Individual Weighted Excess and Least Square Values

Tinbergen Institute Discussion Paper 2020-033/II

24 Pages Posted: 14 Jul 2020

See all articles by Xia Zhang

Xia Zhang

Northwestern Polytechnical University, China

Rene van den Brink

Vrije Universiteit Amsterdam, School of Business and Economics

Arantza Estevez-Fernandez

VU University Amsterdam - Department of Econometrics and Operations Research; Tinbergen Institute

Hao Sun

Northwestern Polytechnical University, China

Date Written: June 20, 2020

Abstract

This work deals with the weighted excesses of players in cooperative games which are obtained by summing up all the weighted excesses of all coalitions to which they belong. We first show that lexicographically minimizing the individual weighted excesses of players gives the same minimal weighted excess for every player. Moreover, we show that the associated payoff vector is the corresponding least square value. Second, we show that minimizing the variance of the players' weighted excesses on the preimputation set, again yields the corresponding least square value. Third, we show that these results give rise to lower and upper bounds for the core payoff vectors and, using these bounds, we define the weighted super core as a polyhedron that contains the core. It turns out that the least square values can be seen as a center of this weighted super core, giving a third new characterization of the least square values. Finally, these lower and upper bounds for the core inspire us to introduce a new solution for cooperative TU games that has a strong similarity with the Shapley value.

Keywords: Individual weighted excess, Prenucleolus, Least square value, Weighted super core, Shapley value

JEL Classification: C71

Suggested Citation

Zhang, Xia and van den Brink, Rene and Estevez-Fernandez, Arantza and Sun, Hao, Individual Weighted Excess and Least Square Values (June 20, 2020). Tinbergen Institute Discussion Paper 2020-033/II, Available at SSRN: https://ssrn.com/abstract=3631753 or http://dx.doi.org/10.2139/ssrn.3631753

Xia Zhang (Contact Author)

Northwestern Polytechnical University, China ( email )

127# YouYi Load
XiAn, Shaanxi 710072
China

Rene Van den Brink

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081HV
Netherlands

Arantza Estevez-Fernandez

VU University Amsterdam - Department of Econometrics and Operations Research ( email )

De Boelelaan 1105
1081 HV Amsterdam
Netherlands

Tinbergen Institute ( email )

Gustav Mahlerplein 117
Amsterdam, 1082 MS
Netherlands

Hao Sun

Northwestern Polytechnical University, China ( email )

127# YouYi Load
XiAn, Shaanxi 710072
China

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