# A Note on the Core of TU-cooperative Games with Multiple Membership Externalities

## Abstract

**:**

## 1. Introduction

## 2. A Worked Example

_{1}, …, f

_{n}}, competes in a multimarket industry, K = {1, …, m}, by setting production quantities. Each firm f is described by a vector of specializations, ${s}_{f}=\{{s}_{f}^{1},...,{s}_{f}^{m}\}$, where each ${s}_{f}^{k}$ is a real number representing firm f’s constant marginal costs in market k when no merger occurs.

**Fixed costs of merger.**${x}_{S}^{k}\left(\mathcal{M}\right)$, the fixed cost of merging S in market k ,is a real-valued function that depends on $\mathcal{M}$ in the following way:

**Marginal costs of production.**Given any merger $S\subseteq N$ in market k, the firms in S select the lowest marginal cost firm to be the only active firm amongst them in market k. Hence, the marginal cost of production of S, ${c}_{S}^{k}$, as a result of the merger is given by

**Demands.**The demand of any product is the same in all markets (normalized to be equal-sized). Products are neither substitutes nor complements, meaning that all markets can be described by identical and independent linear demands. (These markets could be countries for example.) For any market k, therefore,

#### 2.1. Oligopoly Externalities

#### 2.2. Two-firm, Two-market Numerical Illustration

Industry configuration | Profits (scaled $\times 81$) | |||
---|---|---|---|---|

market 1 | market 2 | market 1 | market 2 | |

mergers: | ||||

none | (1),(2) | (1),(2) | 4, 1 | 1, 4 |

market 1 | (1,2) | (1),(2) | 4.75 | $(1+\alpha \times 1.\overline{7})$, $(4-\alpha \times 1.\overline{2})$ |

market 2 | (1),(2) | (1,2) | $(4-\alpha \times 1.\overline{2})$, $(1+\alpha \times 1.\overline{7})$ | 4.75 |

full merger | (1,2) | (1,2) | 5.25 | 5.25 |

## 3. The Model

#### 3.1. Externalities

**Partition externality.**

**PCross externality.**

**Partition-cross externality.**

#### 3.2. Feasible Deviations

- Max rule [10]: $(N\backslash S)$, taking $\rho \left(S\right)$ as given, organizes itself to $\rho (N\backslash S)$ in order to maximize $(N\backslash S)$’s total worth
- Optimistic [23]: $(N\backslash S)$ organizes and forms $\rho (N\backslash S)$ in order to maximize S’s total worth
- Projective [20]: all $C\in \rho $ such that $C\cap S=\varnothing $ remain organized in the same way, all other coalitions ${C}^{\prime}$ from which members in S deviated form coalitions amongst the remaining $({C}^{\prime}\backslash S)$

**Feasible deviations.**

**Conjectured worth function.**

#### 3.3. Superadditivity

**Superadditivity:**

## 4. Coalitional Stability and the Core

#### 4.1. Core Stability

**Z-core:**- Given Z, the Z-core of forming the efficient society of $G(v,K,N)$ with total payoff allocation x is$$\zeta (G(v,K,N);Z)=\{x\in {\mathbf{R}}^{n};\phantom{\rule{4pt}{0ex}}\sum _{f\in N}{x}_{f}\le z\left(N\right)\mathrm{and}\sum _{f\in S}{x}_{f}\ge z\left(S\right)\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{4pt}{0ex}}(S\subseteq N)\}.$$
**Theorem.**- The Z-core of $G(v,K,N)$ is nonempty if, and only if, its conjectured worth function z is balanced.

**Characteristic 1:**- If the cores of a superadditive MMG layer-by-layer separately are nonempty, the Z-core of the whole MMGs is also nonempty.While z is always additive over coalitions and layers, v does not need to be additive when externalities are present. In every layer, superadditivity implies that it is beneficial for members of any $S\subseteq N$ to form the largest possible coalition $\left\{S\right\}$. Hence, whenever x is in a Z-core, ${\sum}_{f\in N}{x}_{f}=z\left(N\right)$. Now, ${z}_{k}$ describes the game described by the conjectured worth function of layer k, i.e., the conjectured CFG view of layer k. Given any ${z}_{k}$, a core stable allocation of forming the grand coalition in that layer exists if, and only if, every ${z}_{k}$ is balanced. Since the sum of balanced games is balanced, the Z-core of $G(v,k,N)$ is, therefore forcedly, nonempty when all ${z}_{k}$s are balanced.
**Characteristic 2:**- In the presence of cross externalities but without partition and partition-cross externalities, the core is unambiguously defined (independent of conjecture).In the absence of partition and partition-cross externalities, in a society $\mathcal{M}$ that is separable into ${\mathcal{M}}_{S}$ and ${\mathcal{M}}_{N\backslash S}$, the worth of any $C\subseteq S$ is independent of ${\mathcal{M}}_{N\backslash S}$ in all layers: ${v}_{k}(C;\mathcal{M})={v}_{k}(C;{\mathcal{M}}^{\prime})$ for all coalitions, layers and societies provided ${\mathcal{M}}_{S}={\mathcal{M}}_{S}^{\prime}$, $(C\in {\rho}_{k}\in \mathcal{M})$ and $(C\in {\rho}_{k}^{\prime}\in {\mathcal{M}}^{\prime})$. Therefore, one unique game described by a characteristic worth function is derived, which implies one unambiguous definition of the core. This unambiguity is independent of the existence of cross externalities that are not partition-cross because deviators endogenize all other cross external variations that may still exist and affect them. The need to conjecture is therefore inherent to the presence of PFG-type (partition and partition-cross) externalities. The core of example 1, for instance, is unambiguously defined.
**Characteristic 3:**- In the presence of positive cross externalities, the core of the MMG may be nonempty even if coalition formation in any of the layers is, ceteris paribus, never beneficial.Example 1 as described by Table 1 illustrates this.
**Characteristic 4:**- In the presence of negative cross externalities, the core of forming the grand coalition in any layer of the MMG may be empty even if coalition formation in all layers is, ceteris paribus, always beneficial.

- Example 2:
- Holding the coalition structure of one layer fixed, any coalition formation in the other layer is beneficial. However, due to the negative cross externality of coalition formation in one layer on the other, the total worth of all coalitions is reduced as coalitions form. The core of forming the grand coalition in one or both of the layers of example 2 is empty: $z\left(1\right)+z\left(2\right)=z\left(N\right)=({v}_{1}\left(1\right)+{v}_{2}\left(1\right))+({v}_{1}\left(2\right)+{v}_{2}\left(2\right))=4\times 1=4>3=0+0+3=({v}_{1}\left(1\right)+{v}_{1}\left(2\right))+{v}_{2}\left(N\right)>2=1+1={v}_{1}\left(N\right)+{v}_{2}\left(N\right)$.

Society | Coalition worth | ||

layer 1 | layer 2 | layer 1 | layer 2 |

(1),(2) | (1),(2) | 1, 1 | 1, 1 |

(1,2) | (1),(2) | 3 | 0, 0 |

(1),(2) | (1,2) | 0, 0 | 3 |

(1,2) | (1,2) | 1 | 1 |

**Characteristic 5:**- Multiple membership may facilitate cooperation not because of cross external effects but because the layers “balance each other": Even in the complete absence of externalities when all layers have empty cores, the core of an MMG may be nonempty.(See [13] “Examples 1 and 2” for a 4- and related 5-player examples.)

- Example 3:
- Let $n=5$, $k=2$ and let there be no externalities so that the MMG is described by two 5-player CFGs, ${v}_{1}$ and ${v}_{2}$. Let ${v}_{1}\left(N\right)=1$, ${v}_{1}\left(C\right)=4/5+\epsilon $ (where ε is small) if $\left|C\right|=4$ and ${v}_{1}\left(C\right)=0$ otherwise. Let ${v}_{2}\left(N\right)=1$, ${v}_{2}\left(C\right)=3/5+\epsilon $ if $\left|C\right|=3,4$ and ${v}_{2}\left(C\right)=0$ otherwise.${v}_{1}$ is unbalanced: for the balanced collection of the 5 coalitions of size 4, ${\zeta}_{\left|4\right|}=\{(1,2,3,4),...,(2,3,4,5)\}$, with balancing weights ${\lambda}_{\left|4\right|}=(1/4,...,1/4)$, $5\times 1/4\times {v}_{1}(i,j,k,l)=5\times 1/4\times (4/5+\epsilon )=1+5/4\times \epsilon >1={v}_{1}\left(N\right)$. ${v}_{2}$ is unbalanced: for the balanced collection of the 10 coalitions of size 3, ${\zeta}_{\left|3\right|}^{\prime}=\{(1,2,3),...,(3,4,5)\}$, with balancing weights ${\lambda}_{\left|3\right|}^{\prime}=(1/6,...,1/6)$, $10\times 1/6\times {v}_{2}(i,j,k)=10\times 1/6\times (3/5+\epsilon )=1+5/3\times \epsilon >1={v}_{2}\left(N\right)$. However, it is easy to verify that $x=(2/5,2/5,2/5,2/5,2/5)$ is a core allocation of v: z associates $z\left(N\right)=2$, $z\left(C\right)=7/5+2\times \epsilon $ if $\left|C\right|=4$, $z\left(C\right)=3/5+\epsilon $ if $\left|C\right|=3$ and $z\left(C\right)=0$ otherwise.

**Characteristic 6:**- The presence of positive (or negative) cross and/or partition externalities may lead to inefficient herding.

- Example 4:
- Let $n=3$, $k=2$ and ${v}_{k}(N;\{\left\{N\right\},\left\{N\right\}\})=1$ for all k, ${v}_{k}(1;\{{\rho}_{1},{\rho}_{2}\})=2\phantom{\rule{4pt}{0ex}}\forall i$ if ${\rho}_{1}={\rho}_{2}=\{\left(1\right),(2,3)\}$ and ${v}_{k}(C;M)=0$ otherwise.The Pessimistic-core of forming the inefficient grand coalition in both layers is nonempty because player 1 expects to receive 0 from being the singleton in both layers, e.g., $x=(2/3,2/3,2/3)$ is such a Pessimistic-core allocation. Inefficient herding results from the positive externality: the formation of the coalition of (2,3) in both layers creates worth for player 1, but player 1 is too pessimistic to agree to stay separate. The same effect may be due to negative externalities as a simple variation of v illustrates: consider, for example, ${v}^{\prime}$ with ${v}_{k}^{\prime}(N;\{\left\{N\right\},\left\{N\right\}\})=1$ for all k, ${v}_{k}^{\prime}(1;\{{\rho}_{1},{\rho}_{2}\})=2\phantom{\rule{4pt}{0ex}}\forall k$ if ${\rho}_{1}={\rho}_{2}=\{\left(1\right),\left(2\right),\left(3\right)\}$ and ${v}_{k}^{\prime}(C;M)=0$ otherwise.

## 5. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

- von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior; Princeton University Press: Princeton, NJ, USA, 1944. [Google Scholar]
- Lucas, W.; Thrall, R. n-Person Games in Partition Function Form. Nav. Res. Log.
**1963**, 10, 281–298. [Google Scholar] - Maskin, E. Bargaining, coalitions and externalities. In Presidential Address to the Econometric Society; Institute for Advanced Study: Princeton, NJ, USA, 2003. [Google Scholar]
- Haas, E.B. Why Collaborate? Issue-Linkage and International Regimes. World Polit.
**1980**, 32, 357–405. [Google Scholar] [CrossRef] - Charnovitz, S. Linking Topics in Treaties. Univ. Pa. J. Int. Econ. Law
**1998**, 19, 329–345. [Google Scholar] - Le Breton, M.; Moreno-Ternero, J.D.; Savvateev, A.; Weber, S. Stability and Fairness in Models with a Multiple Membership. Int. J. Game Theory
**2013**, 42, 242–258. [Google Scholar] [CrossRef] - Gillies, D.B. Solutions to general non-zero-sum games. In Contributions to the Theory of Games 4; Kuhn, H., Tucker, A., Eds.; Princeton University Press: Princeton, NJ, USA, 1959; pp. 47–85. [Google Scholar]
- Shapley, L.S. Notes on the n-person game III: Some variants of the von Neumann-Morgenstern definition of solution. In Rand Memorandum; The Rand Corporation: Santa Monica, CA, USA, 1952. [Google Scholar]
- Hafalir, I.E. Efficiency in Coalition Games with Externalities. Games Econ. Behav.
**2007**, 61, 242–258. [Google Scholar] [CrossRef] - Bloch, F.; van den Nouweland, A. Expectation formation rules and the core of partition function games. Games Econ. Behav.
**2014**, in press. [Google Scholar] [CrossRef] - Bondareva, O.N. Applications of Linear Programming Methods to the Theory of Cooperative Games. Probl. Kybern.
**1963**, 10, 119–139. (in Russian). [Google Scholar] - Shapley, L.S. On Balanced Sets and Cores. Nav. Res. Log.
**1967**, 14, 453–460. [Google Scholar] [CrossRef] - Bloch, F.; de Clippel, G. Cores of Combined Games. J. Econ. Theory
**2010**, 145, 2424–2434. [Google Scholar] [CrossRef] - Diamantoudi, E.; Macho-Stadler, I.; Perez-Castrillo, D.; Xue, L. Sharing the surplus in games with externalities within and across issues. In IAE Working Paper 880; Barcelona Graduate School of Economics: Barcelona, Spain, 2011. [Google Scholar]
- Shapley, L.S. A value for n-person games. In Contributions to the Theory of Games 2; Kuhn, H., Tucker, A., Eds.; Princeton University Press: Princeton, NJ, USA, 1953; pp. 307–317. [Google Scholar]
- Tijs, S.; Branzei, R. Additive Stable Solutions on Perfect Cones of Cooperative Games. Int. J. Game Theory
**2002**, 31, 469–474. [Google Scholar] [CrossRef] - Bloch, F. Sequential formation of coalitions in games with externalities and fixed payoff division. Games Econ. Behav.
**1996**, 14, 90–123. [Google Scholar] [CrossRef] - Ray, D.; Vohra, R. A Theory of Endogenous Coalition Structures. Games Econ. Behav.
**1999**, 26, 286–336. [Google Scholar] [CrossRef] - Aumann, R. A survey of cooperative games without side payments. In Essays in Mathematical Economics; Shubik, M., Ed.; Princeton University Press: Princeton, NJ, USA, 1967. [Google Scholar]
- Hart, S.; Kurz, M. Endogenous formation of coalitions. Games Econ. Behav.
**1983**, 51, 1047–1064. [Google Scholar] [CrossRef] - de Clippel, G.; Serrano, R. Marginal contributions and externalities in the value. Econometrica
**2008**, 76, 1413–1436. [Google Scholar] [CrossRef] - Chander, P.; Tulkens, H. The core of an economy with multilateral environmental externalities. Int. J. Game Theory
**1997**, 26, 379–401. [Google Scholar] [CrossRef] - Shenoy, P. On coalition formation: A game theoretical approach. Int. J. Game Theory
**1979**, 8, 133–164. [Google Scholar] [CrossRef]

© 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nax, H.H.
A Note on the Core of TU-cooperative Games with Multiple Membership Externalities. *Games* **2014**, *5*, 191-203.
https://doi.org/10.3390/g5040191

**AMA Style**

Nax HH.
A Note on the Core of TU-cooperative Games with Multiple Membership Externalities. *Games*. 2014; 5(4):191-203.
https://doi.org/10.3390/g5040191

**Chicago/Turabian Style**

Nax, Heinrich H.
2014. "A Note on the Core of TU-cooperative Games with Multiple Membership Externalities" *Games* 5, no. 4: 191-203.
https://doi.org/10.3390/g5040191