Key Player Policies When Contextual Effects Matter

Abstract

We consider a model where the criminal decision of each individual is affected by not only her own characteristics, but also by the characteristics of her friends (contextual effects). We determine who the key player is, i.e., the criminal who once removed generates the highest reduction in total crime in the network. We propose a new measure, the contextual intercentrality measure, that generalizes the one proposed by Ballester, Calvó-Armengol, and Zenou (2006) by taking into account the change in contextual effects following the removal of the key player. We also provide an example showing that the key player can be different whether contextual effects are taken into account or not. This means that the planner may target the wrong person if it ignores the effect of the “context” when removing a criminal from a network.

Keywords:

• contextual effects
• crime
• key players
• peer effects

ACKNOWLEDGMENTS

We are grateful to Phillip Bonacich as well as two anonymous referees for helpful comments.

Notes

¹There is a growing literature on networks in economics. See, in particular, Goyal (2007), Jackson (2008), and Jackson and Zenou (2014).

²Contextual effects are important, especially for the empirical measure of peer effects in crime. In the standard linear-in-means models, Manski (1993, 2000) has put forward the importance of the reflection problem, which is the difficulty of separating the contextual effect from the endogenous peer effect on own behavior. Recent empirical papers have used the network topology to separate these two effects and to show the importance of contextual effects in education (Calvó-Armengol, Patacchini, & Zenou, 2009; Lin, 2010), obesity (Cohen-Cole & Fletcher, 2008b), and crime (Patacchini & Zenou, 2012).

³Researchers have also defined “group” players so that the planner can remove more than one key player at a time. See, in particular, Ortiz-Arroyo (2010) and Ballester et al. (2010). Ballester et al. (2010) show that the key group problem is NP -hard from the combinatorial perspective. This means that there is no possible sophisticated algorithm such that, given any network, will return the exact key group in reasonable time. They show, however, that the key group problem can be approximated in polynomial-time by the use of a greedy algorithm, where, at each step, the key player formula of Ballester et al. (2006) is used to remove a player. They also show that the error of approximation of using a greedy algorithm instead of solving directly the key group problem is at most 36.79%. As a result, in the present article, we can still use our new intercentrality formula, which takes into account contextual effects, in the greedy algorithm to tackle the issue of group players.

⁴Boldface lowercase letters refer to vectors while boldface capital letters refer to matrices.

⁵Let ρ(G) be the spectral radius of the nonnegative matrix G, that is, the largest absolute value of its eigenvalues. When φ < ρ(G), the inverse [I − φG]⁻¹is well-defined and nonnegative (Debreu & Herstein, 1953).

⁶In fact, uniqueness is a consequence of the spectral condition φρ(G) < 1. Interiority is guaranteed with the additional condition α _i  > 0.

⁷The spectral radius of this graph is 2.17, and thus the condition φμ₁(G) < 1 is satisfied since 2.17 × 0.4 = 0.868 < 1.

⁸In this example, player 1 is both the most active individual and the key player. Ballester et al. (2006) show that, in general, these notions need not to coincide.

⁹The superscripts * and C indicate the Nash equilibrium values without and with contextual effects, respectively.