# Partial Cooperative Equilibria: Existence and Characterization

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Network Formation with Consent

**Proposition 1**Assume $c\in (1,(n-1)/2]$.

- (i)
- Every Nash equilibrium of Γ induces the empty network.
- (ii)
- For each group $C\subseteq I$ such that $\left|C\right|\ge 2c-1$, there exists a partial cooperative Cournot-Nash equilibrium of $(\Gamma ,C)$ that results in the formation of an efficient nonempty network.

## 4. Axiomatization of the Partial Cooperative Cournot-Nash Equilibrium

#### 4.1. More Definitions

#### 4.2. Axioms

**Group rationality**: For any game $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in {G}^{K}$ such that $\left|I\right|=1$ or $I\subseteq K$, it holds that $\varphi (\Gamma )=\{arg{max}_{x\in X}{\sum}_{i\in I}{f}_{i}\left(x\right)\}$.

**Group consistency**: For each $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in {G}^{K}$ such that $I\backslash K\ne \varnothing $, each $x\in \varphi (\Gamma )$ and each $S\subset I$, $S\ne \varnothing $, ${x}_{S}\in \varphi \left({\Gamma}^{S,x}\right)$.

**Group converse consistency**: For each $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in {G}^{K}$ such that $\left|I\right|\ge 2$, $\tilde{\varphi}(\Gamma )\subseteq \varphi (\Gamma )$

#### 4.3. Characterization

**Theorem 1**The PCE${}^{K}$ correspondence is the unique solution on ${G}^{K}$ that satisfies group rationality, group consistency and group converse consistency.

## 5. Existence in Supermodular Games

**Lemma 1**Consider a game $(\Gamma ,C)$, where $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in G$ and $C\subseteq I$. For each $i\in I$, suppose that ${X}_{i}$ is a compact subset of a Euclidean space and that ${f}_{i}$ is continuous. If the correspondence ${D}^{(\Gamma ,C)}:{X}_{C}\u27f6\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\to {X}_{I\backslash C}$ is non-empty valued, then both an optimistic and a pessimistic partial cooperative equilibrium of $(\Gamma ,C)$ exist.

#### 5.1. Supermodular Games

**Lemma 2**Suppose $f:W\times \Theta \u27f6\mathbb{R}$ is supermodular in $(w,\theta )$. Then,

- (i)
- f is supermodular in w for each fixed θ, i.e., for any fixed $\theta \in \Theta $, and for any w and ${w}^{\prime}$ in W, we have$$f(w,\theta )+f({w}^{\prime},\theta )\le f(w\wedge {w}^{\prime},\theta )+f(w\vee {w}^{\prime},\theta )$$
- (ii)
- f satisfies increasing differences in $(w,\theta )$.

**Lemma 3**Let $V\subset {\mathbb{R}}^{n}$. A function $f:V\u27f6\mathbb{R}$ is supermodular on V if and only if f has increasing differences on V.

- ${X}_{i}$ is a sublattice of some Euclidean space;
- ${f}_{i}(\xb7,{x}_{-i})$ is supermodular on ${X}_{i}$ for each ${x}_{-i}\in {X}_{-i}$;
- ${f}_{i}$ has increasing differences in $({x}_{i},{x}_{-i})$.

**Theorem 2**(Zhou, 1994) Consider a supermodular game $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in G$. If ${X}_{i}$ is compact and ${f}_{i}(\xb7,{x}_{-i})$ is upper semi-continuous on ${X}_{i}$ for each fixed ${x}_{-i}$ and each $i\in I$, then the set of Nash equilibria constitutes a non-empty complete lattice, and hence has a maximum and a minimum.

#### 5.2. The Partial Cooperative Stackelberg Equilibrium

**Theorem 3**Consider a supermodular game $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})$. For each $i\in I$, assume that ${X}_{i}$ is compact and ${f}_{i}$ is continuous. Then, for any $C\subseteq I$, both an optimistic and a pessimistic partial cooperative Stackelberg equilibrium of $(\Gamma ,C)$ exist.

**Proposition 2**Consider a supermodular game $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in G$. For each $i\in I$, let ${X}_{i}$ be compact and ${f}_{i}(\xb7,{x}_{-i})$ be upper semi-continuous on ${X}_{i}$ for each fixed ${x}_{-i}\in {X}_{-i}$. Then, for any $C\subseteq I$, both an optimistic and a pessimistic partial cooperative Stackelberg equilibrium of $(\Gamma ,C)$ exist if either of the following conditions hold:

- (a)
- X is finite.
- (b)
- ${X}_{C}$ is finite and, for all ${y}_{C}\in {X}_{C}$, $F(\xb7,{y}_{C})$ is monotone increasing (or monotone decreasing) in ${X}_{I\backslash C}$, i.e., ${x}_{I\backslash C}\ge {z}_{I\backslash C}$ implies $F({x}_{I\backslash C},{y}_{C})\ge F({z}_{I\backslash C},{y}_{C})$ (or $F({x}_{I\backslash C},{y}_{C})\le F({z}_{I\backslash C},{y}_{C})$

#### 5.3. The Partial Cooperative Cournot-Nash Equilibrium

**Lemma 4**Given $C\subseteq I$, a Nash equilibrium of $\Gamma \left(C\right)$ is a partial cooperative Cournot-Nash equilibrium of $(\Gamma ,C)$.

**Theorem 4**Consider a supermodular game $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in G$. For each $i\in I$, suppose that ${X}_{i}$ is compact and ${f}_{i}$ is upper semi-continuous and supermodular on X. For any $C\subseteq I$, a partial cooperative Cournot-Nash equilibrium of $(\Gamma ,C)$ exists.

## 6. Conclusions

## Acknowledgments

## References

- Mallozzi, L.; Tijs, S. Conflict and Cooperation in Symmetric Potential Games. Int. Game Theor. Rev.
**2008a**, 10, 245–256. [Google Scholar] [CrossRef] - Mallozzi, L.; Tijs, S. Partial Cooperation and Non-Signatories Multiple Decision. AUCO Czech Econ. Rev.
**2008b**, 2, 21–27. [Google Scholar] - Mallozzi, L.; Tijs, S. Stackelberg vs. Nash assumption in IEA Models. J. Opt. Appl. (Forthcoming).
- Ray, D.; Vohra, R. Equilibrium Binding Agreements. J. Econ. Theory
**1997**, 73, 30–78. [Google Scholar] [CrossRef] - Ray, D. A Game Theoretic Perspective of Coalition Formation; Oxford University Press: New York, NY, USA, 2007. [Google Scholar]
- Chakrabarti, S.; Gilles, R.P.; Lazarova, E. Partial Cooperation in Symmetric Games; School of Management and Economics, Queen’s University Belfast: Belfast, UK, 2009; (Working Paper). [Google Scholar]
- Chander, P.; Tulkens, H. A Core of an Economy with Multilateral Environmental Externalities. Int. J. Game Theory
**1997**, 26, 379–401. [Google Scholar] [CrossRef] - Deneckere, R.; Davidson, C. Incentives to Form Coalitions with Bertrand Competition. RAND J. Econ.
**1985**, 16, 473–486. [Google Scholar] [CrossRef] - Duggan, J. Non-Cooperative Games among Groups; Department of Political Science and Department of Economics, University of Rochester: Rochester, NY, USA, 2001; (Working Paper). [Google Scholar]
- Salant, S.W.; Switzer, S.; Reynolds, R.J. Losses from Horizontal Merger: The Effects of Exogenous Change in the Industry Structure on Cournot-Nash Equilibrium. Quart. J. Econ.
**1983**, 98, 184–199. [Google Scholar] [CrossRef] - Beaudry, P.; Cahuc, P.; Kempf, H. Is it Harmful to Allow Partial Cooperation. Scand. J. Econ.
**2000**, 102, 1–21. [Google Scholar] [CrossRef] - Carraro, C; Siniscalco, D. The International Dimension of Environmental Policy. Eur. Econ. Rev.
**1992**, 36, 379–387. [Google Scholar] - Carraro, C.; Siniscalco, D. Strategies for International Protection of the Environment. J. Public Econ.
**1993**, 52, 309–328. [Google Scholar] [CrossRef] - Funaki, Y.; Yamato, T. The Core of an Economy with a Common Pool Resource: A Partition Function Form Approach. Int. J. Game Theory
**1999**, 28, 157–171. [Google Scholar] [CrossRef] - Barrett, S. Self-Enforcing International Environmental Agreements. Oxford Econ. Pap.
**1994**, 46, 804–878. [Google Scholar] - D’Aspremont, C.; Jacquemin, A.; Gabszewicz, J.J.; Weymark, J.A. On the Stability of Collusive Price Leadership. Can. J. Econ.
**1983**, 16, 17–25. [Google Scholar] - Diamantoudi, E.; Sartzetakis, E.S. Stable International Environmental Agreements: An Analytical Approach. J. Public Econ. Theory
**2006**, 8, 247–263. [Google Scholar] [CrossRef] - Peleg, B.; Tijs, S. The Consistency Principle for Games in Strategic Form. Int. J. Game Theory
**1996**, 25, 13–34. [Google Scholar] [CrossRef] - Jackson, M.O.; Wilkie, S. Endogenous Games and Mechanisms: Side Payments Among Players. Rev. Econ. Stud.
**2005**, 72, 543–566. [Google Scholar] [CrossRef] - Kalai, A.D.; Kalai, E. Engineering Cooperation in Two-Player strategic Games. Microsoft Research and Kellogg School of Management, Northwestern University: Evanston, IL, USA, 2010; (Working Paper). [Google Scholar]
- Topkis, D.M. Equilibrium Points in Nonzero-sum Submodular Games. SIAM J. Contr. Optimizat.
**1979**, 17, 773–787. [Google Scholar] [CrossRef] - Milgrom, P.; Roberts, J. Rationalizability, Learning and Equilibrium in Games with Strategic Complementarities. Econometrica
**1990**, 58, 1255–1277. [Google Scholar] [CrossRef] - Vives, X. Nash Equilibrium with Strategic Complementarities. J. Math. Econ.
**1990**, 19, 305–329. [Google Scholar] [CrossRef] - Zhou, L. The Set of Nash Equilibria of a Supermodular Game is a Complete Lattice. Game. Econ. Behav.
**1994**, 7, 295–300. [Google Scholar] [CrossRef] - Nash, J.F. Equilibrium Points in N-person Games. Proc. Nat. Acad. Sci.
**1950**, 36, 48–49. [Google Scholar] [CrossRef] [PubMed] - Myerson, R.B. Game Theory: Analysis of Conflict; Harvard University Press: Cambridge, MA, USA, 1991. [Google Scholar]
- Jackson, M.O.; Wolinsky, A. A Strategic Model of Social and Economic Networks. J. Econ. Theory
**1996**, 71, 44–74. [Google Scholar] [CrossRef] - Béal, S.; Quérou, N. Bounded Rationality in Repeated Network Formation. Math. Soc. Sci.
**2007**, 54, 71–89. [Google Scholar] [CrossRef] - Sundaram, R.K. A First Course in Optimization Theory; Cambridge University Press: New York, NY, USA, 1996. [Google Scholar]
- Berge, C. Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity, 1963; reprinted by Dover Publication, Inc.: Mineola, NY, USA, 1997. [Google Scholar]
- Moore, J.C. Mathematical Methods for Economic Theory 2; Springer-Verlag: Heidelberg, Germany, 1999. [Google Scholar]

## Appendix

#### Proof of Proposition 1

**Case (a).**If $\left|C\right|=n$, the result is immediate since the line network is an efficient network by assumption $c\in (1,(n-1)/2]$.

**Case (b).**If $\left|C\right|\le n-2$, let $C=\{2,3,\cdots ,\lfloor |C|/2\rfloor +2,n-\lfloor |C|/2\rfloor ,\cdots ,n-1\}$ if $\left|C\right|$ is odd or $C=\{2,\cdots ,|C|/2+1,n-|C|/2,\cdots ,n-1\}$ if $\left|C\right|$ is even. Consider the strategy profile x such that

**Case (c).**If $\left|C\right|=n-1$, the proof is similar except that one of the two leaves is added to the group.

#### Proof of Lemma 1

**Proposition 3**Consider a non-cooperative game $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in G$ such that, for each $i\in I$, ${X}_{i}$ is a compact subset of a Euclidean space and ${f}_{i}$ is continuous. For any $C\subseteq I$, if the correspondence ${D}^{(\Gamma ,C)}:{X}_{C}\u27f6\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\to {X}_{I\backslash C}$ is upper hemi-continuous and compact-valued.

**Proposition 4**Consider a non-cooperative game $\Gamma =(I,{({X}_{i},{f}_{i})}_{i\in I})\in G$ such that, for each $i\in I$, ${X}_{i}$ is a compact subset of a Euclidean space and ${f}_{i}$ is continuous. For any $C\subseteq I$, functions ${F}^{-}$ and ${F}^{+}$ are well-defined.

**Theorem 5**(Berge, 1963) Let $\varphi :A\u27f6\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\to B$ be an upper hemi-continuous and non-empty valued correspondence and $\psi :A\times B\u27f6\mathbb{R}$ an upper semi-continuous function. Then the function $M:A\u27f6\mathbb{R}$ defined as

^{1.}See also Ray and Vohra [4].^{2.}Note that $(I\backslash S)\cap K=(I\cap K)\backslash (S\cap K)$.^{3.}For completeness, note that the group can simultaneously delete and add links between its members. However any such change in the network configuration cannot improve the value of the network for the group.

© 2010 by the authors; licensee MDPI, 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/3.0/).

## Share and Cite

**MDPI and ACS Style**

Béal, S.; Chakrabarti, S.; Ghintran, A.; Solal, P. Partial Cooperative Equilibria: Existence and Characterization. *Games* **2010**, *1*, 338-356.
https://doi.org/10.3390/g1030338

**AMA Style**

Béal S, Chakrabarti S, Ghintran A, Solal P. Partial Cooperative Equilibria: Existence and Characterization. *Games*. 2010; 1(3):338-356.
https://doi.org/10.3390/g1030338

**Chicago/Turabian Style**

Béal, Sylvain, Subhadip Chakrabarti, Amandine Ghintran, and Philippe Solal. 2010. "Partial Cooperative Equilibria: Existence and Characterization" *Games* 1, no. 3: 338-356.
https://doi.org/10.3390/g1030338