Horizon-K Farsightedness in Criminal Networks

Abstract

We study the criminal networks that will emerge in the long run when criminals are neither myopic nor completely farsighted but have some limited degree of farsightedness. We adopt the horizon-K farsighted set to answer this question. We find that in criminal networks with n criminals, the set consisting of the complete network is a horizon-K farsighted set whenever the degree of farsightedness of the criminals is larger than or equal to $(n-1)$. Moreover, the complete network is the unique horizon-$(n-1)$ farsighted set. Hence, the predictions obtained in case of completely farsighted criminals still hold when criminals are much less farsighted.

1. Introduction

There is empirical evidence suggesting that peer effects and the structure of social interactions matter strongly when explaining an individual’s own criminal or delinquent behavior. (See Patacchini and Zenou [1], among others.) A criminal’s place in the network and his know-how regarding the crime business of his partners determine his criminal opportunities and constraints, as well as his information on these opportunities and constraints. It is, therefore, crucial to understand how such criminal networks are formed and structured, and how they evolve and perform.

Different ways of characterizing which network structures are stable have been proposed in the literature, depending on whether (and how far) agents anticipate that their action may also induce others to change the network relations that they maintain. (Mauleon and Vannetelbosch [2] provide a comprehensive overview of the solution concepts for solving network formation games.) The notion of a pairwise stable network, introduced by Jackson and Wolinsky [3], assumes that agents are able to modify the network one link at a time, and can choose to change the network if the resulting network implies higher payoffs for the deviating agents. As such, pairwise stability involves fully myopic agents in the sense that they do not anticipate that others might react to their actions. At the other end of the spectrum, a number of solution concepts involve perfectly farsighted agents, i.e., agents that fully anticipate the complete sequence of reactions that results from their own actions in the network. However, this assumption of perfect farsightedness, especially when the number of agents becomes large, requires a very high level of foresight on behalf of the agents. Kirchsteiger, Mantovani, Mauleon and Vannetelbosch [4] provide experimental evidence suggesting that subjects are consistent with an intermediate rule of behavior, which can be interpreted as a form of limited farsightedness. Agents only anticipate a limited number of reactions by the other agents in response to the actions they take themselves. (Similar experimental evidence for limited farsightedness is found in van Dolder and Buskens [5]). In this paper, we study the criminal networks that agents form when criminals are neither fully myopic nor completely farsighted, but have some limited degree of farsightedness. In other words, we show how the predictions about stable criminal networks relate to the degree of farsightedness.

There is a large literature on the economics of crime. Calvo-Armengol and Zenou [6] provide a network analysis of criminal behavior. They develop a model where criminals compete with each other in criminal activities but benefit from being friends with other criminals by improving their knowledge of the crime business. Individuals decide first whether to work or to become a criminal, and then they choose the crime effort to exert provided of being a criminal. (Calvo-Armengol and Zenou [6] mostly focus on cases where the network is exogenously given. They show that multiple equilibria with different members of active criminals and levels of involvement in crime business may coexist.) Ballester, Calvo-Armengol and Zenou [7] develop a criminal network game where each delinquent decides how much delinquency effort to exert. The network is determined endogenously by allowing players to join the labor market instead of committing criminal activities. They find that the optimal enforcement policy consists of removing some key player or some key group. Such a policy is complex, since it depends both on the wage and on the network. Indeed, the removal of some players may induce further voluntary moves from other players, who now find it profitable to leave their criminal activities and join the labor market. (See also, Bezin, Verdier and Zenou [8] and Lee, Liu, Patacchini and Zenou [9]).

In this paper, we present a simplified version of the model of Calvo–Armengol and Zenou [6], which places emphasis on the formation of links and keeps the players’ level of criminal activities fixed. For simplicity, we also keep the wage in the labor market low enough, in Calvo-Armengol and Zenou’s model, that all individuals prefer to become a criminal, whatever the social network connecting the criminals. By doing so, we study the networks that will be formed when criminals have the discretion to choose their connections.

Herings, Mauleon and Vannetelbosch [10] introduce the notion of a pairwise farsightedly stable set to study the networks that will be formed by farsighted players. (Other approaches to farsightedness in network formation are suggested by the work of Chwe [11], Xue [12], Herings, Mauleon and Vannetelbosch [13], Mauleon and Vannetelbosch [14], Dutta, Ghosal and Ray [15], Page, Wooders and Kamat [16], Page and Wooders [17], Mauleon, Vannetelbosch and Vergote [18], and Ray and Vohra [19]). Herings, Mauleon and Vannetelbosch [10] analyze a simplified version of the criminal network model of Calvo-Armengol and Zenou [6], and find that, in criminal networks with three players, there may be several pairwise stable networks, but the set consisting of the complete network, where all criminals are linked to each other, is the unique, pairwise, farsightedly stable set. Moreover, they show that the complete network is a pairwise farsightedly stable set for any number of players.

Which criminal networks will emerge in the long run when criminals have a limited degree of farsightedness? We adopt the horizon-K farsighted set of Herings, Mauleon and Vannetelbosch [20] to answer this question. This paper constitutes the first application of this concept and demonstrates its tractability.

A horizon-K farsighted set of networks identifies the networks that emerge in the long run when players have a reasoning horizon equal to $K.$ It is defined as the smallest set of networks, so that there are no deviations by the players to networks outside the set, whereas it is possible to reach at least one network in the set from any initial network. Herings, Mauleon and Vannetelbosch [20] show that a horizon-K farsighted set always exists, and provide easy-to-verify conditions that must be reached for a set of networks to be a horizon-K farsighted set. When K is equal to 1, the concept is identical to the pairwise myopically stable set introduced by Herings, Mauleon and Vannetelbosch [10]. (The myopic stable set of Demuynck, Herings, Saulle, and Seel [21] generalizes the pairwise myopically stable set to a large class of social environments, and shows how it unifies the most important concepts of non-cooperative game theory, such as Nash equilibrium, and cooperative game theory, such as the core.) In that case, it consists of the pairwise stable networks of Jackson and Wolinsky [3], as well as the closed cycles of Jackson and Watts [22]. The value of K reflects the reasoning horizon of the players, and when K tends towards infinity, we reach the limit case of the farsightedly stable set of Herings, Mauleon and Vannetelbosch [10].

Although a horizon-K farsighted set may consist of multiple networks, in our application to criminal networks, the complete network, where all criminals are connected to each other, constitutes a horizon-K farsighted set consisting of a single network whenever the reasoning horizon of a criminal is at least equal to the number of other criminals. When the reasoning horizon of the criminals is exactly equal to this number, then this is the unique horizon-K farsighted set. Such a uniqueness result is striking in the literature on networks, which has provided many examples where a large multiplicity of stable networks is obtained. For example, in both the connections and co-author model in Jackson and Wolinsky [3], there are multiple pairwise stable networks for reasonable specifications of the parameter values.

Hence, we can obtain a very sharp prediction for intermediate degrees of farsightedness (i.e., a degree of farsightedness equal to $n-1$), and show that a limited degree of farsightedness (i.e., at least $n-1$) is sufficient to recover the predictions obtained in the case of completely farsighted criminals. The robustness of the complete network, compared to the other pairwise stable networks, for intermediate degrees of farsightedness reveals the importance of knowing the degree of farsightedness of criminals in order to determine which criminal networks are likely to emerge in the long run. The adequate policy to reduce crime when facing a symmetric, highly connected network, such as the complete network, could differ from the policy which is needed to reduce crime in a different type of network. Patacchini and Zenou [1] investigate whether weak ties (friends of friends) play an important role in providing information about crime. Individuals can have strong and weak ties, and can learn about crime opportunities through these. They find that increasing the percentage of weak ties increases the crime rate in the economy. They conclude that an effective policy should be measured by both the possible crime reduction it implies and the group interaction it engenders. This result confirms that better knowledge of the degree of farsightedness of criminals is, therefore, important in determining which criminal networks are likely to emerge, and how to implement adequate delinquency-reducing policies.

Recently, Lindquist and Zenou [23] present a selective review of several key network studies and findings from the crime literature. They argue that the economic approach providing network-based policies is a valuable complement to the more traditional approach that focuses on the observed structure of the network. The economic approach allows for a comparison of the effectivity of group-based policies (e.g., changing the group norm) versus individual-based policies (e.g., targeting), and when targeting is more appropriate, it specifies who policy-makers should target in order to generate the greatest reduction in crime. Arresting key individuals may dismantle non-hierarchical crime structures, described as loose associations of independent contractors or brokers (such as some Canadian car theft associations and some criminal organizations of human smuggling from China to the United States), but identifying and arresting key groups may be necessary to dismantle strongly hierarchical crime structures (such as the Mafia, where leaders are well-connected) or gangs of well-connected criminals with a high degree of centrality (i.e., a complete network). Hence, once criminals have limited farsightedness, groups of well-connected criminals (such as the Mafia and some types of gang) would emerge. When this is the case, instead of targeting key criminals, group-based policies that make it difficult for criminals to establish mutual links should be implemented.

The paper is organized as follows. In Section 2, we introduce some notations and basic properties of criminal networks. In Section 3, we define the notion of a horizon-K farsighted set. In Section 4, we identify the horizon-K farsighted set of criminal networks. Finally, in Section 5, we conclude.

2. Criminal Networks

Let $N=\{1,\dots ,n\}$ be the finite set of criminals. Throughout the paper, we assume that $n\ge 3.$ A criminal network g is simply a list of which pairs of criminals are linked to each other, and $ij\in g$ indicates that i and j are linked under g. The complete network of the set of criminals $S\subseteq N$ is denoted by $g^S$ and is equal to the set of all subsets of S of size 2. (Throughout the paper, we use the notation ⊆ for weak inclusion and for strict inclusion. Finally, the notation # is used for the cardinality of a set.) It follows that the empty network is denoted by $g^{\varnothing }$. Let $g_{\mid S}=\{ij\in g\mid i,j\in S\}$ be the network found by deleting all links from g, except those between players in S. The set of all possible networks or graphs on N is denoted by $\mathbb{G}$, and consists of all subsets of $g^N$. The cardinality of $\mathbb{G}$ is denoted by $n^{\prime }=2^{n(n-1)/2}$.

The network obtained by adding a link $ij$ to network g is denoted by $g+ij$, and the network that results from deleting link $ij$ from network g by $g-ij$. Let $N\left(g\right)=\{i\in N\mid \exists j\in N$, such that $ij\in g\}$ is the set of criminals who have at least one link in the network g. A path in a network $g\in \mathbb{G}$ between criminals i and j of length $K\ge 1$ is a finite sequence of criminals $i_0,\dots ,i_K$ with $i_0=i$ and $i_K=j$, such that for any $k\in \{0,\dots ,K-1\}$, $i_k{i}_{k+1}\in g$, and such that each criminal in the sequence $i_0,\dots ,i_K$ is distinct. A network g is connected if, for each pair of criminals i and j in $N\left(g\right)$, such that $i\ne j$ there is a path between i and j in g. A non-empty network $h\subseteq g$ is a component of g if h is connected and for any $i\in N\left(h\right)$ and $j\in N\left(g\right)$, $ij\in g$ implies $ij\in h$. The set of components of g is denoted by $C\left(g\right)$. By knowing the components of a network, we can partition the criminals into maximal groups within which criminals are connected. Let $\mathcal{P}\left(g\right)$ denote the partition of N into components and singletons induced by the network g. That is, the set S of players belongs to $\mathcal{P}\left(g\right)$ if, and only if, either there is network h in $C\left(g\right)$, such that $S=N\left(h\right)$, or there exists $i\notin N\left(g\right)$, such that $S=\left\{i\right\}$.

Next, we present a simplified version of the Calvo-Armengol and Zenou model [6]. Given some criminal networks g, the elements of $\mathcal{P}\left(g\right)$ are called criminal groups. Each criminal group S has a positive probability ${\pi }_S\left(g\right)$ of winning the loot $B>0$. It is assumed that the bigger the criminal group, the higher its probability of obtaining the loot. This assumption captures the idea that delinquents learn from other criminals belonging to the same group regarding how to commit crime in a more efficient way, by sharing know-how on the technology of crime. We assume that the probability of winning the loot is given by ${\pi }_S\left(g\right)=\#S/n$.

The network architecture determines how the loot is shared among the criminals in the group. Consider some criminal $i\in N$ and let $S\in \mathcal{P}\left(g\right)$ be the criminal group i which belongs to. Let $d_i\left(g\right)$ denote the degree of criminal i in g; i.e., the number of links criminal i has in g. We define $c_i\left(g\right)={max}_{j\in S}{d}_j\left(g\right)$ as the maximum degree in this criminal group. A criminal i who is part of a group $S\in \mathcal{P}\left(g\right)$ expects a share ${\alpha }_i\left(g\right)$ of the loot given by

$${\alpha }_i\left(g\right)=\left\{\begin{array}{cc}\frac{1}{\#\{j\in S\mid d_j\left(g\right)=c_j\left(g\right)\}},\hfill & \mathrm{if}{d}_i\left(g\right)=c_i\left(g\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

That is, within each criminal group, the criminal that has the highest number of links gets the loot. If two or more criminals have the highest number of links, then they share the loot equally among them.

Criminal i has a probability $q_i\left(g\right)$ of being caught. It is assumed that the higher the number of links a criminal has, the lower his individual probability of being caught. We assume that the probability of being caught is simply given by

$$q_i\left(g\right)=\frac{n-1-d_i\left(g\right)}{n}.$$

In case a criminal is caught, a fraction $\varphi$ of his personal gains from crime are seized. We require $\varphi <n/(n-1)$ to guarantee that expected payoffs are positive for a criminal with the highest degree in his group.

The total payoffs of criminal i belonging to criminal group $S\in \mathcal{P}\left(g\right)$ are, therefore, equal to

$$\begin{array}{ccc}\hfill Y_i\left(g\right)& =& {\pi }_S\left(g\right){\alpha }_i\left(g\right)(1-q_i\left(g\right)\varphi )B\hfill \\ & =& \left\{\begin{array}{cc}\frac{\#S}{n}\frac{1}{1}\#\{j\in S\mid d_j\left(g\right)=c_i\left(g\right)\}(1-\frac{n-1-d_i\left(g\right)}{n}\varphi )B,\hfill & \mathrm{if}{d}_i\left(g\right)=c_i\left(g\right),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

(1)

3. Horizon-$\mathbf{K}$ Farsighted Set

To determine the criminal networks that emerge in the long run when criminals are neither fully myopic nor completely farsighted, but have some limited degree of farsightedness, we use the horizon-K farsighted set as introduced by Herings, Mauleon and Vannetelbosch [20]. In this section, we provide a formal definition of this concept in Definition 3, state a useful result to determine whether a set of networks is a horizon-K farsighted set and establish the uniqueness of a horizon-K farsighted set in Theorem 1.

A farsighted improving path of length $K\ge 1$ from a network g to a network $g^{\prime }$ is a finite sequence of networks $g_0,\dots ,g_K$ with $g_0=g$ and $g_K=g^{\prime }$ such that for any $k\in \{1,\dots ,K-1\}$ either (i) $g_{k+1}=g_k-ij$ for some $ij$, such that $Y_i\left(g_K\right)>Y_i\left(g_k\right)$ or $Y_j\left(g_K\right)>Y_j\left(g_k\right)$, or (ii) $g_{k+1}=g_k+ij$ for some $ij$, such that $Y_i\left(g_K\right)>Y_i\left(g_k\right)$ and $Y_j\left(g_K\right)\ge Y_j\left(g_k\right)$. If there exists a farsighted improving path of length K from g to $g^{\prime }$, then we write $g{\to }_K{g}^{\prime }$. For a given network g and some $K^{\prime }\ge 1$, let $f_{K^{\prime }}\left(g\right)$ be the set of networks that can be reached from g by a farsighted improving path of length $K\le K^{\prime }$. That is, $f_{K^{\prime }}\left(g\right)=\{g^{\prime }\in \mathbb{G}\mid \exists K\le K^{\prime }$ such that $g{\to }_K{g}^{\prime }\}$. Let $f_{\infty }\left(g\right)=\{g^{\prime }\in \mathbb{G}\mid \exists K\in \mathbb{N}$ such that $g{\to }_K{g}^{\prime }\}$ denote the set of networks that can be reached from g by some farsighted improving path. Lemma 1 in Herings, Mauleon and Vannetelbosch [20] shows that, for every $K\ge 1$, for every $g\in \mathbb{G}$, it holds that $f_K\left(g\right)\subseteq f_{K+1}\left(g\right)$, and that for $K\ge n^{\prime }-1$, for every $g\in \mathbb{G}$, it holds that $f_K\left(g\right)=f_{K+1}\left(g\right)=f_{\infty }\left(g\right)$.

An important concept in the analysis of networks is the one of pairwise stability, as was introduced in Jackson and Wolinsky [3]. A network $g\in \mathbb{G}$ is pairwise stable if (i) for every $ij\in g$, $Y_i\left(g\right)\ge Y_i(g-ij)$ and $Y_j\left(g\right)\ge Y_j(g-ij)$, and (ii) for every $ij\notin g$, if $Y_i\left(g\right)<Y_i(g+ij)$, then $Y_j\left(g\right)>Y_j(g+ij)$. We say that a network $g^{\prime }$ is adjacent to g if $g^{\prime }=g+ij$ or $g^{\prime }=g-ij$ for some $ij$. A network $g^{\prime }$defeatsg if either $g^{\prime }=g-ij$ and $Y_i\left(g^{\prime }\right)>Y_i\left(g\right)$ or $Y_j\left(g^{\prime }\right)>Y_j\left(g\right)$, or if $g^{\prime }=g+ij$ with $(Y_i\left(g^{\prime }\right),Y_j\left(g^{\prime }\right))>(Y_i\left(g\right),Y_j\left(g\right))$. (We use the notation $(Y_i\left(g^{\prime }\right),Y_j\left(g^{\prime }\right))>(Y_i\left(g\right),Y_j\left(g\right))$ for $Y_i\left(g^{\prime }\right)\ge Y_i\left(g\right)$ and $Y_j\left(g^{\prime }\right)\ge Y_j\left(g\right)$ with at least one inequality holding strictly, $(Y_i\left(g^{\prime }\right),Y_j\left(g^{\prime }\right))\ge (Y_i\left(g\right),Y_j\left(g\right))$ for $Y_i\left(g^{\prime }\right)\ge Y_i\left(g\right)$ and $Y_j\left(g^{\prime }\right)\ge Y_j\left(g\right)$, and $(Y_i\left(g^{\prime }\right),Y_j\left(g^{\prime }\right))\gg (Y_i\left(g\right),Y_j\left(g\right))$ for $Y_i\left(g^{\prime }\right)>Y_i\left(g\right)$ and $Y_j\left(g^{\prime }\right)>Y_j\left(g\right)$.) A network is pairwise stable if, and only if, it is not defeated by another network. It is also easy to see that $g^{\prime }\in f_1\left(g\right)$ if and only if $g^{\prime }$ defeats g. We can, therefore, define the pairwise stable networks $P_1$ as those $g\in \mathbb{G}$, for which $f_1\left(g\right)=\varnothing$. For $K\ge 1$, let $P_K=\{g\in \mathbb{G}\mid f_K\left(g\right)=\varnothing \}$ denote the set of horizon-Kpairwise stable networks. (Jackson [24] defines a network as farsightedly pairwise stable if there is no farsighted improving path emanating from it. This concept reverts to $P_{\infty }$ and refines the set of pairwise stable networks.) Clearly, if $K^{\prime }>K,$ then $P_{K^{\prime }}\subseteq P_K.$ For values of K above 1, the use of $P_K$ as a solution concept suffers from a number of drawbacks. Not only are these sets often empty, at a more fundamental level, they are problematic as a solution concept as they allow for deviations to networks that are not stable themselves, and do not ultimately result in a stable network.

A refinement of pairwise stability is obtained when we require the network g to defeat every other adjacent network, so $g\in f_1\left(g^{\prime }\right)$ for every network $g^{\prime }$ adjacent to g. We call such a network g pairwise dominant. For $K\ge 1$, a network $g\in \mathbb{G}$ is horizon-Kpairwise dominant if, for every $g^{\prime }$ adjacent to g, it holds that $g\in f_K\left(g^{\prime }\right)$. The set of horizon-Kpairwise dominant networks is denoted by $D_K$.

The set $f_K^2\left(g\right)=f_K\left(f_K\left(g\right)\right)=\{g^{\prime \prime }\in \mathbb{G}\mid \exists g^{\prime }\in f_K\left(g\right)$ such that $g^{\prime \prime }\in f_K\left(g^{\prime }\right)\}$ consists of those networks that can be reached by a composition of two farsighted improving paths of length, at most K from g. We extend this definition and, for $m\in \mathbb{N}$, we define $f_K^m\left(g\right)$ as those networks that can be reached from g by means of m compositions of farsighted improving paths of a length of, at most, K. Let $f_K^{\infty }$ denote the set of networks that can be reached from g by means of any number of compositions of farsighted improving paths of length at most K. Lemma 2 in Herings, Mauleon and Vannetelbosch [20] shows that, for every $K\ge 1$, and for every $g\in \mathbb{G}$, it holds that $f_K^{\infty }\left(g\right)\subseteq f_{K+1}^{\infty }\left(g\right)$, and that for $K\ge n^{\prime }-1$, for every $g\in \mathbb{G}$, it holds that $f_K^{\infty }\left(g\right)=f_{K+1}^{\infty }\left(g\right)=f_{\infty }^{\infty }\left(g\right)$.

Jackson and Watts [22] have defined the notion of a closed cycle. A set of networks C is a cycle if, for any $g^{\prime }\in C$ and $g\in C\setminus \left\{g^{\prime }\right\}$, there is a sequence of improving paths of length 1, connecting g to $g^{\prime }$, i.e., $g^{\prime }\in f_1^{\infty }\left(g\right)$. A cycle C is a maximal cycle if it is not a proper subset of a cycle. A cycle C is a closed cycle if $f_1^{\infty }\left(C\right)=C$, so there is no sequence of improving paths of length 1 starting at some network in C and leading to a network that is not in C. A closed cycle is necessarily a maximal cycle. For every pairwise stable network $g\in P_1$, the set $\left\{g\right\}$ is a closed cycle. The set of networks belonging to a closed cycle is non-empty.

The notion of a horizon-K farsighted set is based on two main requirements: horizon-K deterrence of external deviations and horizon-K external stability.

A set of networks G satisfies the horizon-K deterrence of external deviations if all possible deviations from any network $g\in G$ to a network outside G are deterred by a threat of ending worse-off or equally well-off. (We use the notational convention that $f_{-1}\left(g\right)=\varnothing$ for every $g\in \mathbb{G}$.)

Definition 1.

For $K\ge 1$, a set of networks $G\subseteq \mathbb{G}$ satisfies horizon-K deterrence of external deviations if for every $g\in G,$

(a) 

∀$ij\notin g$ such that $g+ij\notin G$,

$\exists g^{\prime }\in [f_{K-2}(g+ij)\cap G]\cup [f_{K-1}(g+ij)\setminus f_{K-2}(g+ij)]$ such that

$(Y_i\left(g^{\prime }\right),Y_j\left(g^{\prime }\right))=(Y_i\left(g\right),Y_j\left(g\right))$ or $Y_i\left(g^{\prime }\right)<Y_i\left(g\right)$ or $Y_j\left(g^{\prime }\right)<Y_j\left(g\right)$,

(b) 

∀$ij\in g$ such that $g-ij\notin G$,

$\exists g^{\prime },g^{\prime \prime }\in [f_{K-2}(g-ij)\cap G]\cup [f_{K-1}(g-ij)\setminus f_{K-2}(g-ij)]$ such that

$Y_i\left(g^{\prime }\right)\le Y_i\left(g\right)$ and $Y_j\left(g^{\prime \prime }\right)\le Y_j\left(g\right)$.

Condition (a) in Definition 1 captures the addition of a link $ij$ to a network $g\in G$, which leads to a network $g+ij$ outside of G, is deterred by the threat of ending in $g^{\prime }$. Here, $g^{\prime }$ is such that either there is a farsighted improving path of length smaller than or equal to $K-2$, from $g+ij$ to $g^{\prime }$, and $g^{\prime }$ belongs to G, or there is a farsighted improving path of length equal to $K-1$ from $g+ij$ to $g^{\prime }$, and there is no farsighted improving path from $g+ij$ to $g^{\prime }$ of smaller length. Condition (b) is a similar requirement, but for the case where a link is severed. (Since the degree of farsightedness of players is equal to $K,$ Herings, Mauleon and Vannetelbosch [20] distinguish farsighted improving paths of length less than or equal to $K-2$ after a deviation from g to $g+ij$, and farsighted improving paths of length equal to $K-1$. In the former case, the reasoning capacity of the players is not yet reached, and the threat of ending in $g^{\prime }$ is only credible if it belongs to the set G. In the latter case, the only way to reach $g^{\prime }$ from g requires K steps of reasoning or even more; one step in the deviation to $g+ij$ and at least $K-1$ additional steps in any farsighted improving path from $g+ij$ to $g^{\prime }$. Since this exhausts the reasoning capacity of the players, the threat of ending in $g^{\prime }$ is credible, irrespective of whether it belongs to G or not.)

A set of networks G satisfies horizon-K external stability if, from any network outside of G, there is a sequence of farsighted improving paths of lengths smaller than or equal to K, leading to some network in $G.$

Definition 2.

For $K\ge 1$, a set of networks $G\subseteq \mathbb{G}$ satisfies horizon-K external stability if, for every $g^{\prime }\in \mathbb{G}\setminus G$, $f_K^{\infty }\left(g^{\prime }\right)\cap G\ne \varnothing$.

This requirement implies that if we allow players with a degree of farsightedness equal to K to successively create or delete links, they will ultimately reach the set G irrespective of the initial network.

Definition 3.

For $K\ge 1$, a set of networks $G_K\subseteq \mathbb{G}$ is a horizon-K farsighted set if it is a minimal set satisfying horizon-K deterrence of external deviations and horizon-K external stability.

Herings, Mauleon, and Vannetelbosch [20] prove that a horizon-K farsighted set of networks exists. For $K=1$, Theorem 3 of Herings, Mauleon, and Vannetelbosch [20] show that there is a unique horizon-1 farsighted set consisting of all networks that belong to a closed cycle. This result does not carry over to higher levels of K.

As shown by Herings, Mauleon, and Vannetelbosch [20], the collection of horizon-K farsighted sets is independent of K when $K\ge n^{\prime }+1$. Moreover, for every pairwise farsightedly stable set $G_{\infty }$ defined by Herings, Mauleon and Vannetelbosch [10], there is a set $G^{\prime }\subseteq G_{\infty }$ such that $G^{\prime }$ is a level-$(n^{\prime }+1)$ farsighted set. (Herings, Mauleon and Vannetelbosch [10] define a pairwise farsightedly stable set as a set $G_{\infty }$ of networks satisfying horizon-∞ deterrence of external deviations and minimality, but with horizon-∞ external stability replaced by the requirement that, for every $g^{\prime }\in \mathbb{G}\setminus G_{\infty }$, $f_{\infty }\left(g^{\prime }\right)\cap G_{\infty }\ne \varnothing$.)

The following theorem of Herings, Mauleon and Vannetelbosch [20] will be used in the next section to identify the horizon-K farsighted set of criminal networks.

Theorem 1

(Herings, Mauleon and Vannetelbosch [20]). Consider some $K\ge 2$. If $g\in D_J$ for some $J<K$ and for every $g^{\prime }\in \mathbb{G}\setminus \left\{g\right\}$, it holds that $g\in f_K^{\infty }\left(g^{\prime }\right)$, then $\left\{g\right\}$ is a horizon-K farsighted set. If, moreover, $g\in P_K$, then $\left\{g\right\}$ is the unique horizon-K farsighted set.

Theorem 1 requires that $g\in D_J$ for some $J<K$, so we have to show that $g\in f_J\left(g^{\prime }\right)$ for all $g^{\prime }$ adjacent to g. The higher J, the weaker this requirement, so we could replace the requirement $g\in D_J$ for some $J<K$ by $g\in D_{K-1}$. To show that $g\in f_K^{\infty }\left(g^{\prime }\right)$ for all $g^{\prime }\ne g$, we have to find a sequence of farsighted improving paths of K at most, which connect $g^{\prime }$ to g. Very often, the analysis of farsighted improving paths of small lengths is already sufficient. The higher K, the easier it is to satisfy the conditions of Theorem 1 and to find a singleton horizon-K farsighted set. it ic necessary, to show that $g\in P_K$ requires that $f_K\left(g\right)=\varnothing$. This requirement is more difficult to satisfy for increasing values of K.

4. Horizon-K Farsighted Set of Criminal Networks

Throughout this section, we assume $n\ge 3$. Figure 1 presents the pay-offs for three-player criminal networks with $B=9$ and $\varphi =1$ in expression (1).

Table 1. The elements of $f_K\left(g\right)\setminus \left\{g\right\}$ for 3-player criminal networks with $B=9$ and $\varphi =1$.

g	${\mathit{f}}_1\left(\mathit{g}\right)$	${\mathit{f}}_2\left(\mathit{g}\right)$	${\mathit{f}}_{\mathit{K}}\left(\mathit{g}\right),\mathit{K}\ge 3$
$g^0$	$g^1,g^2,g^3$	$g^1,g^2,g^3$	$g^1,g^2,g^3,g^7$
$g^1,g^2,g^3$		$g^7$	$g^7$
$g^4$	$g^1,g^2,g^7$	$g^1,g^2,g^7$	$g^1,g^2,g^3,g^7$
$g^5$	$g^1,g^3,g^7$	$g^1,g^3,g^7$	$g^1,g^2,g^3,g^7$
$g^6$	$g^2,g^3,g^7$	$g^2,g^3,g^7$	$g^1,g^2,g^3,g^7$
$g^7$

shows the farsighted improving paths for the different possible values of K. It can be verified that the farsighted improving paths for the three-player case do not depend on the specific choices for B and $\varphi$.

For the three-player case, we compute the closed cycles and use Theorem 3 in Herings, Mauleon and Vannetelbosch [20] to conclude that $G_1=P_1=\{g^1,g^2,g^3,g^7\}$ is the horizon-1 farsighted set, so $G_1$ consists of all pairwise stable networks. There are many networks that are stable when players are myopic.

For $K\ge 2$, we apply Theorem 1 to show that $G_K=\left\{g^7\right\}$ is the unique horizon-K farsighted set. It holds that $g^7\in D_1$ and $g^7\in f_2^{\infty }\left(g\right)$ for every $g\ne g^7$, so $\left\{g^7\right\}$ is a horizon-K farsighted set. Since $g^7\in P_K$, it follows from Theorem 1 that $\left\{g^7\right\}$ is the unique horizon-K farsighted set. If criminals behave myopically, they may not go beyond forming a single link in the three player case. However, with a degree of farsightedness of at least 2, the complete criminal network emerges as the unique prediction.

The remainder of this section is devoted to the analysis of criminal networks with a general number n of players. As in the three-criminal case, there are many networks that are pairwise stable in the n-person case. The complete network is easily verified as pairwise stable. The generalization of the networks $g^1$, $g^2$, and $g^3$ for the three-criminal case to the n-criminal case would be any network consisting of complete components, where no two components have the same degree. However, any network with a single component where all players have a degree at least equal to two, and one player has a degree that is at least two times higher than the degree of any other player, is pairwise stable.

We will next argue that $\left\{g^N\right\}$ is a horizon-K farsighted set whenever $K\ge n-1$.

We first show that the complete network is pairwise dominant.

Lemma 1.

For criminal networks, it holds that $g^N\in D_1$.

Proof. 

Consider the network $g^N-ij$ for some $ij$. It holds that

$$d_i(g^N-ij)=d_j(g^N-ij)<c_i(g^N-ij)=c_j(g^N-ij),$$

so

$$Y_i(g^N-ij)=Y_j(g^N-ij)=0<Y_i\left(g^N\right)=Y_j\left(g^N\right),$$

and $g^N\in f_1(g^N-ij)$. We have shown that $g^N\in D_1$. □

We show next that the complete network can be reached from any starting network by repeated application of, at most, $n-1$ degrees of farsightedness.

Lemma 2.

For criminal networks it holds for every $g\in \mathbb{G}\setminus \left\{g^N\right\}$ that $g^N\in f_{n-1}^{\infty }\left(g\right)$.

Proof. 

Step 1. If g has a component which is not complete, then there is $g^{\prime }\in f_{n-1}\left(g\right)$, such that $g⊊g^{\prime }$.

Let $S\in \mathcal{P}\left(g\right)$ be a criminal group, such that some internal links are missing, $g_{\mid S}\ne g^S$.

If, for every $i\in S$, it holds that $d_i\left(g\right)=c_i\left(g\right)$, so all players in S have the same degree, then any two unlinked players i and j in S create a link to form the network $g+ij$ and improve their payoffs, since the increase in their degree increases the share in the loot and lowers the probability of being caught for both players, ${\alpha }_i(g+ij)>{\alpha }_i\left(g\right)$, ${\alpha }_j(g+ij)>{\alpha }_j\left(g\right)$, $q_i(g+ij)<q_i\left(g\right)$, and $q_j(g+ij)<q_j\left(g\right)$, so $Y_i(g+ij)>Y_i\left(g\right)$ and $Y_j(g+ij)>Y_j\left(g\right)$. We have that $g{\to }_{1}g+ij$, so clearly $g+ij\in f_{n-1}\left(g\right)$.

If the players in S do not all have the same degree, let $i\in S$ be a player with $d_i\left(g\right)=c_i\left(g\right).$ If $c_i\left(g\right)<\#S-1,$ then Player i links with any Player j, such that $ij\notin g$ to form the network $g+ij.$ It holds that $Y_i(g+ij)>Y_i\left(g\right)>0$, since ${\alpha }_i(g+ij)\ge {\alpha }_i\left(g\right)$ and $q_i(g+ij)<q_i\left(g\right),$ whereas $Y_j(g+ij)\ge Y_j\left(g\right).$ We have that $g{\to }_{1}g+ij$, so clearly $g+ij\in f_{n-1}\left(g\right)$.

If the players in S do not all have the same degree and there is a player in S with degree $\#S-1,$ then let $i\in S$ be a player with $d_i\left(g\right)<\#S-1.$ Player i consecutively links to all players $j\in S$, such that $ij\notin g$, thereby forming a network $g^{\prime }$, where he has degree $\#S-1$. The payoffs of Player i are, in every step, equal to $Y_i\left(g\right)=0$ until the final step, where his payoffs increase to $Y_i\left(g^{\prime }\right)>0$. Every player j that i links to has a degree below $\#S-1$ and, therefore, payoffs equal to $0\le Y_j\left(g^{\prime }\right)$. We have that $g^{\prime }\in f_{\#S-2}\left(g\right)\subseteq f_{n-1}\left(g\right)$ by Lemma 1 in Herings, Mauleon and Vannetelbosch [20].

Step 2. If all components of g are complete and $g\ne g^N$, then there is $g^{\prime }\in f_{n-1}\left(g\right)$, such that $g⊊g^{\prime }$.

The assumptions of Step 2 imply that g consists of at least two criminal groups. Let $S^1$ and $S^2$ be two criminal groups in $\mathcal{P}\left(g\right)$.

If $\#S^1=\#S^2$, then form a link between a Player $i\in S^1$ and a Player $j\in S^2$. Since $q_i\left(g\right)>q_i(g+ij)$, we have that

$$Y_i\left(g\right)=\frac{1}{n}(1-q_i\left(g\right)\varphi )B<\frac{\#S^1}{n}(1-q_i(g+ij)\varphi )B=Y_i(g+ij).$$

By the same calculation, it follows that $Y_j\left(g\right)<Y_j(g+ij)$, so $g{\to }_{1}g+ij$, and, therefore, $g+ij\in f_{n-1}\left(g\right)$.

Otherwise, it holds, without loss of generality, that $\#S^1<\#S^2$. Select some player $i\in S^1$ and a set J consisting of $\#S^2+1-\#S^1$ players in $S^2$, who are consecutively linked to Player i to form network $g^{\prime }$. The resulting finite sequence of networks is denoted $g_0,\dots ,g_K$ with $g_0=g$ and $g_K=g^{\prime }$. Notice that $K\le n-1$. We next show that, for every $k\in \{0,\dots ,K-1\}$, $(Y_i\left(g_k\right),Y_{j_k}\left(g_k\right))<(Y_i\left(g_K\right),Y_{j_k}\left(g_K\right))$, where $j_k\in J$ is such that $g_{k+1}=g_k+i{j}_k$, thereby proving that $(g_0,\dots ,g_K)$ is a farsighted improving path and completing the proof of Step 2.

For every player $j\in J$, we have

$$d_j\left(g_K\right)=d_i\left(g_K\right)=c_i\left(g_K\right),$$

and for all other players, the degree is strictly less than $c_i\left(g_K\right)$, so

$$Y_j\left(g_K\right)=Y_i\left(g_K\right)=\frac{\#S^1+\#S^2}{n}\frac{1}{\#S^2+2-\#S^1}(1-q_i\left(g_K\right)\varphi )B.$$

For $k=0$, we have

$$\begin{array}{ccc}\hfill Y_i\left(g_0\right)& =& \frac{1}{n}(1-q_i\left(g\right)\varphi )B<Y_i\left(g_K\right),\hfill \\ \hfill Y_{j_0}\left(g_0\right)& =& \frac{1}{n}(1-q_{j_0}\left(g\right)\varphi )B<Y_{j_0}\left(g_K\right),\hfill \end{array}$$

where we use $q_i\left(g_0\right)>q_i\left(g_K\right)$ and $q_{j_0}\left(g_0\right)>q_{j_0}\left(g_K\right)$ to obtain the strict inequalities.

For $k=1,\dots ,K-1$, it holds that Player i is connected to Player $j_0$, so $d_i\left(g_k\right)<d_{j_0}\left(g_k\right)=c_i\left(g_k\right)$, so ${\alpha }_i\left(g_k\right)=0$ and $0=Y_i\left(g_k\right)<Y_i\left(g_K\right)$. Similarly, it holds that Player $j_k$ is connected to Player $j_0$, so $d_{j_k}\left(g_k\right)<d_{j_0}\left(g_k\right)=c_{j_k}\left(g_k\right)$, so ${\alpha }_{j_k}\left(g_k\right)=0$ and $0=Y_{j_k}\left(g_k\right)<Y_{j_k}\left(g_K\right)$.

Step 3. For every $g\in \mathbb{G}\setminus \left\{g^N\right\}$, it holds that $g^N\in f_{n-1}^{\infty }\left(g\right)$.

By combining the results of Step 1 and Step 2, we have that for every $g\in \mathbb{G}\setminus \left\{g^N\right\}$, there is $g^{\prime }\in f_{n-1}\left(g\right)$ with strictly more links than g. Since the complete network $g^N$ has $n(n-1)/2$ links, we find that $g^N\in f_{n-1}^{n(n-1)/2}\left(g\right)\subseteq f_{n-1}^{\infty }\left(g\right)$. □

Using Theorem 1, we now prove that the complete network $\left\{g^N\right\}$ is a horizon-K farsighted set for every $K\ge n-1$. (Herings, Mauleon and Vannetelbosch [10] show that, in the example of criminal networks with n players, the complete network $\left\{g^N\right\}$ is a pairwise farsightedly stable set.) Notice that the level of farsightedness needed to sustain the complete network $\left\{g^N\right\}$ is quite small when compared to the number of potential networks and the maximum length of paths. (Once the network connecting delinquents is endogenous, Calvo-Armengol and Zenou [6] find that all complete networks, where all players in the pool of criminals are linked to each other, are pairwise stable. Notice that the size of the pool of criminals depends on the wage on the labor market.)

Theorem 2.

For criminal networks, it holds that $\left\{g^N\right\}$ is a horizon-K farsighted set for every $K\ge n-1$.

Proof. 

By Lemma 1 we have that $g^N\in D_1$. By Lemma 2 we have that for every $g^{\prime }\in \mathbb{G}\setminus \left\{g^N\right\}$ it holds that $g^N\in f_{n-1}^{\infty }\left(g^{\prime }\right)\subseteq f_K^{\infty }\left(g^{\prime }\right)$, where the inclusion follows from Lemma 2 in Herings, Mauleon and Vannetelbosch [20]. We are now in a position to apply Theorem 1 and conclude that $\left\{g^N\right\}$ is a horizon-K farsighted set. □

How about the uniqueness of $\left\{g^N\right\}$ as a horizon-K farsighted set? It is tempting to use the approach of Theorem 1 and show such a result by proving that $g^N\in P_K$. However, consider the case with six players and let $g^{\prime }=g^N-16-26-35-45$. For any value of B and $\varphi$, (We maintain the assumption that $\varphi <n/(n-1).$) we claim that $g^{\prime }\in f_{12}\left(g^N\right)$, so $g^N\notin P_{12}$. Since the network $g^{\prime }$ is connected, $d_1\left(g^{\prime }\right)=d_2\left(g^{\prime }\right)=d_3\left(g^{\prime }\right)=d_4\left(g^{\prime }\right)=4$, and $d_5\left(g^{\prime }\right)=d_6\left(g^{\prime }\right)=3$, it holds for any $i\in \{1,2,3,4\}$ that $Y_i\left(g^{\prime }\right)=(1/4-\varphi /24)B>B/6=Y_i\left(g^N\right)$ and for any $j\in \{5,6\}$ that $Y_j\left(g^{\prime }\right)=0<B/6=Y_j\left(g^N\right)$. The construction of the farsighted improving path is, however, more subtle than simply deleting links 16, 26, 35, and 45 in some order. Indeed, after the deletion of three such links, there are exactly two players with the maximum degree and they would obtain strictly lower payoffs by cutting their link, and would be unwilling to do so. Avoiding this problem requires more farsightedness and involves all players in $\{1,2,3,4\}$, first cutting two of their mutual links, before severing the links with players 5 and 6, and finally restoring their mutual links. One explicit farsighted improving path results from $g^N-12-23-34-41-16-26-35-45+12+23+34+41$ and takes 12 steps. We have denoted the player with an incentive to cut a link first, so $-16$, for instance, means that Player 1 cuts his link with Player 6, whereas $-61$ would mean that Player 6 cuts his link with Player 1. It can be verified that each step in this farsighted improving path is feasible.

We conclude this section by showing that if criminals are not too farsighted, then $g^N\in P_K$, so $\left\{g^N\right\}$ is the unique horizon-K farsighted set. More precisely, we will now consider $K=n-1$. We show first that any network in $f_{n-1}\left(g^N\right)$ has a single component involving all players.

Lemma 3.

For criminal networks it holds for every $g^{\prime }\in f_{n-1}\left(g^N\right)$ that $\mathcal{P}\left(g^{\prime }\right)=\left\{N\right\}$.

Proof. 

Consider the criminal group S of Player 1 in $g^{\prime }$. We show that it contains all players. Suppose it contains only $s\le n-1$ players. Then, starting from $g^N$, those s players have to cut all their links with all other players in $N\setminus S$. This involves at least $s(n-s)$ steps. For a fixed n, the concavity of $s(n-s)$ in s implies that $s(n-s)$ is minimized at $s=1$ or $s=n-1$. Substitution of these values of s shows the minimum to be equal to $n-1$ at both $s=1$ and $s=n-1$. When the s players cut all their links with all other players in $N\setminus S$, all the players in N are strictly worse off, since the probability of being caught has strictly increased and the probability of winning the loot has decreased, contradicting $g^{\prime }\in f_{n-1}\left(g^N\right)$. □

We show next that the complete network $g^N$ is horizon-$(n-1)$ pairwise stable.

Lemma 4.

For criminal networks, it holds that $g^N\in P_{n-1}$.

Proof. 

Suppose $g^{\prime }$ is an element of $f_{n-1}\left(g^N\right)$. Let $g_0,\dots ,g_K$ with $g_0=g^N$ and $g_K=g^{\prime }$ be a farsighted improving path of length $K\le n-1$. By Lemma 3 it holds that $c_i\left(g^{\prime }\right)$ is independent from i, so we denote it by c. Let $M\subseteq N$ be such that $i\in M$ if and only if $d_i\left(g^{\prime }\right)=c$ and denote the cardinality of M by m. It cannot be that $m=n$, since then all players have lower payoffs in $g^{\prime }$ than in $g^N$ because the probability of being caught is higher in $g^{\prime }$ than in $g^N$. Since by Lemma 3 $g^{\prime }$ is connected, it follows that $Y_j\left(g^{\prime }\right)=0$ for all $j\in N\setminus M$. A player $j\in N\setminus M$ will, therefore, not sever a link at any network in the farsighted improving path $g_0,\dots ,g_K$. It follows that

$$\sum _{i\in M}(n-1-d_i\left(g^{\prime }\right))\ge \sum _{j\in N\setminus M}(n-1-d_j\left(g^{\prime }\right)).$$

Since $d_i\left(g^{\prime }\right)>d_j\left(g^{\prime }\right)$ whenever $i\in M$ and $j\in N\setminus M$, we have that $m>n/2$.

Since at least one link $ij$ with $i\in M$ and $j\in N$ is missing in $g^{\prime }$, it follows that the maximum degree in $g^{\prime }$ satisfies $c\le n-2$.

The number K is equal to the number of times a link $ij$ is severed with $i\in M$ and $j\in N\setminus M$ plus the number of times a link $ij$ is cut with $i,j\in M$ plus the number of link additions. We next argue that lower bounds for these three numbers are given by $2(n-m)$, $2m-n-1$, and 1, respectively.

Since all players in $N\setminus M$ experienced the severance of at least two links, and any such link is cut by a player in M, a lower bound for the first number is $2(n-m)$.

For $k=0,\dots ,K$, let $L\left(g_k\right)=\{i\in N\mid d_i\left(g_k\right)=n-1\}$ be the set of players with degree $n-1$ and let $\ell \left(g_k\right)=\#L\left(g_k\right)$ be its cardinality. Clearly, it holds that $\ell \left(g^N\right)=n$ and $\ell \left(g^{\prime }\right)=0$. Let $k^{\prime }$ be the lowest value of k such that $\ell \left(g_k\right)\le m$ for all $k\ge k^{\prime }$. Since $\ell \left(g_k\right)-\ell \left(g_{k+1}\right)\le 2$, we find that $\ell \left(g_{k^{\prime }}\right)=m$ or $\ell \left(g_{k^{\prime }}\right)=m-1$. The sum of the cardinality $\ell \left(g_{k^{\prime }}\right)$ of $L\left(g_{k^{\prime }}\right)$ and the cardinality m of M is, therefore, at least $2m-1$. Since there are only n players, it follows that $\#(L\left(g_{k^{\prime }}\right)\cap M)$, the cardinality of the set of players in $L\left(g_{k^{\prime }}\right)$ that belong to M, is at least $2m-n-1$.

For all $k\ge k^{\prime }$, for all $i\in L\left(g_k\right)$, it holds that $Y_i\left(g_k\right)>Y_i\left(g^{\prime }\right)$, since the loot has to be shared with fewer or the same number of criminals, and the probability of being caught is strictly lower when comparing $g_k$ to $g^{\prime }$. Such a player i will, therefore, never choose to sever a link himself, so whenever a link involving player $i\in L\left(g_k\right)$ is severed when going from $g_k$ to $g_{k+1}$, it must be by a player in $M\setminus L\left(g_k\right)$. It follows that $\ell \left(g_k\right)-\ell \left(g_{k+1}\right)\le 1$. Since $\#(L\left(g_{k^{\prime }}\right)\cap M)\ge 2m-n-1$, we find that going from $g_{k^{\prime }}$ to $g^{\prime }$ involves the deletion of at least $2m-n-1$ links $ij$ with $i,j\in M$.

We next argue that the move from $g_{K-1}$ to $g_K$ involves a link addition. Supposing not, then there is $ij$ with $i\in M$, such that $g_K=g_{K-1}-ij$ and $Y_i\left(g_K\right)>Y_i\left(g_{K-1}\right).$ Since $d_i\left(g_{K-1}\right)=c_i\left(g_{K-1}\right)>c_i\left(g_K\right)=d_i\left(g_K\right)$, it follows that at $g_K$, i has to share the loot with more criminals and has a higher probability of being caught than at $g_{K-1}$, so $Y_i\left(g_K\right)<Y_i\left(g_{K-1}\right)$, leading to a contradiction. Consequently, the move from $g_{K-1}$ to $g_K$ involves a link addition.

We have proven that $K\ge 2(n-m)+2m-n-1+1=n$, which contradicts our original supposition that $K\le n-1$. Consequently, it holds that $f_{n-1}\left(g^N\right)=\varnothing$. □

Using Theorem 1 we now prove that the complete network $\left\{g^N\right\}$ is the unique horizon-$(n-1)$ farsighted set.

Theorem 3.

For criminal networks, it holds that $\left\{g^N\right\}$ is the unique horizon-$(n-1)$ farsighted set.

Proof. 

By Lemma 1 we have that $g^N\in D_1$. By Lemma 2 we have that for every $g^{\prime }\in \mathbb{G}\setminus \left\{g^N\right\}$ it holds that $g^N\in f_{n-1}^{\infty }\left(g^{\prime }\right)$. By Lemma 4 it holds that $g^N\in P_{n-1}$. We are now in a position to apply Theorem 1 and conclude that $\left\{g^N\right\}$ is the unique horizon-$(n-1)$ farsighted set. □

We have found that in criminal networks with n criminals, the set consisting of the complete network is a horizon-K farsighted set whenever the degree of farsightedness of the criminals is larger thanor equal to $(n-1)$. Moreover, the complete network is the unique horizon-$(n-1)$ farsighted set. Hence, we obtain a very sharp prediction for intermediate degrees of farsightedness (i.e., a degree of farsightedness equal to $n-1$), and show that a limited degree of farsightedness (i.e., at least $n-1$) is sufficient to recover the predictions obtained in case of completely farsighted criminals. It therefore seems important to acquire knowledge about the degree of farsightedness of criminals, to determine which criminal networks are likely to emerge in the long run.

A better knowledge of the structural properties of criminal networks will help in understanding the impact of peer influence on delinquent behavior and addressing adequate and novel delinquency-reducing policies. A first line of reasoning would be that crime could be reduced by increasing the value of the parameter $\varphi ,$ the fraction of the personal gains of a criminal that are seized when the criminal is caught. On the other hand, higher values of $\varphi$ make it more costly to be caught and, as a response, criminals will establish networks where they have higher degrees and are therefore less likely to be convicted. Our analysis points towards the complete network as emerging in the long run, irrespective of the value of $\varphi .$ The value of $\varphi$ is, therefore, completely ineffective as a policy instrument.

5. Conclusions

We study the criminal networks that will emerge in the long run when criminals are neither fully myopic nor completely farsighted, but have some limited degree of farsightedness. We adopt the horizon-K farsighted set of Herings, Mauleon and Vannetelbosch [20] to show how the predictions regarding stable criminal networks relate to the degree of farsightedness. A horizon-K farsighted set always exists. We find that in criminal networks with n criminals, the set consisting of the complete network is a horizon-K farsighted set whenever the degree of farsightedness of the criminals is larger than or equal to $(n-1)$. Moreover, the complete network is the unique horizon-$(n-1)$ farsighted set. Hence, a limited degree of farsightedness is sufficient to recover the predictions obtained in the case of completely farsighted criminals.