Negatively Correlated Bandits

We analyze a two-player game of strategic experimentation with two-armed bandits. Each player has to decide in continuous time whether to use a safe arm with a known payoff or a risky arm whose likelihood of delivering payoffs is initially unknown. The quality of the risky arms is perfectly negatively correlated between players. In marked contrast to the case where both risky arms are of the same type, we find that learning will be complete in any Markov perfect equilibrium if the stakes exceed a certain threshold, and that all equilibria are in cutoff strategies. For low stakes, the equilibrium is unique, symmetric, and coincides with the planner's solution. For high stakes, the equilibrium is unique, symmetric, and tantamount to myopic behavior. For intermediate stakes, there is a continuum of equilibria

Strategic Experimentation with Poisson Bandits

We study a game of strategic experimentation with two-armed bandits where the risky arm distributes lump-sum payoffs according to a Poisson process. Its intensity is either high or low, and unknown to the players. We consider Markov perfect equilibria with beliefs as the state variable. As the belief process is piecewise deterministic, payoff functions solve differential-difference equations. There is no equilibrium where all players use cut-off strategies, and all equilibria exhibit an `encouragement effect' relative to the single-agent optimum. We construct asymmetric equilibria in which players have symmetric continuation values at sufficiently optimistic beliefs yet take turns playing the risky arm before all experimentation stops. Owing to the encouragement effect, these equilibria Pareto dominate the unique symmetric one for sufficiently frequent turns. Rewarding the last experimenter with a higher continuation value increases the range of beliefs where players experiment, but may reduce average payoffs at more optimistic beliefs. Some equilibria exhibit an `anticipation effect': as beliefs become more pessimistic, the continuation value of a single experimenter increases over some range because a lower belief means a shorter wait until another player takes over

Breakdowns

We study a continuous-time game of strategic experimentation in which the players try to assess the failure rate of some new equipment or technology. Breakdowns occur at the jump times of a Poisson process whose unknown intensity is either high or low. In marked contrast to existing models, we find that the cooperative value function does not exhibit smooth pasting at the efficient cut-off belief. This finding extends to the boundaries between continuation and stopping regions in Markov perfect equilibria. We characterize the unique symmetric equilibrium, construct a class of asymmetric equilibria, and elucidate the impact of bad versus good Poisson news on equilibrium outcomes

Market Experimentation in a Dynamic Differentiated-Goods Duopoly

We study the evolution of prices in a symmetric duopoly where firms are uncertain about the degree of product differentiation. Customers sometimes perceive the products as close substitutes, sometimes as highly differentiated. Firms learn about their competitive environment from the quantities sold and a background signal. As the information of the market outcomes increases with the price differential, there is scope for active learning. In a setting with linear demand curves, we derive firms' pricing strategies as payoff-symmetric mixed or correlated Markov perfect equilibria of a stochastic differential game where the common posterior belief is the natural state variable. When information has low value, firms charge the same price as would be set by myopic players, and there is no price dispersion. When firms value information more highly, on the other hand, they actively learn by creating price dispersion. This market experimentation is transient, and most likely to be observed when the firms' environment changes sufficiently often, but not too frequently.Duopoly experimentation, Bayesian learning, stochastic differential game, Markov-perfect equilibrium, mixed strategies, correlated equilibrium.

Housing Market Dynamics: On the Contribution of Income Shocks and Credit Constraints (Revised Version)

We propose a life-cycle model of the housing market with a property ladder and a credit constraint. We focus on equilibria which replicate the facts that credit constraints delay some households' first home purchase and force other households to buy a home smaller than they would like. The model helps us identify a powerful driver of the housing market: the ability of young households to afford the down payment on a starter home, and in particular their income. The model also highlights a channel whereby changes in income may yield housing price overshooting, with prices of trade-up homes displaying the most volatility, and a positive correlation between housing prices and transactions. This channel relies on the capital gains or losses on starter homes incurred by credit-constrained owners. We provide empirical support for our arguments with evidence from both the U.K. and the U.S

Negatively Correlated Bandits

We analyze a two-player game of strategic experimentation with two-armed bandits. Each player has to decide in continuous time whether to use a safe arm with a known payoff or a risky arm whose likelihood of delivering payoffs is initially unknown. The quality of the risky arms is perfectly negatively correlated between players. In marked contrast to the case where both risky arms are of the same type, we find that learning will be complete in any Markov perfect equilibrium if the stakes exceed a certain threshold, and that all equilibria are in cutoff strategies. For low stakes, the equilibrium is unique, symmetric, and coincides with the planner’s solution. For high stakes, the equilibrium is unique, symmetric, and tantamount to myopic behavior. For intermediate stakes, there is a continuum of equilibria.

Heterogeneity within Communities: A Stochastic Model with Tenure Choice

Standard explanations for the income heterogeneity within neighborhoods rely on differences of preferences across households and heterogeneity of the housing stock. We propose an alternative and complementary explanation. We construct a stochastic equilibrium sorting model where (1) income is the sole dimension of household heterogeneity, (2) households form state-contingent housing location plans that may involve moves over their lifetimes, (3) households choose whether to own or rent depending on the housing expenditure risk associated with each tenure mode, and (4) there is a probability that newcomer households move in and compete for homes with native households. Income mixing within neighborhood arises for two reasons. First, allowing natives to form state-contingent housing location plans breaks the indivisibility of housing consumption implicit in the literature where households choose their location once and for all. Second, natives can insure themselves against rent fluctuations by buying their home prior to the realization of the population shock; newcomers cannot. As a result, poorer natives stay in the more desirable communities and only richer newcomers move in these communities. Evidence from U.S. metropolitan areas supports the effects predicted by the model

Negatively Correlated Bandits

We analyze a two-player game of strategic experimentation with two-armed bandits. Each player has to decide in continuous time whether to use a safe arm with a known payoff or a risky arm whose likelihood of delivering payoffs is initially unknown. The quality of the risky arms is perfectly negatively correlated between players. In marked contrast to the case where both risky arms are of the same type, we find that learn- ing will be complete in any Markov perfect equilibrium if the stakes exceed a certain threshold, and that all equilibria are in cutoff strategies. For low stakes, the equilib- rium is unique, symmetric, and coincides with the planner's solution. For high stakes, the equilibrium is unique, symmetric, and tantamount to myopic behavior. For inter- mediate stakes, there is a continuum of equilibria.Strategic Experimentation; Two-Armed Bandit; Exponential Distribution; Poisson Process; Bayesian Learning; Markov Perfect Equilibrium

Homeownership

We develop a dynamic stochastic equilibrium model of two locations within a city where heterogeneous households make joint location and tenure mode decisions. To investigate the effect of homeownership on equilibrium prices and allocations, we compare the response of this model economy to a labor shock with that of a rental-only version. This comparison yields three results. First, homeownership enables more households to remain in the more desirable location at the expense of newcomers. Second, homeownership adds to the volatility of the housing market. Third, homeownership may amplify the dispersion of household income within a location. Homeownership raises distributional issues. The households who consume the most housing gain the most from the ability to own their home. Newcomers to the city are the main losers

Strategic Experimentation with Poisson Bandits

We study a game of strategic experimentation with two-armed bandits where the risky arm distributes lump-sum payoffs according to a Poisson process. Its intensity is either high or low, and unknown to the players. We consider Markov perfect equilibria with beliefs as the state variable. As the belief process is piecewise deterministic, payoff functions solve differential-difference equations. There is no equilibrium where all players use cut-off strategies, and all equilibria exhibit an `encouragement effect' relative to the single-agent optimum. We construct asymmetric equilibria in which players have symmetric continuation values at sufficiently optimistic beliefs yet take turns playing the risky arm before all experimentation stops. Owing to the encouragement effect, these equilibria Pareto dominate the unique symmetric one for sufficiently frequent turns. Rewarding the last experimenter with a higher continuation value increases the range of beliefs where players experiment, but may reduce average payoffs at more optimistic beliefs. Some equilibria exhibit an `anticipation effect': as beliefs become more pessimistic, the continuation value of a single experimenter increases over some range because a lower belief means a shorter wait until another player takes over.Strategic Experimentation; Two-Armed Bandit; Poisson Process; Bayesian Learning; Piecewise Deterministic Process; Markov Perfect Equilibrium; Differential-Difference Equation