# 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.

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 but only partially effective vaccine can make can reduce everyone’s welfare. 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.

Draft Date: Nov 6, 2017

I define a new solution concept—the Nash bargaining solution with Endogenous Outside Options (NEOO, pronounced as “new”)—for stationary environments in which individuals form coalitions to jointly produce output. NEOO is a payoff profile x that gives each player her biggest Nash bargaining payoff—over all possible coalitions—when others’ outside options are given by x. I show that NEOO always exists and is unique, and I describe an infinite-horizon bargaining model that uniquely implements it in stationary strategies. The set of feasible coalitions—that is, those whose output is not smaller than the sum of their members’ NEOO payoffs—has a tier structure, with shares determined from the top tier down: Output in each feasible coalition is shared according to the Nash bargaining solution subject to the endogenous outside option of its members in higher tiers. Preference and productivity shocks propagate from higher to lower tiers, but not vice versa.

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.

Draft Date: May 26, 2018