Robust Predictions in Dynamic Games

University of Minnesota Ph.D. dissertation. August 2018. Major: Economics. Advisors: Aldo Rustichini, David Rahman. 1 computer file (PDF); ix, 53 pages.This dissertation is about understanding the robustness property of predictions to misspecification of higher-order beliefs in dynamic games with payoff uncertainty. In particular, it asks: Which simplifying assumptions about beliefs provide robust predictions in dynamic games? The most important result of this dissertation, presented in the second chapter, is to show that lack of robustness is a generic property of predictions consistent with (interim) sequential rationalizability (ISR) unless the prediction is unique. I consider this to be an essential and novel contribution to the literature of robustness in game theory since it challenges the validity of the standard approach to modeling uncertainty in dynamic games because it gives rise, for almost every model of uncertainty, to spurious predictions. Typically, when analyzing a model, different parameters represent different assumptions of the model, and therefore, predictions from the model are sensitive to the specification of those parameters. For example, it is well known that in the standard Bayesian approach to games with incomplete information, a crucial parameter that requires to be specified, and at the same time is neither observable nor verifiable without any error from the point of view of researchers, is players beliefs and hierarchies of beliefs; hence, because of the previous observation, it happens that in many applications hierarchies of beliefs encode strong (informational) assumptions, and as I already mentioned, behavioral predictions (e.g., in the form of Perfect Bayesian Equilibrium, Interim Sequential Rationalizability, among others) depend on those assumptions; moreover, in some cases, this dependence can be very sensitive at the tails of the hierarchies of beliefs specified in the model. The robustness property refers, in this case, to the possibility of guarantee that slight changes in the specification of the parameters do not lead to significant changes in predictions, since at least from a methodological point of view, if the property holds generically it would provide a justification for the validity of the standard approach to model uncertainty. One approach to understanding the robustness property of set-valued solution concepts in games is to ask: Which predictions remain valid after all common certain-belief assumptions are relaxed? Penta (2012) have shown that in (finite) dynamic games with incomplete information the only predictions that remain valid after relaxing all the assumptions about beliefs and hierarchies of beliefs are those consistent with (interim) sequential rationalizability. In other words, ISR is the strongest solution concept such that, for every model of beliefs it is possible to guarantee that an outcome that was ruled out by ISR is ruled out for every approximation of the model. This result implies lack of robustness of any refinement of ISR as, for example, any of the familiar equilibrium concepts. In this dissertation, a stronger notion of robustness is considered, that is if, in addition, it is possible to guarantee that there are no spurious predictions, in the sense that for every predicted outcome of a solution concept there is no approximation ruling that outcome out. This last notion is formalized through a notion of full continuity of predictions with respect to beliefs and hierarchies of beliefs. An approach to this question for static games was given in Ely and Peski (2011). They introduced the concept of critical types as precisely those assumptions on beliefs that are vulnerable to misspecification, that is, as those types to which there are spurious predictions consistent with rationalizability. They showed that critical types are non-generic (rare). The key argument in Ely and Peski's result exploits the fact that, in static games, rationalizability does not depend on the timing of the arrival of players' information. However, in dynamic games, ISR does depend on the timing of information. In particular, players beliefs are restricted only at the beginning of the game, and via conditioning whenever possible. However, at zero probability events, conditional beliefs are unrestricted. I exploit this observation to show that Ely and Peski's result does not hold in dynamic settings: lack of robustness is a generic property of ISR whenever it delivers multiple predictions. As ISR often delivers multiple predictions in applications, this result casts doubts on the interpretation and validity of solution concepts such as Perfect Bayesian Equilibrium, Sequential Equilibrium, and ISR itself. By acknowledging model misspecification of higher-order beliefs, there is no type in Harsanyi's framework at which a researcher can guarantee that no slight perturbation on the modeling assumptions exists which rules some prediction out unless the prediction is unique. Finally, I propose an ongoing research agenda in the problem of robust predictions in dynamic games. In particular, we consider dynamic games with payoff uncertainty and, as in Siniscalchi (2016a,b), assume that players in the game choose strategies according to structural rationality. Players with structural preferences induce, at the beginning of the game, a collection of alternative hypothesis about how the game is going to unfold; and rank any two strategies depending on the expected payoff under those alternative priors in a lexicographic way. A strategy is structurally rational if it is maximal. We propose to study general properties of (weak) interim structural rationalizability (IStR), a solution concept that characterizes the behavioral implications of common certainty in structural rationality. In the case of Bayesian dynamic games with incomplete information, we are particularly interested in the robustness properties of IStR to perturbations of higher-order beliefs. As illustrated by an example, three results are conjecture: a structure theorem of structural rationalizability, a characterization of critical types, and a non-generic result of the set of critical types