# Cartel Formation in Cournot Competition with Asymmetric Costs: A Partition Function Approach

## Abstract

**:**

## 1. Introduction

- We consider stable coalition structures. Although we will elaborate later, a coalition structure is a partition of a player set. Therefore, our analysis includes a situation where multiple cartels can coexist simultaneously.
- We use the concept of a core allocation to define the stability of a coalition structure. For each coalition structure, we assume that the members of each cartel divide their profit among themselves. We say a coalition structure is stable if there exists a feasible core allocation in the coalition structure.
- We introduce asymmetric costs and attempt to solve the so-called “merger paradox”. More specifically, we consider that a firm obtains new technology and a cost advantage $\u03f5\ge 0$ that allow the firm to produce goods at a lower unit cost. We perform comparative statics for $\u03f5$.

## 2. Preliminaries

**Proposition**

**1.**

## 3. Cartel Formation

**Proposition**

**2.**

**Proposition**

**3.**

- As shown in the area $\beta $, if the level of the cost advantage is “moderate”, firm 1 obtains an incentive to lead coalition formation. Moreover, as long as $\u03f5$ retains the moderate level, each merger benefits all participants.
- The area $\gamma $ shows that if the cost advantage is slight and if there are three or more coalitions, then such an advantage $\u03f5$ is too small for firm 1 to lead cartel formation. In this case, the symmetric setting and the asymmetric setting are hardly different.
- In contrast, if the cost advantage is very large as the area $\alpha $ describes, then such “strong” firm 1 no longer needs any cartel. Firm 1 has an incentive to be alone and produce goods by taking full advantage of the very low unit cost.

**Example**

**1.**

**Corollary**

**1.**

## 4. Cartel Stability

**Proposition**

**4.**

- As long as the cost advantage $\u03f5$ is in the “moderate” interval $({\eta}^{-1}\left(n\right),1)$, Corollary 1 suggests that firm 1 leads coalition formation and reaches the grand coalition N.
- Moreover, if $\u03f5$ lies in the particular interval $[h\left(n\right),1)\subseteq ({\eta}^{-1}\left(n\right),1)$, then the grand coalition that is achieved is stable. No coalition has an incentive to deviate from the grand coalition.
- However, if $\u03f5$ is in $({\eta}^{-1}\left(n\right),h\left(n\right))$, the formation of the grand coalition is transient because every allocation in the grand coalition is not a core allocation. Some coalitions endogenously deviate from the grand coalition.

## 5. The Spread of Technology

**Proposition**

**5.**

- (i)
- There is $r\in [0,1]$ such that ${v}_{1}(\u03f5,\rho -1)\ge {v}_{1}^{\prime}(\u03f5,\rho ,r)+{v}_{2}^{\prime}(\u03f5,\rho ,r)$
- (ii)
- It holds that $\u03f5\ge {\eta}^{-1}\left(\rho \right)$.

## 6. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Proposition 1

**Proof.**

#### Appendix A.2. Proof of Proposition 2

**Proof.**

#### Appendix A.3. Proof of Proposition 3

**Proof.**

#### Appendix A.4. Proof of Proposition 4

**Proof.**

#### Appendix A.5. Proof of Proposition 5

**Proof.**

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1. | Ref. [7] discusses the merger paradox in a simple framework: In their model, a merger of firms, in most cases, generates a decline of the joint profits, while, in our settings that are formally introduced later, we assume that a firm obtains cost advantage, and the cost asymmetry gives the firm an incentive to merge with other firms. Moreover, asymmetric settings are also studied by Ref. [8]. They consider a mixed oligopoly that consists of private firms and a public firm. They show that there is a merger between a private firm and the public firm that may benefit the members of the merger. |

2. | For example, in partition $\left\{\right\{1\},\{2\},\{3\left\}\right\}$, firm 1 with the lower unit cost ${c}_{1}$ and firms 2 and 3 with the standard unit cost c compete under the same inverse demand function. |

3. | There is a profit distribution that improves both firms’ profits. |

4. | Note that $v(\u03f5,\rho )$ means the profit of a coalition with the standard cost. |

5. | For the theoretical studies of the core of a partition function form game, see [9,10,11,12,13]. In addition to the core notions for partition function form games, Ref. [14] initially introduced the notions of $\alpha $-core and $\beta $-core. Ref. [15] provides a detailed summary of these core concepts. |

6. | One can define a deviation in another way: ${y}_{j}\ge {x}_{j}$ for every $j\in S$ and ${y}_{i}>{x}_{i}$ for some $i\in S$. These two definitions make no difference in the following results. |

7. | For easy reference, we provide the list of profits for $\u03f5=\frac{1}{4}$. Below, ${S}_{1}$ means the coalition that contains firm 1, and S means a coalition without firm 1: ${v}^{\u03f5}(N,{\mathcal{P}}_{1})=\frac{25}{64}$; ${v}^{\u03f5}({S}_{1},{\mathcal{P}}_{2C})=\frac{1}{4}$, ${v}^{\u03f5}(S,{\mathcal{P}}_{2C})=\frac{1}{16}$; ${v}^{\u03f5}({S}_{1},{\mathcal{P}}_{3B})=\frac{49}{256}$, ${v}^{\u03f5}(S,{\mathcal{P}}_{3B})=\frac{9}{256}$; ${v}^{\u03f5}({S}_{1},{\mathcal{P}}_{4})=\frac{4}{25}$, ${v}^{\u03f5}(S,{\mathcal{P}}_{4})=\frac{9}{400}$. |

8. | Below is the list of profits for $\u03f5=\frac{1}{2}$: ${v}^{\u03f5}(N,{\mathcal{P}}_{1})=\frac{9}{16}$; ${v}^{\u03f5}({S}_{1},{\mathcal{P}}_{2C})=\frac{4}{9}$, ${v}^{\u03f5}(S,{\mathcal{P}}_{2C})=\frac{1}{36}$; ${v}^{\u03f5}({S}_{1},{\mathcal{P}}_{3B})=\frac{25}{64}$, ${v}^{\u03f5}(S,{\mathcal{P}}_{3B})=\frac{1}{64}$; ${v}^{\u03f5}({S}_{1},{\mathcal{P}}_{4})=\frac{9}{25}$, ${v}^{\u03f5}(S,{\mathcal{P}}_{4})=\frac{1}{100}$. |

9. | The denominator 14,400 is the LCM of 16, 9, 36, 64, 25, and 100. |

10. | For a detailed discussion of population monotonicity, see the following papers. A sequence of population monotonic allocations is called a population monotonic allocation scheme (PMAS). This concept was formally introduced by [16], and Ref. [17] studied conditions for a partition function form game to have a PMAS. Ref. [18] weakens some restrictions of PMAS and propose a monotonic core allocation path (MCAP). |

11. | To be more precise, as elaborated in (A7) in the Appendix A, r is “small” if it is less than $\frac{\u03f5-1}{\u03f5\rho}$, and “high enough” if it lies between $\frac{\u03f5-1}{\u03f5\rho}$ and $\frac{(3\u03f5-1){\rho}^{2}+2\rho +(1-\u03f5)}{\u03f5\rho ({\rho}^{2}+1)}$, and “very high” if it exceeds $\frac{(3\u03f5-1){\rho}^{2}+2\rho +(1-\u03f5)}{\u03f5\rho ({\rho}^{2}+1)}$. |

Partition | {{1},{2},{3}} | {{1,j},{k}} | {{2,3},{1}} | $\left\{\right\{1,2,3\left\}\right\}$ | ||||
---|---|---|---|---|---|---|---|---|

Coalition | $\left\{1\right\}$ | $\left\{2\right\}$ | $\left\{3\right\}$ | $\{1,j\}$ | $\left\{k\right\}$ | $\{2,3\}$ | $\left\{1\right\}$ | $\{1,2,3\}$ |

Symmetric costs | $\frac{1}{16}$ | $\frac{1}{16}$ | $\frac{1}{16}$ | $\frac{1}{9}$ | $\frac{1}{9}$ | $\frac{1}{9}$ | $\frac{1}{9}$ | $\frac{1}{4}$ |

Cost advantage $\u03f5$ | $\frac{{(1+3\u03f5)}^{2}}{16}$ | $\frac{{(1-\u03f5)}^{2}}{16}$ | $\frac{{(1-\u03f5)}^{2}}{16}$ | $\frac{{(1+2\u03f5)}^{2}}{9}$ | $\frac{{(1-\u03f5)}^{2}}{9}$ | $\frac{{(1-\u03f5)}^{2}}{9}$ | $\frac{{(1+2\u03f5)}^{2}}{9}$ | $\frac{{(1+\u03f5)}^{2}}{4}$ |

$2\le \mathit{\rho}\le \mathit{n}$ | |||
---|---|---|---|

$1<\u03f5$ | negative | ||

$\u03f5=1$ | zero | ||

$1/3\le \u03f5<1$ | positive | ||

$0\le \u03f5<1/3$ | positive | zero | negative |

if $2\le \rho <\eta \left(\u03f5\right)$ | if $\rho =\eta \left(\u03f5\right)$ | if $\eta \left(\u03f5\right)<\rho \le n$ |

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Abe, T.
Cartel Formation in Cournot Competition with Asymmetric Costs: A Partition Function Approach. *Games* **2021**, *12*, 14.
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Abe T.
Cartel Formation in Cournot Competition with Asymmetric Costs: A Partition Function Approach. *Games*. 2021; 12(1):14.
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Abe, Takaaki.
2021. "Cartel Formation in Cournot Competition with Asymmetric Costs: A Partition Function Approach" *Games* 12, no. 1: 14.
https://doi.org/10.3390/g12010014