# Optimal Deterrence of Cooperation

Stéphane Gonzalez, Aymeric Lardon

► To cite this version:

Stéphane Gonzalez, Aymeric Lardon. Optimal Deterrence of Cooperation. International Journal of Game Theory, 2018, 47 (1), pp.207-227. 10.1007/s00182-017-0584-8. halshs-01660957

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WP 1621 – June 2016

Optimal Deterrence of Cooperation

Stéphane Gonzalez, Aymeric Lardon

**Abstract:**

We introduce axiomatically a new solution concept for cooperative games with transferable utility inspired by the core. While core solution concepts have investigated the sustainability of cooperation among players, our solution concept, called contraction core, focuses on the deterrence of cooperation. The main interest of the contraction core is to provide a monetary measure of the robustness of cooperation into the grand coalition. We motivate this concept by providing optimal fine imposed by competition authorities for the dismantling of cartels in oligopolistic markets. We characterize the contraction core on the set of balanced cooperative games with transferable utility by four axioms: the two classic axioms of non-emptiness and individual rationality, a superadditivity principle and a new axiom of consistency.

**Keywords:**

TU-game, contraction core, optimal fine, Cournot oligopoly, axiomatization

**JEL codes:**

C71, D43
# Optimal Deterrence of Cooperation*

Stéphane Gonzalez† Aymeric Lardon‡

June 14, 2016

## Abstract

We introduce axiomatically a new solution concept for cooperative games with transferable utility inspired by the core. While core solution concepts have investigated the sustainability of cooperation among players, our solution concept, called contraction core, focuses on the deterrence of cooperation. The main interest of the contraction core is to provide a monetary measure of the robustness of cooperation into the grand coalition. We motivate this concept by providing optimal fine imposed by competition authorities for the dismantling of cartels in oligopolistic markets. We characterize the contraction core on the set of balanced cooperative games with transferable utility by four axioms: the two classic axioms of non-emptiness and individual rationality, a superadditivity principle and a new axiom of consistency.

**Keywords:** TU-game, contraction core, optimal fine, Cournot oligopoly, axiomatization

**JEL Classification:** C71, D43

## 1 Introduction

One of the main issues in cooperative game theory concerns the possibility for players to cooperate all together. A well-known solution concept for cooperative games with transferable utility (henceforth TU-games) dealing with the existence of stable cooperative agreements is the core (Gillies, 1953). The classic Bondareva-Shapley theorem establishes that the non-emptiness of the core is characterized by the balancedness property as proved independently in Bondareva (1963) and Shapley (1967). A possible interpretation of the balancedness property which we are interest in is the following: each player must distribute one unit of time among all the coalitions of which she is a member; the balancedness property stipulates that the optimal time allocation for players is to devote all their unit of time into the grand coalition, i.e., the whole set of players.

Even in the case where the core is empty, the literature has investigated the possibility to enforce a stable cooperative agreement by introducing other core solution concepts: the strong and the weak $\epsilon$-cores (Shapley and Shubik, 1966), the least core (Maschler et al., 1979), the aspiration

<footnote>*This work was supported by the Programme Avenir Lyon Saint-Etienne of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007).</footnote>

<footnote>†Univ Lyon, UJM Saint-Etienne, CNRS, GATE L-SE UMR 5824, F-42023 Saint-Etienne, France. e-mail: stephane.gonzalez@univ-st-etienne.fr</footnote>

<footnote>‡Corresponding author. University of Nice Sophia-Antipolis, CNRS, GREDEG UMR 7321, F-06560 Valbonne, France. e-mail: aymeric.lardon@unice.fr</footnote>
core (Albers, 1979; Cross, 1967; Bennett, 1983), the extended core (Bejan and Gómez, 2009), the negociation set (Gonzalez and Grabisch, 2015b) and the d-multicoalitional core (Gonzalez and Grabisch, 2016). All these core extensions are non-empty when applied to non-balanced TU-games and coincide with the core on the set of balanced TU-games.

Until now, solution concepts inspired by the core have restricted attention to the sustainability of cooperation. Nevertheless, in many competitive environments, cooperation is not socially desirable, and players must be discouraged to work all together. For example, horizontal agreements on prices between firms are punished by competition authorities. Similarly, drug cartels are reproved to protect population. In the same vein, the dismantling of terrorist groups appears to be of primary importance for national security. Furthermore, to the best of our knowledge, even when cooperation is efficient, the robustness of stable cooperative agreements has not been studied yet. For example, how much the cohesion of collaborative activities on research and development is sensible to discovery values? Or does the stability of trade agreements depend crucially on transportation costs? To meet these challenges head on, a general solution concept spanning several fields of economics (industrial organization, innovation, international trade, criminology...) appears fundamental in order to provide insight into the deterrence of cooperation.

In this article, we investigate the deterrence of cooperation among players for balanced cooperative TU-games by imposing monetary penalty on the grand coalition. Precisely, we are interested in finding the minimal amount of fine, called the optimal fine, under which cooperation can no longer be sustained. This leads us to consider a new solution concept, called contraction core, which contains all stable cooperative agreements for which any fine increase makes these agreements unstable. In this sense, the contraction core contains all the “weakest” stable cooperative agreements further to the optimal fine imposed on the grand coalition. This fine can be interpreted as a measure of the robustness of cooperation into the grand coalition. In terms of time allocation, this means that authority deters the formation of the grand coalition and that players must devote fractions of their unit of time to any other coalition as a second best time allocation. This notion will be used for the definition of feasibility and efficiency conditions related to our solution concept. Unlike the core solution concepts mentioned above, the contraction core does not contain the core, and so it is not a core extension. Moreover, the contraction core has the advantage of being a singleton on the set of balanced and symmetric TU-games.

Following in the footsteps of previous works (Trockel, 2005; Moulin, 2014) which deal with microeconomics by using cooperative concepts, we propose an illustrative example of oligopolistic markets in order to motivate our solution concept. In economic welfare analysis, it is a well-established and old idea that monopoly power often negatively affects social welfare. Although cooperation on research and development activities may have beneficial welfare effects (D'Aspremont and Jacquemin, 1988), most of horizontal agreements on sales prices are considered as harmful to social welfare. The cooperative approach of oligopoly games is of great interest in order to analyze the stability of cartels which are one of the main preoccupations of competition authorities.<fnref n="1" /> We point out that our analysis does not pay attention to the welfare effects of trade restriction as advocated by the rule of reason in antitrust law, but focuses on the deterrence of the monopoly power which leads, a priori, to welfare losses. Thus, the contraction core constitutes an effective tool to

<footnote n="1">¹The developing theory of oligopoly TU-games comprises many contributions such as Zhao (1999), Norde et al. (2002), Driessen and Meinhardt (2005), Lardon (2012) and Lekeas and Stamatopoulos (2014) among others.</footnote>
prevent the formation of cartel. Precisely, we consider the set of Cournot oligopoly TU-games in
γ-characteristic function form (Hart and Kurz, 1983) which is plausible in the context of oligopoly
industries. Under this approach, the worth of any coalition (the cartel profit) is enforced by a
competition setting in which any cartel faces external firms acting individually. We assume that
the inverse demand function is linear and firms operate at constant and identical marginal costs.
These assumptions ensure that the balancedness property holds on this set of Cournot oligopoly
TU-games as shown by Lardon (2012), and so the contraction core is well-defined. After having
determined the worth of any coalition, we compute the contraction core and provide an expression
of the optimal fine imposed by competition authorities in order to deter the grand coalition which
corresponds, in the present case, to the cartel comprising all the firms. Surprisingly, this expression
differs depending on the number of firms and leads to distinguish markets of small size (less than
five firms) and those of medium and large size (more than six firms).

Beyond this economic application, in order to get a better grasp of the contraction core, we provide
an axiomatic characterization of this new solution concept on the set of balanced TU-games. We
invoke the two classic axioms of non-emptiness and individual rationality as well as two new other
axioms of superadditivity and consistency. The original superadditivity and consistency properties
(Peleg, 1986) used to characterize the core, implicitly depend on grand coalition feasibility. We
replace them with similar properties based on a new definition of feasibility derived from non-
trivial coalition formation which relies on second best time allocation for players.<fnref n="2" /> We impose this
feasibility requirement on our superadditivity principle. Consistency principle is based on an appro-
priated reduced games property. Traditional reduced games (Davis and Maschler, 1965) used by
Peleg (1986) make an exception to the grand coalition in order to ensure grand coalition feasibility.
Bejan and Gómez (2012) use a more general version (Moldovanu and Winter, 1994) that treats
all coalitions in the same way. Our new axiom of consistency is based on a new modified version
of reduced games which make again an exception to the grand coalition. Precisely, unlike any
other coalition, the grand coalition of any reduced game is not allowed to cooperate with the com-
plementary coalition. Moreover, given any second best time allocation in the original TU-game,
we provide a formula to compute the corresponding second best time allocation in any reduced
game. Our axioms of superadditivity and consistency do not coincide with those of Peleg (1986)
and their generalized versions in Bejan and Gómez (2012) on the set of balanced TU-games.

The article is organized as follows. Section 2 presents the contraction core as well as some of its
properties. Section 3 gives an illustrative example of oligopolistic markets for the deterrence of
the monopoly power. In Section 4, we provide an axiomatic characterization of the contraction
core. Section 5 deals with a natural extension of the contraction core to the set of all TU-games.

2 Cooperatives games and the contraction core

2.1 Cooperatives games with transferable utility

A cooperative TU-game is an ordered pair (N, v) consisting of a finite set of players N and a
characteristic function v : 2<sup>N</sup> —> R such that v(Ø) = 0 where 2<sup>N</sup> denotes the power set of
N. Subsets of N are called coalitions, and we call v(S) the worth of coalition S. The size of

<footnote n="2">²Bejan and Gómez (2012) use a more relaxed feasibility condition based on first best time allocation.</footnote>
coalition S is denoted by $s = |S|$. Let $\Gamma$ denote the set of TU-games.

Later in the paper, we will use both simple and symmetric TU-games. A TU-game $(N, v)$ is **simple** if for any coalition $S \in 2^N \setminus \{\emptyset, N\}$, we have $v(S) \in \{0, 1\}$ and $v(N) = 1$. A coalition $S$ such that $v(S) = 1$ is called a **winning coalition**. A player $i \in N$ is called a **veto player** if she belongs to any winning coalition. A TU-game $(N, v)$ is **symmetric** if there exists a mapping $f: \mathbb{N} \to \mathbb{R}$ such that for any coalition $S \in 2^N \setminus \{\emptyset\}$, we have $v(S) = f(s)$.

Let $\mathcal{B} \subseteq 2^N \setminus \{\emptyset\}$ be a collection of coalitions. Then $\mathcal{B}$ is said to be a **balanced collection of coalitions** if for every $S \in \mathcal{B}$ there exists a balancing weight $\delta_S \in \mathbb{R}_+$ such that $\sum_{S \in \mathcal{B}: i \in S} \delta_S = 1$ for every $i \in N$. Consider $\delta_S$ as an amount of time allocated to coalition $S$ by any of its members. When each player has one unit of time, the requirement that $\sum_{S \in \mathcal{B}: i \in S} \delta_S = 1$ is then a time feasibility condition. We denote by $\Lambda(N)$ the set of balanced collections and $\Lambda^*(N)$ the set of balanced collections not containing the grand coalition where $n \ge 2$. By convention, $\Lambda^*(N) = \Lambda(N)$ when $n = 1$. A TU-game $(N, v)$ is balanced if for every balanced collection $\mathcal{B} \in \Lambda(N)$ it holds that $\sum_{S \in \mathcal{B}} \delta_S v(S) \le v(N)$. Let $\Gamma_c$ denote the subset of balanced TU-games. On the set $\Gamma_c$ the best time allocation for players is to devote all their unit of time to the grand coalition.

## 2.2 Feasibility as second best time allocation

We now introduce the appropriate notion of feasibility which will be useful for the definition of the contraction core. In a TU-game $(N, v) \in \Gamma$, every player $i \in N$ may receive a **payoff** $x_i \in \mathbb{R}$. A vector $x \in \mathbb{R}^n$ is a **payoff vector**. For any coalition $S \in 2^N \setminus \{\emptyset\}$ and any payoff vector $x \in \mathbb{R}^n$, we define $x(S) = \sum_{i \in S} x_i$ and we denote by $x^S \in \mathbb{R}^s$ the vector such that $x_i^S = x_i$ for all $i \in S$.

Generally speaking, feasibility is a restriction on players' payoffs and can be interpreted in terms of time allocation. The classic feasibility condition, called the grand coalition feasibility, is defined as the set of payoff vectors, denoted by $X(N, v)$, that are feasible when players allocate their unit of time to the grand coalition, i.e.:

$$X(N, v) = \{x \in \mathbb{R}^n : x(N) \le v(N)\}.$$

A more relaxed feasibility condition which considers non-trivial coalition formation, is defined as the set of payoff vectors, denoted by $X_\Lambda(N, v)$, that are feasible when players can devote fractions of their time to any coalition, not just the grand coalition, i.e.:

$$X_\Lambda(N, v) = \{x \in \mathbb{R}^n : x(N) \le \sum_{S \in \mathcal{B}} \delta_S v(S) \text{ for some } \mathcal{B} \in \Lambda(N)\}.$$

On the set $\Gamma_c$, both conditions of feasibility are equivalent since the best time allocation for players is to form the grand coalition. Now, suppose authority prevents the formation of the grand coalition. The feasibility condition which we are interested in becomes some possible arrangements of players devoting fractions of their time to any coalition except the grand coalition.

**Definition 2.1** For any TU-game $(N, v) \in \Gamma_c$, the set of feasible payoff vectors of $(N, v)$, denoted by $X_{\Lambda^*}(N, v)$, is defined as:
$$X_{\Lambda^*}(N, v) = \{x \in \mathbb{R}^n : x(N) \le \sum_{S \in \mathcal{B}} \delta_S v(S) \text{ for some } \mathcal{B} \in \Lambda^*(N)\},$$

On the set $\Gamma_c$, this feasibility condition relies on coalition formation of players as second best time allocation. This leads to define the associated efficiency condition.

**Definition 2.2** For any TU-game $(N, v) \in \Gamma_c$, the set of efficient payoff vectors of $(N, v)$, denoted by $X_{\Lambda^*}^*(N, v)$, is defined as:

$$X_{\Lambda^*}^*(N, v) = \arg\max\{x(N) : x \in X_{\Lambda^*}^*(N, v)\}.$$

On the set $\Gamma_c$, any efficient payoff vector is exactly achieved by a second best time allocation for players.

## 2.3 Contraction core

The new feasibility and efficient conditions related to second best time allocation permit to define the main object of our study on the set $\Gamma_c$, the contraction core.

**Definition 2.3** For any TU-game $(N, v) \in \Gamma_c$, the contraction core, denoted by $CC(N, v)$, is defined as:

$$CC(N, v) = \{x \in X_{\Lambda^*}^*(N, v) : \forall S \subset N, x(S) \ge v(S)\}.$$

The contraction core contains all efficient payoff vectors<fnref n="3" /> achieved by any second best time allocation that satisfy a relaxed coalitional stability condition for which the grand coalition is not taken into account.

The following are the definitions of the core and the aspiration core for which we will make comparisons with the contraction core.<fnref n="4" /> The core (Gillies, 1953) of a TU-game $(N, v) \in \Gamma$, denoted by $C(N, v)$, is defined as:

$$C(N, v) = \{x \in X(N, v) : \forall S \subseteq N, x(S) \ge v(S)\}.$$

Bondareva (1963) and Shapley (1967) showed that any TU-game $(N, v) \in \Gamma_c$ if and only if $C(N, v) \ne \emptyset$. The aspiration core (Albers, 1979; Cross, 1967; Bennett, 1983) of a TU-game $(N, v) \in \Gamma$, denoted by $AC(N, v)$, is defined as:

$$AC(N, v) = \{x \in X_{\Lambda}(N, v) : \forall S \subseteq N, x(S) \ge v(S)\}.$$

Both the core and the aspiration core contain all feasible payoff vectors (with the understanding that we consider grand coalition feasibility for the former and feasibility as first best time allocation for the latter) that satisfy the classic coalitional stability condition.

<footnote n="3">³We need to use the set of efficient payoff vectors in the definition of the contraction core in order to deal with the one-player case.</footnote>

<footnote n="4">⁴While the contraction core is defined on the subset $\Gamma_c$, the core and the aspiration core are defined on the set $\Gamma$.</footnote>
## 2.4 Deterrence of cooperation

We now show that the contraction core is relevant in order to deal with the deterrence of cooperation. Given any TU-game $(N, v) \in \Gamma_c$ and any $t \in \mathbb{R}_+$, its **t-contraction** is the TU-game, denoted by $(N, v^t)$, such that $v^t(S) = v(S)$ for any $S \subset N$, and $v^t(N) = v(N) - t$. In particular, we assign real number $\underline{t}(N, v)$, called the **optimal fine**, to any TU-game $(N, v) \in \Gamma_c$, which is defined as:

$$ \underline{t}(N, v) = \begin{cases} \inf\{t \in \mathbb{R} : \forall k > t, (N, v^k) \text{ is not balanced}\} & \text{if } n \ge 2; \\ 0 & \text{if } n = 1. \end{cases} $$

The $\underline{t}(N, v)$-contraction corresponds to the original TU-game $(N, v)$ for which the grand coalition must pay optimal fine $\underline{t}(N, v)$. This optimal fine gives the minimal amount for which any fine increase makes cooperation into the grand coalition unstable. It can be considered as a measure of the robustness of stable cooperative agreements.<fnref n="5" /> An alternative formula of the optimal fine easier to compute is the following:

$$ \underline{t}(N, v) = v(N) - \max_{B \in \Lambda^*(N)} \sum_{S \in B} \delta_S v(S). $$

We show that the contraction core of any TU-game $(N, v) \in \Gamma_c$ is equal to the core of its $\underline{t}(N, v)$-contraction.

**Proposition 2.4** For any TU-game $(N, v) \in \Gamma_c$, it holds that $CC(N, v) = C(N, v\underline{t}(N,v))$.

**Proof:** First, we prove that $C(N, v\underline{t}(N,v)) \subseteq CC(N, v)$. Take any $x \in C(N, v\underline{t}(N,v))$. Then, it holds that:

$$ \begin{aligned} x(N) &= v\underline{t}(N,v)(N) \\ &= v(N) - \underline{t}(N,v) \\ &= \max_{B \in \Lambda^*(N)} \sum_{S \in B} \delta_S v(S). \end{aligned} $$

Hence $x \in X_{\Lambda^*}^*(N, v)$. Moreover, for any $S \in 2^N \setminus \{\emptyset, N\}$ we have:

$$ \begin{aligned} x(S) &\ge v\underline{t}(N,v)(S) \\ &= v(S), \end{aligned} $$

which proves that $x \in CC(N, v)$.

Second, we prove that $CC(N, v) \subseteq C(N, v\underline{t}(N,v))$. Take any $x \in CC(N, v)$. Since $x \in X_{\Lambda^*}^*(N, v)$ the above equalities imply that $x(N) = v\underline{t}(N,v)(N)$. Moreover, it holds that for any $S \in 2^N \setminus \{\emptyset, N\}$, $x(S) \ge v(S) = v\underline{t}(N,v)(S)$. Hence $x \in C(N, v\underline{t}(N,v))$. ■

<footnote n="5">⁵Observe that $\underline{t}(N, v) = 0$ for the one-player case since no cooperation occurs.</footnote>
The contraction core contains all the “weakest” stable cooperative agreements further to the optimal fine imposed on the grand coalition. This means that authority deters the formation of the grand coalition which compels players to find another almost unstable agreement in the contraction core.

## 2.5 Properties of the contraction core

One of the main advantages of the contraction core is to be a singleton on the set of balanced and symmetric TU-games. In order to prove this result, we first introduce the concept of autonomous coalition (Gonzalez and Grabisch, 2015a). Given any TU-game $(N, v) \in \Gamma_c$, a coalition $S \in 2^N \setminus \{\emptyset\}$ is autonomous for $(N, v)$ if for any payoff vector $x \in C(N, v)$, it holds that $x(S) = v(S)$.

**Proposition 2.5 (Gonzalez and Grabisch, 2015a)** For any TU-game $(N, v) \in \Gamma_c$, the following statements are equivalent:

1. There exists a coalition $S \in 2^N \setminus \{\emptyset, N\}$ which is autonomous for $(N, v)$.

2. For all $t > 0$, it holds that $C(N, v^t) = \emptyset$.

Furthermore, Gonzalez and Grabisch (2015a) prove that the set of autonomous coalitions is a balanced collection.

**Proposition 2.6** For any symmetric TU-game $(N, v) \in \Gamma_c$, the contraction core $CC(N, v)$ is a singleton.

**Proof:** It follows from the symmetry of $(N, v)$ that its $\mathfrak{t}(N, v)$-contraction $(N, v^{\mathfrak{t}(N,v)})$ is also symmetric. By Proposition 2.4, it holds that $CC(N, v) = C(N, v^{\mathfrak{t}(N,v)})$. It is well-known that payoff vector $x \in \mathbb{R}^n$ such that $x_i = v^{\mathfrak{t}(N,v)}(N)/n$ for all $i \in N$ is a core element of any symmetric and balanced TU-game. Moreover, it follows from Proposition 2.5 that there exists an autonomous coalition $K \subset N$ of size $k < n$. The symmetry of $(N, v)$ implies that any coalition $S$ of size $k$ is also autonomous. The collection of all coalitions of size $k$, denoted by $\mathcal{B}$, is a balanced collection with weight $\delta_S = \binom{n-1}{k-1}$ for any $S \in \mathcal{B}$. We define payoff vector $x' \in \mathbb{R}^n$ such that $x'_i = v(K)/k$ for all $i \in N$. Hence, it holds that:

$$ 
\begin{aligned}
x'(N) &= \sum_{S \in \mathcal{B}} \binom{n-1}{k-1} x'(S) \\
&= \sum_{S \in \mathcal{B}} \binom{n-1}{k-1} v(S) \\
&= v^{\mathfrak{t}(N,v)}(N),
\end{aligned}
$$

where the last equality follows from Proposition 2.5. We conclude that $x_i = x'_i$ for all $i \in N$, and so $x' \in CC(N, v)$.

It remains to show that $x' \in CC(N, v)$ is the unique element of the contraction core. Suppose by contradiction that there exists $y \in CC(N, v)$ such that $y \neq x'$. Then, there exists a player $j \in N$ such that $y_j > v(K)/k$ and a player $i \in N$ such that $y_i < v(K)/k$. Since $k < n$, there exists an autonomous coalition $T$ of size $k$ such that $j \in T$ and $i \notin T$. Hence, it holds that $\sum_{r \in T \setminus \{j\}} y_r < (v(K)/k) \times (k-1)$, and so $\sum_{r \in (T \cup \{i\}) \setminus \{j\}} y_r < v(K)$, a contradiction since $(T \cup \{i\}) \setminus \{j\}$ is also an autonomous coalition. ■

Next, we provide a subset of simple TU-games in which the contraction core is not a singleton.
**Proposition 2.7** For any simple TU-game $(N, v) \in \Gamma$ with at least two veto players and at least two winning coalitions, the contraction core is not a singleton.

**Proof:** It is known that the core of any simple TU-game contains any payoff vector that distributes all the gains of the grand coalition among veto players. Take any simple TU-game $(N, v)$ with at least two veto players and at least two winning coalitions. It holds that $v(N) = 1$ and $v(S) = 1$ for some $S \in 2^N \setminus \{\emptyset, N\}$ which implies that $CC(N, v) = C(N, v)$. Moreover, since there are at least two veto players, the above-mentioned result on the core permits to conclude that the contraction core is not a singleton. ■

The following example shows that the contraction core may not be a singleton even on the set of all TU-games.

**Example 2.8** Consider the TU-game $(N, v) \in \Gamma_c$ such that $N = \{1, 2, 3\}$, $v(\{1\}) = v(\{2\}) = 0$, $v(\{3\}) = 3$, $v(\{1, 2\}) = 6$, $v(\{1, 3\}) = v(\{2, 3\}) = 0$, and $v(\{1, 2, 3\}) = 15$. Then, it holds that $\underline{v}(N, v) = 6$ and $CC(N, v) = \text{convex hull}(\{6, 0, 3\}; (0, 6, 3))$.

### 3 Illustrative example

In this section, we propose to apply the contraction core to oligopolistic markets in order to compute the optimal fine imposed by competition authorities for cartel deterrence. We analyze a quantity competition between $n$ firms. Every firm $i \in N$ produces **quantity** $q_i \in \mathbb{R}_+$ of a homogeneous good. Furthermore, we consider the linear **inverse demand function**:

$$p(Q) = a - bQ,$$

where $a$ is the intercept of demand, $b$ is the slope of $p$ and $Q = \sum_{i \in N} q_i$ is the total output of the market. Each firm produces at constant **average and marginal cost** $c \in \mathbb{R}_+$. **Profits** for the $i$th producer in terms of quantities, $\pi_i$, is expressed as:

$$\pi_i(q) = (p(Q) - c)q_i.$$

Without loss of generality, we assume that $c = 0$.

Following Hart and Kurz (1983) and Chander and Tulkens (1997), we consider the situation in which any subset of firms $S$ form a cartel (coalition) while the others continue to act independently. Cartel members are assumed to act as a single firm maximizing their joint profit by correlating their strategies. This leads to consider the set of Cournot oligopoly TU-games in $\gamma$-characteristic function form defined as:

$$v_{\gamma}(S) = \sum_{i \in S} \pi_i(q_i^*, \tilde{q}_j),$$

where $(q_i^*, \tilde{q}_j)$ is the Cournot-Nash equilibrium between $S$ and the other players with the understanding that each player $i \in S$ produces identical quantity $q_i^*$ and each outsider $j \in N \setminus S$ chooses
the same quantity $\tilde{q}_j$.<fnref n="6" /> Under these considerations, we can compute the worth of any coalition as established in the following proposition.

**Proposition 3.1** Let $(N, v_\gamma)$ be an oligopoly TU-game in $\gamma$-characteristic function form. Then for any coalition $S \in 2^N \setminus \{\emptyset\}$, it holds that:

$$v_\gamma(S) = \frac{1}{b} \left( \frac{a}{n-s+2} \right)^2.$$

**Proof:** Take any $S \in 2^N \setminus \{\emptyset\}$. Cartel members and outsiders' optimal quantities are characterized by the first order conditions:

$$\forall i \in S, \frac{\partial}{\partial q_i} \sum_{i \in S} \pi_i(q) = 0 \iff 2b \sum_{i \in S} q_i^* = a - b \sum_{j \in N \setminus S} q_j,$$

and

$$\forall j \in N \setminus S, \frac{\partial}{\partial q_j} \pi_j(q) = 0 \iff 2b \tilde{q}_j = a - b \sum_{k \in N \setminus \{j\}} q_k,$$

respectively. Since the inverse demand function is linear and firms operate at the same marginal cost, any Cournot-Nash equilibrium implies that identical parties must choose identical strategies (quantities), i.e., for any $i, k \in S$, $q_i^* = q_k^*$ and for any $j, l \in N \setminus S$, $\tilde{q}_j = \tilde{q}_l$. From this remark, the intersection of the two above reaction functions yields:

$$q_i^* = \frac{a}{sb(n-s+2)} \quad \text{and} \quad \tilde{q}_j = \frac{a}{b(n-s+2)},$$

which permits to compute the worth of coalition $S$ as:

$$v_\gamma(S) = \sum_{i \in S} \pi_i(q_i^*, \tilde{q}_j) \\ = \frac{1}{b} \left( \frac{a}{n-s+2} \right)^2.$$

This concludes the proof.

Proposition 3.1 shows that any Cournot oligopoly TU-game $(N, v_\gamma)$ is symmetric. The worth $v_\gamma(S)$ of any coalition $S$ is increasing with the intercept of demand $a$ and the size $s$ of coalition $S$. Moreover, it is decreasing with the slope $b$ and the number of outsiders $n-s$. It follows from Lardon (2012) that $(N, v_\gamma) \in \Gamma_c$.

It is now possible to provide the optimal fine imposed by competition authorities in order to deter the grand coalition.

<footnote n="6">⁶This is a consequence of the symmetric cost assumption.</footnote>
**Proposition 3.2** For any Cournot oligopoly TU-game $(N, v_\gamma)$, it holds that:

$$ \underline{t}(N, v_\gamma) = \begin{cases} \frac{1}{b} \left( \frac{a(n-1)}{2(n+1)} \right)^2 & \text{if } n \le 5; \\ \frac{a^2}{b} \left( \frac{5n-9}{36(n-1)} \right) & \text{if } n > 6. \end{cases} $$

**Proof** : Since the Cournot oligopoly TU-game $(N, v_\gamma)$ is symmetric, the non-emptiness of the core is characterized by the following condition:

$$ \forall S \in 2^N \setminus \{\emptyset\}, \quad \frac{v_\gamma(S)}{s} \le \frac{v_\gamma(N)}{n}. $$

It follows that the optimal penalty $\underline{t}(N, v_\gamma)$ can be computed as:

$$ \begin{aligned} \underline{t}(N, v_\gamma) &= v_\gamma(N) - n \max_{S \subseteq N} \frac{v_\gamma(S)}{s} \\ &= \frac{a^2}{4b} - n \max_{s \in \{1, \dots, n-1\}} \frac{a^2}{sb(n-s+2)^2}. \end{aligned} $$

It remains to find the size $s$ which minimizes the function $f(s) = s(n-s+2)^2$ defined on $[1; n-1]$.

We deduce from $f'(s) = (n-s+2)(n-3s+2)$ and $f''(s) = -4n+6s-8$ that $f$:

- attains its maximum at point $s^* = (n+2)/3$ where $1 < s^* < n-1$ for any $n \ge 3$;

- is strictly increasing on $[1; s^*]$ and strictly decreasing on $[s^*; n-1]$.

Hence it holds that $\arg\min_{s \in [1, \dots, n-1]} f(s) \subseteq \{1; n-1\}$. We distinguish two cases:

- first, if $n=2$ it trivially holds that $f$ attains its minimum at $s=1$.

- second assume that $n \ge 3$. It follows from $f(1) = (n+1)^2$ and $f(n-1) = 9(n-1)$ that:

$$ \arg \min_{s \in [1, \dots, n-1]} f(s) = \begin{cases} \{1\} & \text{if } 3 < n < 5; \\ \{1; n-1\} & \text{if } n = 5; \\ \{n-1\} & \text{if } n > 5. \end{cases} $$

Thus, when $2 \le n \le 5$ it holds that:

$$ \begin{aligned} \underline{t}(N, v_\gamma) &= \frac{a^2}{4b} - n \frac{a^2}{b(n+1)^2} \\ &= \frac{1}{b} \left( \frac{a(n-1)}{2(n+1)} \right)^2. \end{aligned} $$

Moreover, when $n \ge 5$ it holds that:
$$ 
\begin{aligned}
\underline{t}(N, v_{\gamma}) &= \frac{a^2}{4b} - n \frac{a^2}{9b(n-1)} \\
&= \frac{a^2}{b} \left( \frac{5n-9}{36(n-1)} \right),
\end{aligned}
$$

which concludes the proof.

<figure><img src="image_2.png" /></figure>

Proposition 3.2 shows that the optimal fine imposed by competition authorities is increasing with the intercept of demand *a* and the number of firms *n*. Moreover, it is decreasing with the slope *b*. Surprisingly, the expression of the optimal fine leads to distinguish markets of small size (*n* ≤ 5) and those of medium and large size (*n* ≥ 6) for the deterrence of the monopoly power.

We know by Propositions 2.6 and 3.1 that the contraction core is a singleton. Proposition 3.2 permits to go further by providing an expression of the contraction core.

**Corollary 3.3** For any Cournot oligopoly TU-game $(N, v_\gamma)$, the contraction core is expressed as:

$$ CC(N, v_{\gamma}) = \begin{cases} \left\{ \frac{1}{b} \left( \frac{a}{n+1} \right)^2 \times e \right\} & \text{if } n \le 5; \\ \left\{ \left( \frac{a^2}{9b(n-1)} \right) \times e \right\} & \text{if } n \ge 6; \end{cases} $$

where $e = (1, \dots, 1)$.

In both market types, each individual payoff in the contraction core is increasing with the intercept of demand *a* and decreasing with the slope *b* and the number of firms *n*.

# 4 Axiomatization of the contraction core

In this section, we provide an axiomatic characterization of the contraction core on the set of balanced TU-games.

Let $\Gamma_0$ be any arbitrary subset of $\Gamma$. A **solution** on $\Gamma_0$ is a mapping $\sigma$ that assigns a (possibly empty) set $\sigma(N, v) \subseteq X_{\Lambda^*}(N, v)$ to any TU-game $(N, v)$.

## 4.1 Axioms

We now present the axioms relevant to our analysis. The first two are classic in the literature on core axiomatizations.

**Definition 4.1 Non-emptiness (NE)** A solution $\sigma$ on $\Gamma_0$ satisfies NE if for any $(N, v) \in \Gamma_0$, $\sigma(N, v) \neq \emptyset$.
**Definition 4.2 Individual rationality (IR) A solution σ on Γ₀ satisfies IR if for any (N, v) ∈ Γ₀,**
every x ∈ σ(N, v), and every i ∈ N, xᵢ ≥ v({i}).

Both of these axioms are satisfied by all core extensions discussed in the introduction, and so are
useful in characterizing them.

Next, we introduce three versions of reduced games and their corresponding consistency axioms
in order to make core comparisons. Take any (N, v) ∈ Γ, any S ∈ 2<sup>N</sup>\{Ø} and any x ∈ R<sup>n</sup>.
The first reduced game type makes a special treatment to the grand coalition and permits to
characterize the core (Peleg, 1986). The **DM-reduced game** (Davis and Maschler, 1965) of
(N, v) with respect to S and x is the game (S, v<sub>S,x</sub>) ∈ Γ defined for any T ∈ 2<sup>S</sup> as:

$$
v_{S,x}(T) = \begin{cases} 0 & \text{if } T = \emptyset; \\ v(N) - x(N \setminus S) & \text{if } T = S; \\ \max\{v(T \cup Q) - x(Q) : Q \subseteq N \setminus S\} & \text{otherwise.} \end{cases}
$$

**Definition 4.3 DM-consistency (DM-CON) A solution σ on Γ₀ satisfies DM-CON if for any**
**(N, v) ∈ Γ, every S ∈ 2<sup>N</sup>\{Ø} and every x ∈ σ(N, v), then (S, v<sub>S,x</sub>) ∈ Γ₀ and x<sup>S</sup> ∈ σ(S, v<sub>S,x</sub>).**

The second version is more general and treats all coalitions in the same way and permits to
characterize the aspiration core. The **modified DM-reduced game** (Bejan and Gómez, 2012) of
(N, v) with respect to S and x is the game (S, v<sub>*</sub><sup>S,x</sup>) ∈ Γ defined for any T ∈ 2<sup>S</sup> as:

$$
v_{*}^{S,x}(T) = \begin{cases} 0 & \text{if } T = \emptyset; \\ \max\{v(T \cup Q) - x(Q) : Q \subseteq N \setminus S\} & \text{otherwise.} \end{cases}
$$

**Definition 4.4 MDM-consistency (MDM-CON) A solution σ on Γ₀ satisfies MDM-CON if**
**for any (N, v) ∈ Γ, every S ∈ 2<sup>N</sup>\{Ø} and every x ∈ σ(N, v), then (S, v<sub>*</sub><sup>S,x</sup>) ∈ Γ₀ and x<sup>S</sup> ∈**
**σ(S, v<sub>*</sub><sup>S,x</sup>).**

We can verify that the contraction core does not satisfies MDM-CON on Γ<sub>c</sub>.

**Example 4.5** Consider the TU-game (N, v) ∈ Γ<sub>c</sub> such that N = {1, 2, 3}, v({1}) = v({2}) = v({3}) = 0, v({1, 2}) = 4, v({1, 3}) = v({2, 3}) = 2, and v({1, 2, 3}) = 10. It holds that t(N, v) = 6 and CC(N, v) = C(N, v<sup>t</sup>(N, v)) = {(2, 2, 0)}. When S = {1} and x = (2, 2, 0), the modified DM-reduced game is given by v<sub>*</sub><sup>{1},x</sup>({1}) = v({1, 2, 3}) - 2 - 0 = 8. Thus, 2 ∉ CC({1}, v<sub>*</sub><sup>{1}},<sup>x</sup>) = {8} so that the contraction core does not satisfied MDM-CON.

The third version which is relevant for our results makes again a special treatment to the grand
coalition of any reduced game which is not allowed to cooperate with the complementary coalition.
This permits to satisfy the feasibility condition related to second best time allocation. The **new**
**modified DM-reduced game** of (N, v) with respect to S and x is the game (S, v<sub>S,x</sub><sup>*</sup>) ∈ Γ defined
for any T ∈ 2<sup>S</sup> as:

$$
v_{S,x}^*(T) = \begin{cases} 0 & \text{if } T = \emptyset; \\ \max\{v(T \cup Q) - x(Q) : Q \subseteq N \setminus S\} & \text{if } T = S; \\ \max\{v(T \cup Q) - x(Q) : Q \subseteq N \setminus S\} & \text{otherwise.} \end{cases}
$$
**Definition 4.6 NMDM-consistency (NMDM-CON) A solution σ on Γ₀ satisfies NMDM-CON if for any (N, v) ∈ Γ, every S ∈ 2N \ {Ø} and every x ∈ σ(N, v), then (S, v*<S,x>) ∈ Γ₀ and x* ∈ σ(S, v*<S,x).**

Observe that the three axioms of consistency defined above satisfy the following logical equality:
DM-CON ∨ NMDM-CON = MDM-CON.

The last axiom differs from the classic superadditivity axiom on the feasibility requirement.

**Definition 4.7 Conditional Superadditivity (C-SUPA) A solution σ on Γ₀ satisfies C-SUPA if for any (N, vA), (N, vB) ∈ Γ₀, every xA ∈ σ(N, vA) and every xB ∈ σ(N, vB), then xA + xB ∈ σ(N, vA + vB) whenever (N, vA + vB) ∈ Γ₀ and xA + xB is feasible for (N, vA + vB), i.e., xA + xB ∈ XΛ*(N, vA + vB).**

While the feasibility requirement related to first best time allocation is redundant on the set Γc,
ours is not trivially satisfied since the grand coalition is deterred.

**4.2 Axiomatization**

Before characterizing the contraction core, we first need the following lemma.

**Lemma 4.8** *Take any *B* ∈ Λ\*(N) where *n* ≥ 2 with balanced weights (δ_H)_{H∈*B*. For any *S* ∈ 2N \ {Ø} where *s* ≥ 2, define:*

$$
\mathcal{B}^S = \{ T \subset S : T = H \cap S \neq \emptyset \text{ for some } H \in \mathcal{B} \}
$$

and for every $T \in \mathcal{B}^S$:

$$
\hat{\delta}_T = \left( 1 - \sum_{H \in \mathcal{B}: H \cap S = S} \delta_H \right)^{-1} \sum_{T = H \cap S} \delta_H.
$$

Then, $\mathcal{B}^S \in \Lambda^*(S)$ with balanced weight $(\hat{\delta}_T)_{T \in \mathcal{B}^S}$.

**Proof:** First, it follows from $\sum_{H \in \mathcal{B}: i \in H} \delta_H = 1$, $s \ge 2$ and $N \notin \mathcal{B}$ that $\sum_{H \in \mathcal{B}: H \cap S = S} \delta_H < 1$.
Second, for each $i \in S$ it holds that:
$$
\begin{align*}
\left(1 - \sum_{H \in \mathcal{B}: H \cap S = S} \delta_H\right) \sum_{T \in \mathcal{B}^S: i \in T} \hat{\delta}_T &= \sum_{T \in \mathcal{B}^S: i \in T} \sum_{T=H \cap S} \delta_H \\
&= \sum_{\substack{H \in \mathcal{B}: \\ H \cap S \subseteq S \\ i \in H}} \delta_H \\
&= \sum_{\substack{H \in \mathcal{B}: \\ H \cap S \subseteq S \\ i \in H}} \delta_H - \sum_{\substack{H \in \mathcal{B}: \\ H \cap S = S \\ i \in H}} \delta_H \\
&= \sum_{\substack{H \in \mathcal{B}: \\ i \in H}} \delta_H - \sum_{\substack{H \in \mathcal{B}: \\ H \cap S = S \\ i \in H}} \delta_H \\
&= 1 - \sum_{\substack{H \in \mathcal{B}: \\ H \cap S = S}} \delta_H,
\end{align*}
$$

which concludes the proof.

<figure><img src="image_3.png" /></figure>

Given any second best time allocation in a TU-game, Lemma 4.8 provides a formula to compute
the corresponding second best time allocation in any of its reduced game. It clearly implies the
result on NMDM-CON in the following proposition.

**Proposition 4.9** The contraction core satisfies NE, IR, NMDM-CON and C-SUPA on the set Γ<sub>c</sub>.

**Proof:** It is straightforward to verify that NE, IR and C-SUPA are satisfied. It remains to prove that the contraction core satisfies NMDM-CON. Let (N, v) ∈ Γ<sub>c</sub>, S ∈ 2<sup>N</sup>\{∅} and x ∈ CC(N, v). We distinguish two cases:

- assume that $s \ge 2$. Take $\mathcal{B} \in \Lambda^*(N)$ with balanced weights $(\delta_H)_{H \in \mathcal{B}}$ such that $x(N) = \sum_{H \in \mathcal{B}} \delta_H v(H)$. Then, by Lemma 4.8 it holds that $\mathcal{B}^S \in \Lambda^*(S)$ with balanced weight $(\hat{\delta}_T)_{T \in \mathcal{B}^S}$. Now, we prove that $x(T) \le v_{S,x}^*(T)$ for each $T \in \mathcal{B}^S$. Given $T \in \mathcal{B}^S$, there exists $H \in \mathcal{B}$ such that $T = H \cap S$. From $x(N) = \sum_{H \in \mathcal{B}} \delta_H v(H)$ and $x(S) \ge v(S)$ for each $S \in 2^N \setminus \{\emptyset, N\}$, it holds that $x(H) = v(H)$, hence $x(T) = v(H) - x(H \setminus T)$. Since $H \setminus T \subseteq N \setminus S$, it holds that $x(T) \le \max\{v(T \cup Q) - x(Q) : Q \subseteq N \setminus S\} = v_{S,x}^*(T)$.

Then, we prove that $x(T) \ge v_{S,x}^*(T)$ for each $T \in 2^S \setminus \{\emptyset, S\}$. By contradiction, assume that there exists $T \in 2^S \setminus \{\emptyset, S\}$ such that $x(T) < v_{S,x}^*(T)$. Hence there exists $y^T \in \mathbb{R}^t$ such that $y(T) = v_{S,x}^*(T)$ and $y(T) > x(T)$. Thus, it holds that $y(T) = v(T \cup Q) - x(Q)$ for some $Q \subseteq N \setminus S$. Hence, $y(T) + x(Q) = v(T \cup Q)$ and so, $x(T) + x(Q) < v(T \cup Q)$, a contradiction with $x \in CC(N, v)$ since $T \cup Q \subset N$. We conclude that $x(T) \ge v_{S,x}^*(T)$ for each $T \in 2^S \setminus \{\emptyset, S\}$.
Thus, $x(T) = v_{S,x}(T)$ for each $T \in \mathcal{B}^S$, and so $x(S) = \sum_{T \in \mathcal{B}^S} \hat{\delta}_T x(T) = \sum_{T \in \mathcal{B}^S} \delta_T v_{S,x}^*(T)$.
Moreover, $x(T) \ge v_{S,x}^*(T)$ for each $T \in 2^S \setminus \{\emptyset, S\}$ implies that $x^S \in CC(S, v_{S,x}^*)$.

- assume that $s = 1$. Take $\mathcal{B} \in \Lambda^*(N)$ with balanced weights $(\delta_H)_{H \in \mathcal{B}}$ such that $x(N) = \sum_{H \in \mathcal{B}} \delta_H v(H)$. Now, we prove that $x^S \le v_{S,x}^*(S)$. Given $S = \{i\}$, there exists $H \in \mathcal{B}$ such
that $i \in H$. From $x(N) = \sum_{H \in \mathcal{B}} \delta_H v(H)$ and $x(S) \ge v(S)$ for each $S \in 2^N \setminus \{\emptyset, N\}$, it holds that $x(H) = v(H)$, hence $x(S) = v(H) - x(H \setminus S)$. Since $H \setminus S \subset N \setminus S$, it holds that $x(S) \le \max\{v(S \cup Q) - x(Q) : Q \subset N \setminus S\} = v_{S,x}^*(S)$.

Then, we prove that $x^S \ge v_{S,x}^*(S)$. By contradiction, assume that $x^S < v_{S,x}^*(S)$. Hence, there is $y^S \in \mathbb{R}$ such that $y^S = v_{S,x}^*(S)$ and $y^S > x^S$. Thus, it holds that $y^S = v_{S,x}^*(S) = v(S \cup Q) - x(Q)$ for some $Q \subset N \setminus S$. Hence, $y(S) + x(Q) = v(S \cup Q)$ and so, $x(S) + x(Q) < v(S \cup Q)$, a contradiction with $x \in CC(N, v)$ since $S \cup Q \subset N$. We conclude that $x^S \in CC(S, v_{S,x}^*)$. ■

**Proposition 4.10** Let $\sigma$ be a solution concept on $\Gamma_0 \subseteq \Gamma$ satisfying IR and NMDM-CON. If $(N, v) \in \Gamma_0$ and $x \in \sigma(N, v)$ then $x(S) \ge v(S)$ for any $S \in 2^N \setminus \{\emptyset, N\}$.

**Proof:** Let $\sigma$ be a solution concept on $\Gamma_0 \subseteq \Gamma$ satisfying IR and NMDM-CON. Let $x \in \sigma(N, v)$, $S \in 2^N \setminus \{\emptyset, N\}$ and $i \in S$. By NMDM-CON, $x_i \in \sigma\{i\}, v_{\{i\},x}^*$. By IR, it holds that:

$$ 
\begin{aligned}
x_i &\ge v_{\{i\},x}^*(\{i\}) \\
&= \max\{v(\{i\} \cup Q) - x(Q) : Q \subset N \setminus \{i\}\} \\
&\ge v(S) - x(S \setminus \{i\}),
\end{aligned}
$$

which proves that $x(S) \ge v(S)$ as desired. ■

**Proposition 4.11** If $\sigma$ is a solution concept defined on $\Gamma_0 \subseteq \Gamma_c$ that satisfies IR and NMDM-CON, then for any $(N, v) \in \Gamma_0$, any payoff vector $x \in \sigma(N, v)$ is efficient, i.e., $x(N) = \max_{B \in \Lambda^*(N)} \sum_{S \in \mathcal{B}} \delta_S v(S)$ (or $x \in X_{\Lambda^*}^*(N, v)$).

**Proof:** Let $\sigma$ be a solution concept on $\Gamma_0 \subseteq \Gamma_c$ satisfying IR and NMDM-CON. Assume that $(N, v) \in \Gamma_0$ and take any $x \in \sigma(N, v)$ and any $y \in X_{\Lambda^*}^*(N, v)$. Then, there is $B \in \Lambda^*(N)$ such that $y(N) \le \sum_{S \in \mathcal{B}} \delta_S v(S)$. It follows from $B \in \Lambda^*(N)$ and Proposition 4.10 that:

$$ 
\begin{aligned}
x(N) &= \sum_{S \in \mathcal{B}} \delta_S x(S) \\
&\ge \sum_{S \in \mathcal{B}} \delta_S v(S) \\
&\ge y(N).
\end{aligned}
$$

We conclude that $x \in X_{\Lambda^*}^*(N, v)$. ■

**Proposition 4.12** If $\sigma$ is a solution concept defined on $\Gamma_c$ satisfying IR and NMDM-CON, then $\sigma(N, v) \subseteq CC(N, v)$ for any $(N, v) \in \Gamma_c$.

**Proof:** Take any $x \in \sigma(N, v)$. By Proposition 4.10, it holds that $x(S) \ge v(S)$ for every $S \in 2^N \setminus \{\emptyset, N\}$. Moreover, by Proposition 4.11, $x \in X_{\Lambda^*}^*(N, v)$. So, $x \in CC(N, v)$. ■

**Proposition 4.13** If a solution concept defined on $\Gamma_c$ satisfies NE, IR, NMDM-CON and C-SUPA, then $CC(N, v) \subseteq \sigma(N, v)$ for any $(N, v) \in \Gamma_c$.
**Proof:**$^7$ Let $x \in CC(N, v)$ and define $(N, w) \in \Gamma_c$ as:

$$w(S) = \begin{cases} x(S) & \text{if } |S| \ge 2; \\ v(S) & \text{if } |S| = 1. \end{cases}$$

It holds that $C(N, w) = \{x\}$. By Proposition 4.12, $\sigma(N, w) \subseteq CC(N, w) = C(N, w) = \{x\}$. By NE, it holds that $x \in \sigma(N, w)$.

Consider the game $(N, z) \in \Gamma_c$ defined as:

$$\forall S \in 2^N, z(S) = v(S) - w(S).$$

Hence, $z(S) \le 0$ if $2 \le |S| < n$, $z(\{i\}) = 0$ for every $i \in N$ and $z(N) \in \mathbb{R}$. Note that $0 \in CC(N, z)$ since $0 = \max_{B \in \Lambda^*(N)} \sum_{S \in B} \delta_S z(S) = \sum_{i \in N} z(\{i\})$. By Proposition 4.11, for every $y \in CC(N, z)$ it holds that $y(N) = 0$. Since $z(\{i\}) = 0$ for every $i \in N$, we have $y_i \ge 0$ by IR and so, $y = 0$. Thus, $CC(N, z) = \{0\}$. By Proposition 3.8, it holds that $\sigma(N, z) \subseteq CC(N, z) = \{0\}$. By NE, $0 \in \sigma(N, z)$.

Note that $x(N) + 0 = \max_{B \in \Lambda^*(N)} \sum_{S \in B} \delta_S v(S) = \max_{B \in \Lambda^*(N)} \sum_{S \in B} \delta_S (w + z)(S)$. So, $x + 0 \in X_{\Lambda^*}(N, w + z)$, i.e., $x + 0$ is feasible for $(N, w + z)$. Thus, by C-SUPA it follows from $x \in \sigma(N, w)$ and $0 \in \sigma(N, z)$ that $x + 0 \in \sigma(N, w + z)$, hence $x \in \sigma(N, v)$. ■

**Theorem 4.14** *The contraction core is the only solution concept on $\Gamma_c$ that satisfies NE, IR, NMDM-CON and C-SUPA.*

**Proof:** Combine Propositions 4.9, 4.12 and 4.13. ■

## 4.3 Independence of the axioms

The following examples show that the axioms used in the characterization of the contraction core are logically independent on the set $\Gamma_c$, i.e., none is implied by the others.

**Example 4.15** Consider the solution concept $\sigma_1$ on $\Gamma_c$ such that for any $(N, v) \in \Gamma_c$, $\sigma_1(N, v) = \emptyset$. Obviously, $\sigma_1$ violates NE but vacuously satisfies IR, NMDM-CON and C-SUPA.

**Example 4.16** Consider the solution concept $\sigma_2$ on $\Gamma_c$ such that for any $(N, v) \in \Gamma_c$, $\sigma_2(N, v) = X_{\Lambda^*}(N, v)$. We know that $\sigma_2$ satisfies NE because $X_{\Lambda^*}(N, v) \supseteq CC(N, v) \neq \emptyset$. It satisfies C-SUPA by definition. It follows from Lemma 4.8 and a similar argument used in the proof of Proposition 4.9 that NMDM-CON is also satisfied. It should be clear that $\sigma_2$ violates IR.

**Example 4.17** Consider the solution concept $\sigma_3$ on $\Gamma_c$ such that for any $(N, v) \in \Gamma_c$, $\sigma_3(N, v) = \{x \in X_{\Lambda^*}(N, v) : x_i \ge v(\{i\})\}$. Clearly, $\sigma_3$ satisfies NE, IR and C-SUPA. The previous result implies that $\sigma_3$ does not satisfy NMDM-CON.

<footnote>$^7$Our proof is inspired from that in Peleg and Sudhölter (2003) in the case where $n \ge 3$. Nevertheless, the main difference is that we don't need to distinguish cases $n = 2$ and $n \ge 3$.</footnote>
**Example 4.18** For every $(N, v) \in \Gamma_c$, every $S \in 2^N \setminus \{\emptyset\}$ and every $x \in \mathbb{R}^n$, the excess of $S$ from $x$ in $(N, v)$ is given by the quantity $e(S, x, v) = v(S) - x(S)$. The excess $e(S, x, v)$ gives the amount of dissatisfaction of coalition $S$ from $x$ in $(N, v)$. We define the vector $\theta(x) = (\theta_1(x), \dots, \theta_{2^{n-1}}(x))$ whose components are the numbers $(e(S, x, v))_{S \in 2^N \setminus \{\emptyset\}}$ arranged in non-increasing order. For any TU-game $(N, v) \in \Gamma_c$, the contraction nucleolus, denoted by $CN(N, v)$, is defined as:

$$CN(N, v) = \{x \in X_{\Lambda^*}^*(N, v) : \theta(y) \geq_L \theta(x) \text{ for all } y \in X_{\Lambda^*}^*(N, v)\},$$

where $\geq_L$ is the lexicographical ordering. First, since $X_{\Lambda^*}^*(N, v)$ is non-empty, compact and convex, it follows from corollary 5.1.10 in Peleg and Sudhölter (2003) that $CN(N, v)$ consists of a single point. Hence, the contraction nucleolus satisfies NE. Second, the contraction nucleolus also satisfies IR since it belongs to the contraction core. The proof is left as an exercise to the reader. Finally, the contraction nucleolus complies with NMDM-CON. The proofs are omitted because they are similar to those in Peleg and Sudhölter (2003) in order to show that the nucleolus satisfies DM-CON. Hence, it follows from our axiomatization that the contraction nucleolus does not satisfy C-SUPA.

# 5 Concluding remarks: extended contraction core

We have introduced a new solution concept, the contraction core, that serves as a basis for the investigation of the deterrence of cooperation. This solution concept has permitted to provide a measure of the robustness of cooperation which, as far as we know, has not been analyzed in the literature. We have successfully applied the contraction core to oligopolistic markets and we have provided optimal fine imposed by competition authorities for cartel deterrence. We can be convinced that there are many other potential applications of the contraction core.

More generally, we have also provided an axiomatic characterization of the contraction core in order to better understand it. In particular, this has permitted to make comparisons with the core and the aspiration core. We have defined the contraction core on the set of balanced TU-games in order to be consistent with our objective to study the deterrence of cooperation. We argue that it is possible to define an "extended" contraction core by applying the feasibility condition related to second best time allocation on the set of all TU-games. The "extended" contraction core is then non-empty on the set of all TU-games and coincides with the aspiration core on the set of non-balanced TU-games. In this case, it will be a straightforward exercise for the reader to check that the axiomatization given in Section 4 still holds on the set of all TU-games.

# References

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