# PUBLISHED

**Uncertain Rationality, Depth of Reasoning and Robustness in Games with Incomplete Information****,* **with Fabrizio Germano (Universitat Pompeu Fabra and Barcelona GSE) and Jonathan Weinstein (Washington University in St. Louis)

*Theoretical Economics** *15 (2020), 89—122

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.

*Previous versions of the paper circulated under the title ''Approximate Rationalizability in Games with Incomplete Information'' (extremely beefed-up new incarnation of Chapter 2 of my PhD dissertation)

**Strategic Cautiousness as an Expression of Robustness to Ambiguity****, **with Gabriel Ziegler (University of Edinburgh)

*Games and Economic Behavior* 119 (2020), 197—215

Economic predictions often hinge on two intuitive premises: agents rule out the possibility of others choosing unreasonable strategies ('strategic reasoning'), and prefer strategies that hedge against unexpected behavior ('cautiousness'). These two premises conflict and this undermines 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 this via a model of incomplete preferences, where (i) each player's strategic uncertainty is represented by a possibly non-singleton set of beliefs and (ii) a rational player chooses a strategy that is a best-reply to every belief in this set. We show that the interplay between these two features precludes the conflict between strategic reasoning and cautiousness and therefore solves the inclusion-exclusion problem raised by Samuelson (1992). Notably, our approach provides a simple foundation for the iterated elimination of weakly dominated strategies.

**Uncertain Information Structures and Backward Induction*******

*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.

*Previous versions of the paper circulated under title ''Incomplete Imperfect Information and Backward Induction'' (Chapter 3 of my PhD dissertation)

**Bounded Rationality and Correlated Equilibria****,* **with Fabrizio Germano

*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.

*Previous versions of the paper circulated under the title ''Approximate Knowledge of Rationality and Correlated Equilibria'' (Chapter 1 of my PhD dissertation)

**Games with Perception****,** with Elena Iñarra (University of the Basque Country) and Annick Laruelle (Ikerbasque and University of the Basque Country)

*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.

# COMPLETED

**Rationalizability, Observabillity and Common Knowledge****,* **with Antonio Penta (ICREA, Universitat Pompeu Fabra and Barcelona GSE)

Revision requested by *Review of Economic Studies*

We study the strategic impact of players' higher order uncertainty over the observability of actions in general two-player games. More specifically, we consider the space of all belief hierarchies generated by the uncertainty over whether the game will be played as a static game or with perfect information. Over this space, we characterize the correspondence of a solution concept which represents the behavioral implications of Rationality and Common Belief in Rationality (RCBR), where 'rationality' is understood as \emph{sequential} whenever a player moves second. We show that such a correspondence is generically single-valued, and that its structure supports a robust refinement of rationalizability, which often has very sharp implications. For instance: (i) in a class of games which includes both zero-sum games with a pure equilibrium and coordination games with a unique efficient equilibrium, RCBR generically ensures efficient equilibrium outcomes; (ii) in a class of games which also includes other well-known families of coordination games, RCBR generically selects components of the Stackelberg profiles; (iii) if common knowledge is maintained that player 2's action is not observable (e.g., because 1 is commonly known to move earlier, etc.), in a class of games which includes of all the above RCBR generically selects the equilibrium of the static game most favorable to player 1.

*Previous versions of the paper were presented under the title ''Extensive-Form Uncertainty of the Higher-Order: Robust Predictions, Refinements and Coordination''

**Higher Orders of Rationality and the Structure of Games****,*** with Francesco Cerigioni (Universitat Pompeu Fabra and Barcelona GSE), Fabrizio Germano and Pedro Rey-Biel (Universitat Ramon Llull, ESADE)

Submitted

Identifying higher orders of rationality is crucial to the understanding of strategic behavior. Nonetheless, the identification of a subject's actual order of rationality from observed behavior in games remains highly elusive. Games may significantly impact and hence invalidate the identified order. To tackle this fundamental problem, we introduce an axiomatic approach that singles out a new class of games, the e-ring games. We then present results from a within subject experiment comparing individuals' classification across e-ring games and standard games previously used in the literature. The results show that satisfying the axioms introduced significantly reduces errors and contributes towards a more reliable identification.

*Previous versions of the paper were presented under the title ''Rationality and Observed Behavior''

**Failures of Contingent Thinking**, with Evan Piermont (Royal Holloway, University of London)

In this paper, we provide a theoretical framework to analyze an agent who misinterprets or misperceives the true decision problem she faces. Within this framework, we show that a wide range of behavior observed in experimental settings manifest as failures to perceive implications, in other words, to properly account for the logical relationships between various payoff relevant contingencies. We present behavioral characterizations corresponding to several benchmarks of logical sophistication and show how it is possible to identify which implications the agent fails to perceive. Thus, our framework delivers both a methodology for assessing an agent's level of contingent thinking and a strategy for identifying her beliefs in the absence full rationality.

**Heterogeneously Perceived Incentives in Dynamic Environments: Rationalization, Robustness and Unique Selections**, with Evan Piermont

Available *very* soon!

In dynamic setting economic agents can elicit others' private information by rationalizing observed behavior. This elicitation is sensitive to agents' perception of which contingencies other agents will persistently deem as impossible, about which contingencies other agents perceive other agents to persistently deem as impossible, and so on. We formalize the potential heterogeneity of this perception as higher-order disagreements about which the set of payoff-states of a dynamic game is. The repercussion on rationalization and continuation play of apparently negligible disagreements exposes the fragility of some established insights: When knowledge of the state space is only 'almost common', *forward induction* predictions may fail to refine those of backward induction. We also prove that forward induction predictions always admit unique selections à la Weinstein and Yildiz (2007)—also for spaces *not* satisfying richness, and reveal that they are not robust to small disagreements about the state space. Backward induction predictions do not admit unique selections but are robust to small disagreements about the state space.

# ONGOING

**The Foundations of Robustness in Games with Incomplete Information**

On social choice, with Marina Bannikova (Universitat Autònoma de Barcelona) and José Manuel Giménez-Gómez (Universitat Rovira i Virgili)