Targeting the Key-player: An Incentive-based Approach

In many economic situations, agents’ behaviors depend on their peers. Such interactions are well documented for criminal  activities, or for R&D partnerships, or for protective investment against terrorism. These networked interdependencies are exploited by policymakers, either to reduce or to increase the overall network activity. For example, effort reduction may be desirable where criminal activities are concerned, whereas increased effort may be valuable in R&D investment or protection against terrorism. Determining whether, and how, the structure of these interdependencies should be taken into account by public policies is a hot topic.

One possible policy consists in trading effort change against transfers. However, contracting costs often substantially increase with the number of contracts. Given a limited budget, the policymaker may then resort to making deals with a limited subset of agents. We study the problem of a policymaker contracting with a single agent in order to minimize or maximize aggregate effort in presence of networked complementarities. The problem can be solved in two steps: first, studying the optimal contract with any agent, and then selecting the best agent, called the key player. Our analysis shows that, for all agents’ utilities with linear best-responses, it only takes two statistics about the position of each agent on the network to identify the key player: the Bonacich centrality, which counts the (weighted) number of walks from the agent to all agents on the network, and the self-loop centrality, which counts the (weighted) number of closed walks starting from the agent. In more detail, we show that the policy effect is the product of the change in targeted agent effort (what we call the individual component) and the change in aggregate effort following a one-unit change in the effort of the targeted agent (the network component). The latter is a pure network multiplier effect and is equal to the ratio of the Bonacich centrality to the self-loop centrality. The former is a function of both statistics, the budget level, the shape of utility, and whether the policymaker maximizes or minimizes effort. The key player depends on all these parameters, but only the Bonacich centrality and the selfloop centrality are network-dependent. We also further characterize the key player under the standard case of linear quadratic utilities, which reveals that the key player is budget-dependent. We then discuss our modeling assumptions. We first address the issue of contract enforceability. In this model, opportunism is a concern under effort maximization but plays no role under effort minimization. In contrast to enforceable contracts, we find that the key player does not depend on budget level. We also examine the excess-effort linear contract, which is a natural contract given that agents exert effort in the absence of a principal. This contract puts the inter-centrality index in the spotlight, which is reminiscent of the key player analysis of Ballester et al. (2006).

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