Abstract
We relate three solutions for cooperative games, the Shapley value, the nucleolus and the core. We use an empirical case study, provided in Hubert and Orlova (2018) to analyze the liberalization of network access in the European gas market. For these games the Shapley value is not in the core. To obtain a differentiated picture of the (in)stability of an allocation, we propose the n𝜖-core which is a generalization of the strong 𝜖-core, and define three stability measures. We find that the liberalization of network access increases the degree of instability of the Shapley value for all three metrics. The nucleolus is a unique point in the core, hence often used to characterize stable imputations. We show that liberalization compresses the core, but not always the nucleolus corresponds well to the shifts in the minimal and maximal values which players might receive in the core.
This paper is part of larger collaborative research project on the Eurasian gas network which was developed and supervised by Prof. Dr. Franz Hubert and to which Onur Cobanli made essential contributions. The draft of this paper was written in the 2014 at Humboldt University Berlin.
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Notes
- 1.
[11] consider the strategic relevance of various options to expand the network. [10] investigate three pipeline projects in detail: Nord Stream, South Stream, and Nabucco. [1] considers pipeline projects for the Central Asian region. [12] and [13] look at the liberalization of pipeline access within the European Union, with the first paper emphasizing regional effects and cartels, while the second paper’s focus is on customers versus local champions.
- 2.
The development of liberalization process was mainly determined by several consecutive Directives of the EU Commission: Directive 98/30/EC, also known as the First Gas Directive [3], Directive 2003/55/EC, known as the Second Gas Directive [4], Directive 2009/73/EC [5] which refers to the Third Energy Package.
- 3.
Incumbents pointed out the strong import dependency of the EU in the gas industry and argued that
there is a need for a limited number of strong market players in order to deal with the high level of concentration of gas producers outside the European Union [2, Second Phase, p. 207].
- 4.
Two findings about the compression of the range: (i) that the total effect of reform is dominated by the second step only for the EU champions and the customers and (ii) that the losses of the champion and the customers in a region are of the same magnitude, correspond to the results in case of nucleolus. In [13] we find that in case of nucleolus the second step of reform dominates the effect of full liberalization only for the EU champions and the customers, and that full liberalization leads to pure redistribution of power between the champion and the customers in a region.
- 5.
See [9] for a detailed review of studies, applying cooperative game theory to the cost allocation problems.
- 6.
For the game with |N| players there are |N|(|N|− 1)∕2 pairs of players, but each player in a pair has to implement the calculation so that in total |N|(|N|− 1) values will be computed. For the groups of k players the total number of computations is \(kc_k^{|N|}\).
- 7.
The analyzed games are not convex, but the core is never empty for the analyzed games.
- 8.
These are the basic parameter settings in [13].
- 9.
The redistributed amount from the full liberalization is equal to the sum of benefits of those players who gain from two steps of reform. For the estimates of redistribution given nucleolus see [13].
- 10.
The only exception is the champion in Netherlands, for the champion in Center-East the two effects are shown as equal due to rounding.
- 11.
In the integrated market only for the champion in Center the calculated maximal value is less than the respective contribution. But the difference between the respective contribution and the maximum is minor.
- 12.
The contribution of the champion in Netherlands is not affected by the first step of reform.
- 13.
Only in the integrated market for the customers in Center the calculated maximal value is slightly less than the respective contribution.
- 14.
The lowest compression of the span within the group of EU champions and customers is equal to 21.6 percentage points and corresponds to the customers in Center-East region.
- 15.
The difference between the center and the nucleolus is larger than 10% only for Belarus, Belgium, Poland and UK. For these players the nucleolus is shifted to the right endpoint of the min-max range.
- 16.
The coalitions are: {Algeria, Turkey & Balkan, Belarus, customers in Center-East, champion in Center-East, customers in Italy, champion in Italy, Russia, Ukraine} and {Turkey & Balkan, Belarus, customers in Center-East, champion in Center-East, customers in Italy, champion in Italy, Libya, Russia, Ukraine}.
- 17.
Minor modifications in the statements might refer to the maximal values. In the basic scenario for a number of maximal values it holds that the value is equal to the respective contribution. With the change of parameters some of these maximal values become slightly less than the respective contributions.
References
Cobanli, O.: Central Asian gas in Eurasian power game. Energy Policy 68, 348–370 (2014)
DG Competition Report on Energy Sector Inquiry. SEC(2006) 1724, Brussels
Directive 98/30/EC of the European Parliament and of the Council of 22 June 1998 concerning common rules for the internal market in natural gas. Official Journal of the European Communities, L204
Directive 2003/55/EC of the European Parliament and of the Council of 26 June 2003 concerning common rules for the internal market in natural gas and repealing Directive 98/30/EC. Official Journal of the European Union, L176
Directive 2009/73/EC of the European Parliament and of the Council of 13 July 2009 concerning common rules for the internal market in natural gas and repealing Directive 2003/55/EC. Official Journal of the European Union, L211(94)
Engevall, S., Göthe-Lundgren, M., Värbrand, P.: The traveling salesman game: An application of cost allocation in a gas and oil company. Ann. Oper. Res. 82, 203–218 (1998)
Engevall, S., Göthe-Lundgren, M., Värbrand, P.: The heterogeneous vehicle-routing game. Transp. Sci. 38(1), 71–85 (2004)
Felsenthal, D.S., Machover, M.: A priori voting power: What is it all about? Polit. Stud. Rev. 2, 1–23 (2004)
Fiestras-Janeiro, M.G., García-Jurado, I., Mosquera, M.A.: Cooperative games and cost allocation problems. Top 19(1), 1–22 (2011)
Hubert F., Cobanli, O.: Pipeline power: A case study of strategic network investments. Rev. Netw. Econ. 14(2), 75–110 (2016)
Hubert, F., Ikonnikova, S.: Investment options and bargaining power in the eurasian supply chain for natural gas. J. Ind. Econ. 59(1), 85–116 (2011)
Hubert, F., Orlova, E.: Competition or countervailing power for the European gas market. Discussion Paper (2014)
Hubert, F., Orlova, E.: Network access and market power. Energy Econ. 76, 170–185 (2018)
Littlechild, S.: A simple expression for the nucleolus in a special case. Int. J. Game Theory 3(1), 21–29 (1974)
Littlechild, S.C., Owen, G.: A simple expression for the Shapley value in a special case. Manag. Sci. 20(3), 370–372 (1973)
Littlechild, S.C., Thompson, G.F.: Aircraft landing fees: A game theory approach. Bell J. Econ. 8(1), 186–204 (1977)
Megiddo, N.: On the nonmonotonicity of the bargaining set, the kernel and the nucleolus of a game. SIAM J. Appl. Math. 27(2), 355–358 (1974)
Montero, M.: On the nucleolus as a power index. Homo Oeconomicus 22(4), 551–567 (2005)
Shapley, L.S., Shubik, M.: A method for evaluating the distribution of power in a committee system. Am. Polit. Sci. Rev. 48(03), 787–792 (1954)
Shapley, L.S., Shubik, M.: Quasi-cores in a monetary economy with nonconvex preferences. Econometrica 34(4), 805–827 (1966)
Shubik, M.: Incentives, decentralized control, the assignment of joint costs and internal pricing. Manag. Sci. 8(3), 325–343 (1962)
Thomas, L.C.: Dividing credit-card costs fairly. IMA J. Manag. Math. 4(1), 19–33 (1992)
Young, H.P.: Monotonic solutions of cooperative games. Int. J. Game Theory 14(2), 65–72 (1985)
Acknowledgements
I am very grateful to my supervisor Prof. Dr. Franz Hubert, without whom this work would be impossible. We thank Johannes H. Reijnierse for providing us with MATLAB code for calculating the nucleolus.
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Appendix
Appendix
As in [13], we assess the robustness of our results by considering three more variants: a high value of demand intercept and the far-sighted scenario, a low value of intercept and the short-sighted scenario, the low value of intercept and the far-sighted scenario. We will discuss the robustness of our results in the same order as they are reported in the main text.
Power Allocation
We start analysis with the comparison of concepts according to the power allocation (see Figs. 12.4, 12.5, 12.6, 12.7, 12.8, 12.9, 12.10, 12.11, 12.12). With minor modifications, all previous statements from the main text could be repeated for each of three scenarios. For example, for all variants of parameters it holds that the Shapley value assigns more power to all outside producers than the nucleolus and the core. It also holds that Belarus and Ukraine have less power in the Shapley value case as compared to the nucleolus. Moreover, for Belarus the shares are less than the respective minimal values in the core. Only for Ukraine it depends on the scenario and the access regime whether the Shapley value falls into the min-max range.
Different solutions for external producers and transit countries (far-sighted scenario, high intercept). The grey bar presents the min-max range of a player in the core. Blue elliptical disks and red circles present the nucleolus and the Shapley values, respectively. All figures are in percentage of the total surplus
Different solutions for EU champions and customers (far-sighted scenario, high intercept)
Different solutions for EU regions (far-sighted scenario, high intercept)
Different solutions for external producers and transit countries (short-sighted scenario, low intercept). The grey bar presents the min-max range of a player in the core. Blue elliptical disks and red circles present the nucleolus and the Shapley values, respectively. All figures are in percentage of the total surplus
Different solutions for EU champions and customers (short-sighted scenario, low intercept)
Different solutions for EU regions (short-sighted scenario, low intercept)
Different solutions for external producers and transit countries (far-sighted scenario, low intercept). The grey bar presents the min-max range of a player in the core. Blue elliptical disks and red circles present the nucleolus and the Shapley values, respectively. All figures are in percentage of the total surplus
Different solutions for EU champions and customers (far-sighted scenario, low intercept)
Different solutions for EU regions (far-sighted scenario, low intercept)
For all scenarios, in the fragmented and integrated markets, the Shapley values of champions and customers belong to the respective min-max ranges. All results for the liberalized market can be repeated. For all scenarios the Shapley value assigns more power to all champions than the nucleolus and the core. All customers have less power under the Shapley value than under the nucleolus and the core.
In case of all variants the Shapley value tends to allocate less power to other EU regions as compared to the nucleolus. For the low value of demand intercept only in half of the cases it holds that the Shapley value does not belong to the respective min-max range.
Liberalization: Compression of the Core
The impact of liberalization on the minimal values, the maximal values and the range is presented in Tables 12.3, 12.4, 12.5. For all scenarios it holds that the total effect on the range is dominated by the second step of reform only for the EU champions and the customers, but by the first step of reform for all other players. For the champions the compression of the range is determined by the decrease of maximal values resulted from the second step of reform. For the customers the compression is determined by the increase of minimal values from the second step. The statements concerning the influence of liberalization on the minimum and the maximum hold for all scenarios.Footnote 17 For example, in the fragmented and integrated markets the minimal values of all customers are determined by the binding individual rationality constraints so that we do not observe any impact of the first step of reform on the minimal values of customers. In contrast, in the liberalized market the individual rationality constraints become non-binding. In other words, the minimal values increase with the second step of reform.
Liberalization: The Nucleolus and the Core
The main results concerning the relation of the nucleolus and the core are robust to changes of parameters. With the first step of reform the nucleolus and the respective midpoint move into the same direction for 60% or 70% of the players, depending on the variant. Within the group of champions and customers such pattern holds only for two or four players. For other players in this group the values shift into the opposite direction or the min-max range is not affected, but the nucleolus changes. For the players outside EU and for the EU regions without champions and customers the values shift into the same direction for all scenarios. In case of the second step of reform for all variants it holds that for all champions and customers the nucleolus is forced to move into the same direction as the respective midpoint. Among other players we find examples when the values shift into the opposite direction. We also find cases when the min-max range is not affected, but the nucleolus changes.
Liberalization: Degree of Instability of the Shapley Value
Results concerning the degree of instability of the Shapley value are robust to changes of parameters (Tables 12.6, 12.7, 12.8). For all scenarios it holds that liberalization increases the instability of the Shapley value. The minimal costs of establishing a coalition in the fully liberalized market are several times larger than the counterparts in the fragmented and integrated markets. Opening of access to pipelines decreases the minimal number of players in a deviating coalition. In the fully liberalized market only two players are enough to veto the Shapley value. The fraction of deviating coalitions raises with each step of reform. The increase realized from the second step of liberalization is larger than from the first step for all scenarios.
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Orlova, E. (2020). Cooperative Solutions for the Eurasian Gas Network. In: Petrosyan, L.A., Mazalov, V.V., Zenkevich, N.A. (eds) Frontiers of Dynamic Games. Static & Dynamic Game Theory: Foundations & Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-51941-4_12
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