Fixed-equilibrium rationalizability in signaling games

doi:10.1016/0022-0531(90)90035-I

Journal of Economic Theory

Volume 52, Issue 2, December 1990, Pages 304-331

https://doi.org/10.1016/0022-0531(90)90035-I Get rights and content

Abstract

This paper studies equilibrium refinements in signaling games through an examination of rationalizability in derived games obtained by replacing the equilibrium path with a sure outcome that yields the equilibrium payoff to all players. The informed player chooses between the sure payoff and sending an out-of-equilibrium signal from the original game. Whether or not the strategy of choosing the sure payoff is rationalizable is related to the iterated intuitive condition (divinity) when the original game is viewed as having imperfect (incomplete) information. Our results also demonstrate the significance of testing out-of-equilibrium signals as a set rather than individually.

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Cited by (22)

Testing rational hypotheses in signaling games
2023, European Economic Review
We introduce a solution concept for signaling games, called Rational Hypothesis Testing Equilibrium (RHTE). Beliefs are updated via Ortoleva’s (2012) Hypothesis Testing model, allowing for conditioning on information sets off the path. Hypotheses are conjectures by the uninformed player about opponent’s strategies that rationalize sending an unexpected message. Each RHTE is a Perfect Bayesian Equilibrium, but not vice versa. RHTE features a number of desirable properties: First, beliefs are structurally consistent in the spirit of Kreps and Wilson (1982). Second, beliefs are consistent with mutual knowledge of rationality. Third, RHTE can be related to the prominent refinement concepts, including the Intuitive Criterion, strategic stability, and undefeated equilibrium. In the Spence game, RHTE restricts the admissible wages, significantly reducing the number of equilibria. Finally, we show that our equilibrium notion offers an alternative explanation for the experimental results in Brandts and Holt (1992, 1993).
On non-monotonic strategic reasoning
2020, Games and Economic Behavior
Citation Excerpt :
Equilibrium refinements related to strategic stability have been motivated informally with the idea that a deviator will be expected to aim for a higher payoff than the equilibrium payoff. Path restrictions have been used to make this idea precise: see Battigalli and Siniscalchi (2003) for the Iterated Intuitive Criterion (Cho and Kreps, 1987), Catonini (2020) for the elimination of “equilibrium paths that can be upset by a convincing deviation” (Osborne, 1990), and (with different but related techniques) Sobel et al. (1990) for Divine Equilibrium (Banks and Sobel, 1987). Why do path restrictions refine the set of strongly rationalizable outcomes?
Strong-Δ-Rationalizability introduces first-order belief restrictions in the analysis of forward induction reasoning. Without actual restrictions, it coincides with Strong Rationalizability (Battigalli, 2003, Battigalli and Siniscalchi, 2003). These solution concepts are based on the notion of strong belief (Battigalli and Siniscalchi, 2002). The non-monotonicity of strong belief implies that the predictions of Strong-Δ-Rationalizability can be inconsistent with Strong Rationalizability. I show that Strong-Δ-Rationalizability refines Strong Rationalizability in terms of outcomes when the restrictions correspond to belief in a distribution over terminal nodes. Moreover, under such restrictions, the epistemic priority between rationality and belief restrictions is irrelevant for the predicted outcomes.
Payoff information and learning in signaling games
2020, Games and Economic Behavior
We add the assumption that players know their opponents' payoff functions and rationality to a model of non-equilibrium learning in signaling games. Agents are born into player roles and play against random opponents every period. Inexperienced agents are uncertain about the prevailing distribution of opponents' play, but believe that opponents never choose conditionally dominated strategies. Agents engage in active learning and update beliefs based on personal observations. Payoff information can refine or expand learning predictions, since patient young senders' experimentation incentives depend on which receiver responses they deem plausible. We show that with payoff knowledge, the limiting set of long-run learning outcomes is bounded above by rationality-compatible equilibria (RCE), and bounded below by uniform RCE. RCE refine the Intuitive Criterion (Cho and Kreps, 1987) and include all divine equilibria (Banks and Sobel, 1987). Uniform RCE sometimes but not always exists, and implies universally divine equilibrium.
Rationalizability and epistemic priority orderings
2019, Games and Economic Behavior
Citation Excerpt :
In signaling games, Battigalli and Siniscalchi (2003) show that Strong-Δ-Rationalizability yields a non-empty set under belief in an equilibrium outcome that satisfies the iterated intuitive criterion (Cho and Kreps, 1987). Sobel et al. (1990), construct a “fixed-equilibrium rationalizability” procedure that yields a non-empty set when the equilibrium is divine (Banks and Sobel, 1987). Despite of this, such equilibria are not always stable (see Cho and Kreps, 1987 and Banks and Sobel, 1987).
At the beginning of a dynamic game, players may have exogenous theories about how the opponents will play. If these theories are commonly known, players will refine their first-order beliefs and challenge their own theories through strategic reasoning. I propose a new solution concept, Selective Rationalizability, which captures the following hypothesis: when the observed behavior is not compatible with the beliefs in rationality and in the theories of all orders, players keep the beliefs in rationality that are compatible with the observed behavior, and drop the incompatible beliefs in the theories. Thus, Selective Rationalizability captures Common Strong Belief in Rationality (Battigalli and Siniscalchi, 2002) and refines Extensive-Form Rationalizability (Pearce, 1984, Battigalli, 1996), whereas Strong-Δ-Rationalizability (Battigalli, 2003, Battigalli and Siniscalchi, 2003) captures the opposite epistemic priority choice. Selective Rationalizability is extended to encompass richer epistemic priority orderings among different theories of opponents' behavior. This allows to shed some new light on strategic stability (Kohlberg and Mertens, 1986).
Interactive epistemology in games with payoff uncertainty
2007, Research in Economics
We adopt an interactive epistemology perspective to analyse dynamic games with partially unknown payoff functions. We consider solution procedures that iteratively delete strategies conditional on private information about the state of nature. In particular we focus on a weak and a strong version of the $Δ$ -rationalizability solution concept, where $Δ$ represents given restrictions on players’ beliefs about state of nature and strategies [Battigalli, P., 2003. Rationalizability in infinite, dynamic games of incomplete information. Research in Economics 57, 1–38; Battigalli, P., Siniscalchi, M., 2003. Rationalization and incomplete information. Advances in Theoretical Economics 3 (Article 3). http://www.bepress.com/bejte/advances/vol3/iss1/art3]. We first show that weak $Δ$ -rationalizability is characterized by initial common certainty of rationality and of the restrictions $Δ$ , whereas strong $Δ$ -rationalizability is characterized by common strong belief in rationality and the restrictions $Δ$ (cf. [Battigalli, P., Siniscalchi, M., 2002. Strong belief and forward induction reasoning. Journal of Economic Theory 106, 356–391]). The latter result allows us to obtain an epistemic characterization of the iterated intuitive criterion. Then we use the framework to analyse the robustness of complete-information rationalizability solution concepts to the introduction of “slight” uncertainty about payoffs. If the set of conceivable payoff functions is sufficiently large, the set of strongly rationalizable strategies with slight payoff uncertainty coincides with the set of complete-information, weakly rationalizable strategies.
Strong belief and forward induction reasoning
2002, Journal of Economic Theory
We provide a unified epistemic analysis of some forward-induction solution concepts in games with complete and incomplete information. We suggest that forward induction reasoning may be usefully interpreted as a set of assumptions governing the players' belief revision processes, and define a notion of strong belief to formalize these assumptions. Building on the notion of strong belief, we provide an epistemic characterization of extensive-form rationalizability and the intuitive criterion, as well as sufficient epistemic conditions for the backward induction outcome in generic games with perfect information. Journal of Economic Literature Classification Numbers: C72, D82.

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^☆: We thank Valentina Corradi, Eric van Damme, Eddie Dekel, Drew Fudenberg, Robert Gibbons, Matthew Rabin, and a referee for helpful comments. Sobel is grateful to the National Science Foundation and the Sloan Foundation for financial support. A National Science Foundation Graduate Fellowship supported Stole's research.

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Fixed-equilibrium rationalizability in signaling games☆

Abstract

J. Econ. Theory

J. Econ. Theory

J. Econ. Theory

Equilibrium selection in signaling games

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Coordination and the Potential for Self Sacrifice

Rationalizable strategic behavior

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A refinement of sequential equilibrium

Econometrica