# Research

*Simple games*serve as abstract representations of voting systems and other group-decision procedures. A

*stable set*(or von Neumann-Morgenstern solution) of a simple game represents a “standard of behavior” that satisfies certain internal and external stability properties.

*Compound simple games*are built out of

*component*games, which are, in turn, “players” of a

*quotient*game. I describe a method to construct

*fair*(or symmetry-preserving) stable sets of compound simple games from fair stable sets of their quotient and components. This method is closely related to the composition theorem of Lloyd Shapley (1963) and contributes to the answer of a question that he formulated: what is the set G of simple games that have a fair stable set? In particular, this method shows that G includes all simple games whose

*factors*--or quotients in their “unique factorization” of Shapley (1967)--are in G. This suggests a path to characterize the set G of all simple games that have a fair stable set.

Different sellers often sell the same good at different prices. Using a stationary bargaining model, I characterize how the equilibrium price distribution of a good depends on its characteristics—that is, on the interaction between its sellers’ costs, its buyers’ values, and a network capturing various frictions associated with trading it. In contrast to the standard random-matching model, I allow traders to strategically choose whom to make offers to, which qualitatively changes how the network shapes prices: As in the random-matching model, the market decomposes into different submarkets, and—in the limit as bargaining frictions vanish—the law of one price holds within but not across them. But strategic choice of partners changes both how the market decomposes into different submarkets and the determinants of each submarket’s price. In particular, the price in each submarket does not depend on its buyer-to-seller ratio. Rather, it is the price that emerges from Nash bargaining between two of its members. Leveraging this observation, I describe an algorithm that shows how the interaction between sellers’ costs, buyers’ values, and the network determines the structure of price dispersion in stationary markets.

**R&R **at *Games and Economic Behavior. *Draft Date: July 13, 2018.

Different types of agents make non-contractible investments before bargaining over both who matches to whom and the terms of trade. In thin markets, the holdup problem—that is, underinvestment caused by agents receiving only a fraction of the returns from their investments—is ubiquitous. However, we show that holdup is not a problem in markets that attract traders over time—even when only a few traders are present in the market at any point in time. In particular, we characterize the type-symmetric Markov perfect equilibria of a non-cooperative investment and bargaining game with sequential entry, and we show that—in every such equilibrium—all agents receive the full returns from their marginal investments in the limit as they become patient. Intuitively, the option to wait for future market participants creates competition—so even apparently-thin markets can be competitive. This provides non-cooperative foundations for the standard price-taking assumption in the literature investigating investment efficiency in competitive matching markets.

Draft date: July 28, 2018

Vaccines are crucial to curb infectious-disease epidemics. Indeed, one of the highest priorities of the National Institutes of Health (NIH) on the HIV front is the development and delivery of a vaccine that is at least moderately effective. However, risk compensation could undermine the ability of partially-effective vaccines to curb epidemics: Since vaccines reduce the cost of risky interactions, vaccinated agents may optimally choose to engage in more of them and, as a result, may increase everyone’s infection probability. We show that—in contrast to the prediction of standard models—things can be worse than that: A free and perfectly safe but only partially effective vaccine can reduce everyone’s welfare, and hence fail to satisfy—in a strong sense—the fundamental principle of “first, do no harm.” The reason is simple: By reducing the cost of risky interactions, a partially-effective vaccine can destabilize the existing interaction structure in favor of a less efficient one. Because of the strategic complementarities in risky interactions that we show arise when agents strategically choose their partners, the most efficient stable interaction structure after the introduction of a partially-effective vaccine can be much denser and—due to the negative externalities of risky interactions—worse for everyone. The result of this paper underscores the importance of taking into account the effects that different interventions have on social structure, and it suggests that the NIH might want to go big—i.e. deliver a highly-effective vaccine—or go home.

In a wide variety of social settings (e.g. crime, education, political activism, technology adoption), players’ returns to their efforts depend on how much effort others exert. Modeling these situations as a network game with strategic complementarities, we show that a player’s cycle centrality — a weighted sum of the number of network cycles that she is in — determines the extent to which she benefits from her complementarities with others. In contrast to the widely-used Bonacich centrality — which measures how efforts propagate through the network — cycle centrality measures how the variance of efforts propagates through the network. A utilitarian social planner who can incentivize one player’s effort targets the one with the highest cycle centrality.

Draft Date: November 6, 2017

I describe a strategic theory of coalition formation that shows how the Nash bargain- ing solution simultaneously determines both the sharing rule and the relevant outside options in each potential coalition. This provides a framework to investigate how eco- nomic shocks propagate via outside options. The qualitative predictions of this theory contrast with those of standard models of coalition formation that build on the Nash bargaining solution. In particular, by allowing the sharing rule in each coalition to be sensitive to its members’ endogenous outside options, agents do not face the holdup problem that otherwise arises, and they do not necessarily match assortatively.