Some new results on least square values for TU games

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

doi:10.1007/bf02564802

Cited by 16 publications

(15 citation statements)

References 8 publications

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“…dividing the worth of the 'grand coalition' proportional to the Banzhaf values of the players) as axiomatized in van den Brink and van der Laan (1998) does not satisfy collusion neutrality nor linearity, while the additive normalization of the Banzhaf value (i.e. adding or subtracting from the Banzhaf value of every player the same amount to obtain an efficient payoff vector) as considered in Ruiz, Valenciano and Zarzuelo (1998) does not satisfy collusion neutrality nor the null player property.…”

Section: Discussionmentioning

confidence: 99%

Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games

Brink

2009

SSRN Journal

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Three well-known solutions for cooperative TU-games are the Shapley value, the Banzhaf value and the equal division solution. In the literature various axiomatizations of these solutions can be found. Axiomatizations of the Shapley value often use efficiency which is not satisfied by the Banzhaf value. On the other hand, the Banzhaf value satisfies collusion neutrality which is not satisfied by the Shapley value. Both properties seem desirable. However, neither the Shapley value nor the Banzhaf value satisfy both. The equal division solution does satisfy both axioms and, moreover, together with symmetry these axioms characterize the equal division solution. Further, we show that there is no solution that satisfies efficiency, collusion neutrality and the null player property. Finally, we show that a solution satisfies efficiency, collusion neutrality and linearity if and only if there exist exogenous weights for the players such that in any game the worth of the 'grand coalition' is distributed proportional to these weights.

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Section: Discussionmentioning

confidence: 99%

Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games

Brink

2009

SSRN Journal

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“…Lastly, we revisit the example given in Ruiz et al (1996). Next, we briefly describe how the above procedure can be generalized to the class of individually rational least square values (Ruiz et al, 1998a). A map w from {1, .…”

Section: Proposition 1 For Everymentioning

confidence: 99%

“…In case the feasible region of this optimization problem is only the set of efficient allocations, the optimal solution is called a least square value, LS w . Ruiz et al (1998a) proposed a procedure to obtain the IRLS w value quite similar to Algorithm 1. In fact, both procedures differ only in the starting point, being LS w (N, v) the initial allocation used to obtain the IRLS w value.…”

Section: Proposition 1 For Everymentioning

confidence: 99%

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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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“…t51 (E't) ; A[aIIJPu'uonnlos :(fl ruelqord uottuznurrdo aqt raplsuoJ *uU 3ur Jolcan tq8ram e Jo sluauodruoc w pa/y\erl aq r{uru uq8ralr aqJ 'laqunu antltsod of IJ-values have been proved in [11]; among them it has been shown. as in Keane.…”

mentioning

confidence: 96%