Mohamed Belhaj

Faculty ,  Centrale Méditerranée

Econometrics, Finance and mathematical methods
Belhaj
Status
Professor
Research domain(s)
Finance, Game theory and social networks
Thesis
2005, University of Toulouse 1
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Contact
mohamed.belhaj[at]centrale-med.fr
+33(0)4 13 55 25 25
Address

AMU - AMSE
5-9 Boulevard Maurice Bourdet, CS 50498
​13205 Marseille Cedex 1

Abstract We consider agents organized in an undirected network of local complementarities. A principal with a fixed budget offers costly bilateral contracts in order to increase the sum of agents' effort. We study contracts rewarding effort exceeding the effort made in the absence of the principal. First, targeting a subgroup of the whole society becomes optimal under substantial contracting costs, which significantly increases the computational complexity of the principal's problem. In particular, under sufficiently low intensity of complementarities, a correspondence is established between optimal targeting and an NP-hard problem. Second, for any intensities of complementarities, the optimal unit returns offered to all targeted agents are positive for all contracting costs and in general heterogeneous, even though networks are undirected. Yet, heterogeneity never leads to negative returns, which implies that, with these linear payment schemes, coordination is never an issue for the principal.
Keywords Networked synergies, Optimal targeting, Linear scheme
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Abstract A monopoly sells a network good to a large population of consumers. We explore how the monopoly's profit and the consumer surplus vary with the arrival of public information about the network structure. The analysis reveals that, under homogeneous preferences for the good, degree assortativity ensures that information arrival increases both profit and consumer surplus. In contrast, heterogeneous preferences for the good can create a tension between consumer surplus and profit.
Keywords Monopoly, Network effects, Network information, Bonacich centrality, Degree assortativity, Assortative mixing
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Abstract A principal targets agents organized in a network of local complementarities, in order to increase the sum of agents' effort. We consider bilateral public contracts à la Segal (1999). The paper shows that the synergies between contracting and non-contracting agents deeply impact optimal contracts: they can lead the principal to contract with a subset of the agents, and to refrain from contracting with central agents.
Keywords Network, Synergies, Aggregate effort, Optimal group targeting
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Abstract We consider a network game with local complementarities. A policymaker, aiming at minimizing or maximizing aggregate effort, contracts with a single agent on the network to trade effort change against transfer. The policymaker has to find the best agent and the optimal contract to offer. Our study shows that for all 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 and the self-loop centrality. We also characterize key players under linear quadratic utilities for various contractual arrangements.
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Abstract We address the problem of a planner looking for the efficient network when agents play a network game with local complementarities and links are costly. We show that for general network cost functions, efficient networks belong to the class of Nested-Split Graphs. Next, we refine our results and find that, depending on the specification of the network cost function, complete networks, core-periphery networks, dominant group architectures, quasi-star and quasi-complete networks can be efficient.
Keywords Strategic complementarity, Network games, Nested split graphs
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Abstract We consider a society in which each agent has one unit of a resource to allocate between two activities. Agents are organized in a social network, and each activity generates complementarities between neighbors. We find multiplicity of equilibrium for high intensity of interaction, and we characterize equilibria in terms of specialization and polarization. Overall, results reveal the crucial role played by network geometry. The results also suggest that the structure of the social network should be taken into account for the design of a public policy in favor of a specific activity.
Keywords Economie quantitative
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Abstract We study network games under strategic complementarities. Agents are embedded in a fixed network. They choose a positive, continuous action and interact with their network neighbors. Interactions are positive and actions are bounded from above. We first derive new sufficient conditions for uniqueness, covering all concave as well as some non-concave best responses. We then study the relationship between position and action and identify situations where a more central agent always plays a higher action in equilibrium. We finally analyze comparative statics. We show that a shock may not propagate throughout the entire network and uncover a general pattern of decreasing interdependence.
Keywords Uniqueness, Strategic complementarities, Network games, Interdependence, Centrality
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Abstract This paper explores the effect of moral hazard on both risk-taking and informal risk-sharing incentives. Two agents invest in their own project, each choosing a level of risk and effort, and share risk through transfers. This can correspond to farmers in developing countries, who share risk and decide individually upon the adoption of a risky technology. The paper mainly shows that the impact of moral hazard on risk crucially depends on the observability of investment risk, whereas the impact on transfers is much more utility dependent.
Keywords Economie quantitative
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Abstract We consider a model of interdependent efforts, with linear interaction and lower bound on effort. Our setting encompasses asymmetric interaction and heterogeneous agents’ characteristics. We examine the impact of a rise of cross-effects on aggregate efforts. We show that the sign of the comparative static effects is related to a condition of balancedness of the interaction. Moreover, we point out that asymmetry and heterogeneous characteristics are sources of non-monotonic variation of aggregate efforts.
Keywords Strategic interaction, Social network, Heterogeneous char, Asymmetric interaction, Aggregate efforts
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Abstract This paper introduces demotivation in the context of social comparison in networks. Social comparison is modeled as a status effect rewarding or penalizing agents according to their relative performance with respect to local peers. A demotivated agent faces both a reduced marginal return to effort and a psychological cost. In the absence of demotivation, social comparison leads to higher effort levels but reduces equilibrium welfare. Introducing demotivation leads to two main findings. First, it generates a network game of strategic substitutes. Second, despite the individual psychological costs incurred by demotivated agents, it can enhance overall welfare—by alleviating social pressure to exert effort and by generating positive externalities for peers.
Keywords Social Comparison, Demotivation, Networks, Strategic Substitutes, Equilibrium Welfare
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Abstract A set of agents is aware of the existence of an economic opportunity, and compete for the associated prize. We study incentives to communicate about the existence of this economic opportunity to uninformed agents when the winner of the prize shares it with others, through some exogenous sharing rule. Communicating about the opportunity has two conflicting effects: it increases competition, but it can also increase the likelihood of receiving a large share of the prize. We find that, for any sharing rule, there is a minimum equilibrium, which Pareto dominates all other equilibria. We also find that under bilaterally symmetric sharing, more sharing generates more communication. We then discuss these results along several extensions.
Keywords Investment, Communication, Sharing Network, Rival Opportunity
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Abstract We analyze risk-taking regulation when financial institutions are linked through shareholdings. We model regulation as an upper bound on institutions' default probability, and pin down the corresponding limits on risk-taking as a function of the shareholding network. We show that these limits depend on an original centrality measure that relies on the cross-shareholding network twice: (i) through a risk-sharing effect coming from complementarities in risk-taking and (ii) through a resource effect that creates heterogeneity among institutions. When risk is large, we find that the risk-sharing effect relies on a simple centrality measure: the ratio between Bonacich and self-loop centralities. More generally, we show that an increase in cross-shareholding increases optimal risk-taking through the risk-sharing effect, but that resource effect can be detrimental to some banks. We show how optimal risk-taking levels can be implemented through cash or capital requirements, and analyze complementary interventions through key-player analyses. We finally illustrate our model using real-world financial data and discuss extensions toward including debt-network, correlated investment portfolios and endogenous networks.
Keywords Financial Network, Risk-Taking, Prudential Regulation
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Abstract We consider agents organized in an undirected network of local complementarities. A principal with a limited budget offers costly bilateral contracts in order to increase the sum of agents' effort. We study excess-effort linear payment schemes, i.e. contracts rewarding effort in excess to the effort made in absence of principal. The analysis provides the following main insights. First, for all contracting costs, the optimal unit returns offered to every targeted agent are positive and generically heterogeneous. This heterogeneity is due to the presence of outsiders, who create asymmetric interaction between contracting agents. Second, when contracting costs are low, it is optimal to contract with everyone and optimal unit returns are identical for all agents. Third, when contracting costs are sufficiently high, it becomes optimal to target a subset of agents, and optimal targeting can lead to NP-hard problems. In particular, when the intensity of complementarities is sufficiently low, a correspondence is established between optimal targeting and the densest k subgraph problem. Overall, the optimal targeting problem involves a trade-off between centrality and budget spending-central agents are influential, but are also more budget-consuming. These considerations can lead the principal to not target central agents.
Keywords Networked synergies, Aggregate effort, Optimal group targeting, Linear contract
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Abstract A principal targets agents organized in a network of local complementarities, in order to increase the sum of agents' effort. We consider bilateral public contracts à la Segal (1999). The paper shows that the synergies between contracting and non-contracting agents deeply impact optimal contracts: they can lead the principal to contract with a subset of the agents, and to refrain from contracting with central agents.
Keywords Multi-agent contracting, Network, Synergies, Aggregate effort, Optimal group targeting
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Abstract We consider a network game with local complementarities. A policymaker, aiming at minimizing or maximizing aggregate effort, contracts with a single agent on the network to trade effort change against transfer. The policymaker has to find the best agent and the optimal contract to offer. Our study shows that for all 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 and a weighted measure of the number of closed walks originating from the agent. We also characterize key players under linear quadratic utilities for various contractual arrangements.
Keywords Key player, Network, Linear interaction, Incentives, Contract, Limited budget
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Abstract We study the value of network information in a context of monopoly pricing in the presence of local network externalities. We compare a setting in which all players, i.e. the monopoly and consumers, know the network structure and consumers' private preferences with a setting in which players only know the joint distribution of preferences, in-degrees and out-degrees. We give conditions under which network information increases profit or/and consumer surplus. The analysis reveals the crucial role played by four properties: degree assortativity, homophily (in preferences), preference-degree assortativity and preference-Bonacich centrality assortativity.
Keywords Price discrimination, Bonacich centrality, Network information, Degree assortativity, Homophily, Preference-degree assortativity, Preference-Bonacich centrality assortativity, Network effects, Monopoly
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Abstract A principal offers bilateral contracts to a set of agents organized in a network conveying synergies, in a context where agents' efforts are observable and where the principal's objective increases with the sum of efforts. We characterize optimal contracts as a function of agents' positions on the network. The analysis shows that contract enforceability is key to understand optimality. We also examine linear contracting and we analyze the situation where the principal is constrained to contract with a single agent on the network. Last, we extend this setting to network entry.
Keywords Network, Strategic complementarity, Enforceability, Optimal contracting, Multi-agency
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Abstract We consider agents playing a linear network game with strategic complementarities. We analyse the problem of a policy maker who can change the structure of the network in order to increase the aggregate efforts of the individuals and/or the sum of their utilities, given that the number of links of the network has to remain fixed. We identify some link reallocations that guarantee an improvement of aggregate efforts and/or aggregate utilities. With this comparative statics exercise, we then prove that the networks maximising both aggregate outcomes (efforts and utilities) belong to the class of Nested-Split Graphs.
Keywords Network, Linear interaction, Bonacich Centralities, Strategic complementarity, Nested split graphs
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Abstract This paper brings into focus a link between the investment and financing decisions of a firm which has an access to costly debt financing. Our analysis shows that lump-sum debt issuance costs play a prominent role in a determination of the optimal investment strategy. Faced with larger lump-sum debt issuance costs, a firm will optimally set up a higher-scale investment project in order to "compensate" dead-weight financing costs by higher return. Moreover, in the presence of lump-sum debt issuance costs, the optimal investment scale of financially constrained firms exhibits an inverted U-shaped relationship with the firm's borrowing capacity, so that relatively more/less constrained firms will realize smaller investment projects, whereas firms with an intermediate borrowing capacity will undertake larger investment.
Keywords Debt issuance costs, Investment intensity, Real options, Credit constraints
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Abstract We study network games with linear best-replies and strategic complementarities. We assume that actions are continuous but bounded from above. We show that there is always a unique equilibrium. We find that two key features of these games under small network effects may not hold when network effects are large. Action may not be aligned with network centrality and the interdependence between agents' actions may be broken.
Keywords Network games, Strategic complementarities, Supermodular Games, Bonacich centrality
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Abstract Early regulator interventions into problem banks is one of the key suggestions of Basel II. However, no guidance is given on their design. To fill this gap, we outline an incentive-based preventive supervision strategy that eliminates bad asset management in banks. Two supervision techniques are combined: continuous regulator intervention and random audits. Random audit technologies differ as to quality and cost. Our design ensures good management without excessive supervision costs, through a gradual adjustment of supervision effort to the bank's financial health. We also consider preventive supervision in a setting where audits can be delegated to an independent audit agency, showing how to induce agency compliance with regulatory instructions in the least costly way.
Keywords Banking supervision, Random audit, Incentives, Moral hazard, Delegation
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