RESEARCH

 

WORKING PAPERS

 

● "Schelling Redux: An Evolutionary Dynamic Model of Residential Segregation," with W.H. Sandholm, 06/2007.

Abstract: Schelling (1971) introduces a seminal model of the dynamics of residential segregation in an isolated neighborhood. His model combines agent heterogeneity with explicit behavior dynamics; as such it is presented informally, and with the use of “semi-equilibrium” restrictions on out-of-equilibrium play. In this paper, we use recent techniques from evolutionary game theory to introduce a formal version of Schelling’s model, one that dispenses with equilibrium restrictions on the adjustment process. We show that key properties of the resulting infinite-dimensional dynamic can be derived using a simple finite-dimensional dynamic that captures aggregate behavior. We determine conditions for the stability of integrated equilibria, and we derive a strong restriction on out-of-equilibrium dynamics that implies global convergence to equilibrium: along any solution trajectory, one population’s aggregate behavior adjusts monotonically, while the other’s changes direction at most once. We present a variety of examples, and we show how extensions of the basic model can be used to study both policies to promote integration and alternative specifications of agents’ preferences.

 

● " Stochastic Evolution with Perturbed Payoffs and Rapid Play," with W.H. Sandholm, 07/2008.

Abstract: A population of agents are recurrently matched to play a symmetric normal form game. The strategies’ payoffs are subject to i.i.d. shocks, and agents face many rounds of play, and hence many rounds of payoff shocks, during each period. At the end of the period, one agent receives a revision opportunity, and switches to the strategy whose average perturbed payoff during the period was highest. Our stochastic stability analysis concerns long run behavior as the number of rounds per period approaches infinity. We prove that risk dominant equilibria are always selected in 2x2 games, and that efficient equilibria are always selected in nxn pure coordination games played by two agents, but that more generally equilibrium selection depends on the exact nature of the shock distribution.

    To perform this analysis, we define the unlikelihood of choosing strategy i given payoff vector π to be the exponential rate at which the probability of choosing strategy i vanishes as the number of rounds grows large. We characterize unlikelihoods using techniques from large deviations theory, and prove that the unlikelihood of choosing strategy i is a convex function of the payoff vector π. As a byproduct of this analysis, we derive the rates of decay of choice probabilities in the multinomial probit model as the shock variance approaches zero.

 

PUBLICATION

 

● "The Projection Dynamic and the Replicator Dynamic," with W.H. Sandholm and R. Lahkar. Forthcoming, Games and Economic Behavior.

Abstract: We investigate a variety of connections between the projection dynamic and the replicator dynamic. At interior population states, the standard microfoundations for the replicator dynamic can be converted into foundations for the projection dynamic by replacing imitation of opponents with “revision driven by insecurity” and direct choice of alternative strategies. Both dynamics satisfy a condition called inflowoutflow symmetry, which causes them to select against strictly dominated strategies at interior states; still, because it is discontinuous at the boundary of the state space, the projection dynamic allows strictly dominated strategies to survive in perpetuity. The two dynamics exhibit qualitatively similar behavior in strictly stable and null stable games. Finally, the projection and replicator dynamics both can be viewed as gradient systems in potential games, the latter after an appropriate transformation of the state space.

 

 

WORKS IN PROGRESS (with tentative abstracts)

 

○ "Robustness of Bad Reputation" (coming soon!)

I study the long-run relationship between an expert and customers where an expert is a long-run player and customers are short-run players. Many studies in the reputation literature show that the long-run player’s payoff in a repeated game is bounded from below. However, in “Bad Reputation”, Ely and Valimaki (2003) show that the long-run player’s payoff in a repeated game has an extremely small upper bound. I show that this result depends on the information structure of the model. Specifically, the complete absence of information about the expert’s type after certain histories is the key point which leads to the bad reputation effect. By letting the expert to serve several customers in each period instead of serving only one customer per period, I am able to show that there is a Nash equilibrium in which the expert’s payoff per customer is the maximum stage game payoff per customer if the expert is patient enough. Furthermore, this result holds under some conditions in a different setting where the expert serves only one customer per period and customers have two types; informed type with probability 1-ε, and uniformed type with probability ε. If the expert’s patience level is sufficiently high relative to the probability of the uninformed customer, the bad reputation effect vanishes. However, if the probability of uninformed customer is sufficiently small relative to expert’s patience level, the bad reputation effect persists.

 

○ "An Evolutionary Model of Segregation and Inequality"

Empirical studies on segregation and inequality show that racial composition and wealth levels of neighborhoods change over time, and that declines in group inequality have not reduced extreme levels of segregation in major cities. I extend the first essay by allowing agents to have preferences not only over the racial composition of the neighborhood but also over the wealth level of the neighborhood. Since my model employs explicitly specified evolutionary dynamics it is able to track the racial composition and wealth level of the neighborhood over time, and thus it is able to explain these empirical observations.

 

 

○ "The Quantile Best Response Dynamic"

I introduce and analyze a new evolutionary dynamic where each agent is a quantile maximizer when making decisions. I use the model of preferences introduced by Rostek (2006) in which an individual compares uncertain alternatives through a quantile of the induced utility distributions, and which nests maxmin and maxmax but also captures less extreme preferences. In all fundamental evolutionary dynamics agents are expected utility maximizers, and except the best response dynamic all has a mean dynamic which can be described by an ordinary differential equation. On the other hand, when agents are quantile maximizers, the resulting mean dynamic is not an ordinary differential equation but a differential inclusion. There are very large regions in the simplex where there is no unique solution to the mean dynamic. This work is very preliminary at this stage. I plan to establish the mathematical properties of this dynamic as well as explain some economic phenomenon which can’t be explained by standard models.