Journal of Mathematical Economics 71 (2017), 135—151.
In everyday economic interactions, it is not clear whether each agent's sequential choices are visible to other participants or not: agents might be deluded about others' ability to acquire, interpret or keep track of data. Following this idea, this paper introduces uncertainty about players' ability to observe each others' past choices in extensive-form games. In this context, we show that monitoring opponents' choices does not affect the outcome of the interaction when every player expects their opponents indeed to be monitoring. Specifically, we prove that if players are rational and there is common strong belief in opponents being rational, having perfect information and believing in their own perfect information, then, the backward induction outcome is obtained regardless of which of her opponents' choices each player observes. The paper examines the constraints on the rationalization process under which reasoning according to Battigalli's (1996) best rationalization principle always yields the same outcome irrespective of whether players observe their opponents' choices or not. To this respect we find that the obtention of the backward induction outcome crucially depends on tight higher-order restrictions on beliefs about opponents' perfect information. The analysis provides a new framework for the study of uncertainty about information structures and generalizes the work by Battigalli and Siniscalchi (2002) in this direction.
*Job market paper and Chapter 3 of my PhD dissertation; previous versions of the paper circulated under title ''Incomplete Imperfect Information and Backward Induction.''
International Journal of Game Theory 46 (2017), 595—629.
We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players' behavior deviates from rationality. Instead we assume that there exists a probability p such that all players play rationally with at least probability p, and all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information, and they can be applied to observed frequencies of play to derive a measure p that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.
*Chapter 1 of my PhD dissertation; previous versions of the paper circulated under the title ''Approximate Knowledge of Rationality and Correlated Equilibria.''
Journal of Mathematical Psychology 64—65 (2015), 58—65.
We are interested in 2x2 game situations where players act depending on how they perceive their counterpart although this perception is payoff irrelevant. Perceptions concern dichotomous characteristic. The model includes uncertainty as players know how they perceive their counterpart, but not how they are perceived. We study whether the mere possibility of playing differently depending on the counterpart's perception generates new equilibria. We analyze equilibria in which strategies are contingent on perception. We show that the existence of this discriminatory equilibrium depends on the characteristic in question and on the class of game.
Uncertain Rationality, Depth of Reasoning and Robustness in Games with Incomplete Information,* with Fabrizio Germano and Jonathan Weinstein (Washington University in St. Louis).
Second revision requested by Theoretical Economics.
Predictions under common knowledge of payoffs may differ from those under arbitrarily, but finitely, many orders of mutual knowledge; Rubinstein's (1989) Email game is a seminal example. Weinstein and Yildiz (2007) showed that the discontinuity in the example generalizes: for types with multiple rationalizable (ICR) actions, there exist similar types with unique rationalizable action. In consequence, equilibrium predictions are non-robust to misspecifications of higher-order beliefs. This paper studies how departures from common belief in rationality (CBR) impact on Weinstein and Yildiz's discontinuity. We weaken ICR to ICR-λ, where λ is a sequence whose n-th term is the probability players attach to n-th order belief in rationality. Weinstein and Yildiz's discontinuity is found to hold when higher-order belief in rationality remains above some threshold (constant λ), and to fail when higher-order belief in rationality eventually becomes low enough (λ converging to 0). Thus, when CBR breaks down almost completely at high orders, the intuitive continuity of behavior with respect to perturbations in higher-order beliefs is restored.
*Extremely beefed-up new incarnation of Chapter 2 of my PhD dissertation; previous versions of the paper circulated under the title ''Approximate Rationalizability in Games with Incomplete Information.''
Revision requested by Econometrica.
We study the strategic impact of players' higher order uncertainty over whether their actions are observable to their opponent. We characterize the "robust predictions" of Rationality and Common Belief in Rationality (RCBR), i.e. those which do not depend on the restrictions on players' infinite order beliefs over the extensive form. We show that RCBR is generically unique, and that the robust predictions often support a robust refinement of rationalizability. For instance, in unanimity games, the robust predictions of RCBR rule out any inefficient equilibrium action; in zero-sum games, they support the maxmin solution, solving a classical tension between RCBR and the maxmin logic; in common interest games, RCBR generically ensures efficient coordination of behavior, thereby showing that higher order uncertainty over the extensive form serves as a mechanism for equilibrium coordination on purely eductive grounds.
We also characterize the robust predictions in settings with asynchronous moves, but in which the second mover does not necessarily observe the first mover's action. In these settings, higher order uncertainty over the observability of the earlier choice yields particularly sharp results: in "Nash-commitment games," for instance, RCBR generically selects the equilibrium of the static game which is most favorable to the earlier mover. This means that a first-mover advantage arises whenever higher-order beliefs do not rule out it might exist. Hence, in the presence of extensive form uncertainty, timing alone may determine the attribution of the strategic advantage, independent of the actual observability of choices.
*Previous versions of the paper were presented under the title ''Extensive-Form Uncertainty of the Higher-Order: Robust Predictions, Refinements and Coordination.''
Despite its relevance for understanding behavior in strategic interactions, very little consensus has been reached on how to identify subjects' levels of higher order rationality. To address this, we propose a novel class of two-player games with incomplete information, the e-ring games, that builds on the ring games of Kneeland (2015) and the email game of Rubinstein (1989). We check within subject consistency of identified rationality levels across standard games and find that our games yield rationality levels exhibiting the highest correlation both with the most frequent classification at the individual level, and with an independent measure of cognitive ability.
Strategic Cautiousness as an Expression of Robustness to Ambiguity, with Gabriel Ziegler (Northwestern University).
Economic predictions often hinge on two intuitive premises: agents rule out the possibility of others choosing unreasonable strategies ('strategic reasoning'), and prefer strategies that may provide hedge against unexpected behavior ('cautiousness'). This simultaneous dismissal and consideration of opponents' strategies is conflicting and questions the compatibility of usual economic predictions with reasoning-based foundations. This paper proposes a new take on this classical tension by interpreting cautiousness as robustness to ambiguity. We formalize the latter via a model of incomplete preferences à la Bewley (2002), so that: (i) each player's strategic uncertainty is represented by a possibly non-singleton set of beliefs and (ii) a rational player best-responds to every belief in her set. We show that the interplay between these two features precludes the conflict between strategic reasoning and cautiousness, and provide unifying foundations for rationalizability, the Dekel-Fudenberg procedure and strict rationalizability in terms of rationality, ambiguity and related belief constraints. Furthermore, the framework also covers the iterated elimination of weakly dominated strategies, which is shown to be founded in terms of rationality, ambiguity and related assumption constraints.
- Strategic Behavior and Robustness to Model Misspecification in Dynamic Games with Incomplete Information, with Evan Piermont (Royal Holloway, University of London). Available soon!
- The Foundations of Robustness in Games with Incomplete Information.