The analysis of threshold contagion processes in large networks is challenging. While the lack of accurate network data is often a major obstacle, finding optimal interventions is computationally intractable even in well-measured large networks. To obviate these issues we consider threshold contagion over networks sampled from a graphon—a flexible stochastic network formation model—and show that in this case the contagion outcome can be predicted by only exploiting information about the graphon. To this end, we exploit a second interpretation of graphons as graph limits to formally define a threshold contagion process on a graphon for infinite populations. We then show that contagion in large but finite sampled networks is well approximated by graphon contagion. This convergence result suggests that one can design interventions for large sampled networks by first solving the equivalent problem for an infinite population interacting according to the limiting graphon. We show that, under suitable regularity assumptions, the latter is a tractable problem and we provide analytical characterizations for the extent of contagion and for optimal seeding policies in graphons with both finite and infinite agent types.


We study a model of network externalities transmitted over links wherein the externalities stem from links themselves rather than nodes. For example, bilateral investments can fail and trigger cascading defaults of firms. Joint research projects can succeed and innovations can be diffused over the network. We characterize stable and efficient networks. Under negative externalities, disjoint cliques are stable and efficient. Under positive externalities complete networks and star networks are stable. Efficient networks feature are a mix: pineapple networks which consist of one large clique and a star network appended to each other.

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This paper introduces a model of endogenous network formation and systemic risk. In it, firms form joint ventures called ‘links’ which are subsequently subjected to either good or bad shocks. Bad shocks incentivize default. Links yield full benefits only if the counter-party does not subsequently default on the project. Accordingly, defaults triggered by bad shocks render firms insolvent and defaults propagate via links. The model yields three insights. First, stable networks with ex-ante identical agents exhibit a core-periphery structure. Second, an increase in the probability of good shocks increases systemic risk. Third, because the network formed depends on the correlation between shocks to links, an observer who misconceives the correlation will underestimate the probability of system-wide default by a factor of a half.


Optimal regulatory restrictions on banks have to solve a delicate balance. Tighter regulations reduce the likelihood of banks’ distress. Looser regulations foster the allocation of funds towards productive investments. With multiple banks, optimal regulation becomes even more challenging. Banks form partnerships in the interbank lending market in order to face liquidity needs and to meet investment possibilities. We show that the interbank network can suddenly collapse when regulations are pushed beyond a critical level, with a discontinuous increase in systemic risk as the cross-insurance of banks collapses.


Dillenberger (2010) introduced the negative certainty independence (NCI) axiom, which captures the certainty effect phenomenon. He left open the question of whether there are continuous and monotone preference relations over simple lotteries that satisfy NCI but do not belong to the betweenness class of preferences considered by Chew (1989) and Dekel (1986). We answer this question in the affirmative.


This paper studies the dynamic implications of the endogenous rate of time preference depending on the stock of capital, in a one-sector growth model. The planner’s problem is presented and the optimal paths are characterized. We prove that there exists a critical value of initial stock, in the vicinity of which, small differences lead to permanent differences in the optimal path. Indeed, we show that a development trap can arise even under a strictly convex technology. In contrast with the early contributions that consider recursive preferences, the critical stock is not an unstable steady state so that if an economy starts at this stock, an indeterminacy will emerge. We also show that even under a convex–concave technology, the optimal path can exhibit global convergence to a unique stationary point. The multipliers system associated with an optimal path is proven to be the supporting price system of a competitive equilibrium under externality and detailed results concerning the properties of optimal (equilibrium) paths are provided. We show that the model exhibits globally monotone capital sequences yielding a richer set of potential dynamics than the classic model with exogenous discounting.


Various blockchain systems have been designed for dynamic networked systems. Due to the nature of the systems, the notion of “time” in such systems is somewhat subjective; hence, it is important to understand how the notion of time may impact these systems. This work focuses on an adversary who attacks a Proof-of-Work (POW) blockchain by selfishly constructing an alternative longest chain. We characterize optimal strategies employed by the adversary when a difficulty adjustment rule alà Bitcoin applies.

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[Article version] Longest-chain Attacks: Difficulty Adjustment and Timestamp Verifiability

We study an adversary who attacks a Proof-of-Work (POW) blockchain by selfishly constructing an alternative longest chain. We characterize optimal strategies employed by the adversary when a difficulty adjustment rule alà Bitcoin applies. As time (namely the times-tamp specified in each block) in most permissionless POW blockchains is somewhat subjective, we focus on two extreme scenarios: when time is completely verifiable, and when it is completely unverifiable. We conclude that an adversary who faces a difficulty adjustment rule will find a longest-chain attack very challenging when timestamps are verifiable. POW blockchains with frequent difficulty adjustments relative to time reporting flexibility will be substantially more vulnerable to longest-chain attacks. Our main fining provides guidance on the design of difficulty adjustment rules and demonstrates the importance of timestamp verifiability.


We analyze a threshold contagion process in a graphon. We interpret the graphon as a stochastic network formation model. We investigate whether contagion in networks sampled from a graphon can be predicted by only exploiting information about the graphon. Our main results show that contagion in large but finite networks sampled from a graphon is well approximated by contagion in the graphon. We illustrate our results by providing analytical characterizations of contagion and optimal seeding policies in graphons with finite and with infinite types.