zarzuelo, 1998). For each of the …rst three extensions, we also provide an axiomatic characterization of a particular value for partition function form games.…”

Section: Introductionmentioning

confidence: 99%

Values for environments with externalities – The average approach

Macho‐Stadler

Pérez‐Castrillo

Wettstein

2018

Games and Economic Behavior

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Values for Environments with Externalities -The Average Approach Terms of use: Documents in EconStor may AbstractWe propose the "average approach," where the worth of a coalition is a weighted average of its worth for different partitions of the players' set, as a unifying method to extend values for characteristic function form games. Our method allows us to extend the equal division value, the equal surplus value, the consensus value, the λ-egalitarian Shapley value, and the least-square family. For each of the first three extensions, we also provide an axiomatic characterization of a particular value for partition function form games. And for each of the last two extensions, we find a family of values that satisfy the properties.JEL-Codes: D620, C710.

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“…Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus. Lastly, the result is extended to the broader family of Individually Rational Least Square values (Ruiz et al, 1998b). …”

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confidence: 98%

“…Finally, Ruiz et al (1998b) pointed out that the additive normalization of a semivalue corresponds to a certain least square value. Hence, we can say that the truncated normalization of a semivalue corresponds to a certain individually rational least square value.…”

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confidence: 99%

The least square nucleolus is a normalized Banzhaf value

2014

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In this note we study a truncated additive normalization of the Banzhaf value. We are able to show that it corresponds to the Least Square nucleolus (LS-nucleolus), which was originally introduced as the solution of a constrained optimization problem (Ruiz et al., 1996). Thus, the main result provides an explicit expression that eases the computation and contributes to the understanding of the LS-nucleolus. Lastly, the result is extended to the broader family of Individually Rational Least Square values (Ruiz et al., 1998b).

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The Family of Least Square Values for Transferable Utility Games

Cited by 103 publications

References 8 publications

“Procedural” values for cooperative games

“Procedural” values for cooperative games

Values for environments with externalities – The average approach

The least square nucleolus is a normalized Banzhaf value

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