Jump to ContentJump to Main Navigation
Show Summary Details
More options …

The B.E. Journal of Theoretical Economics

Editor-in-Chief: Schipper, Burkhard

Ed. by Fong, Yuk-fai / Peeters, Ronald / Puzzello , Daniela / Rivas, Javier / Wenzelburger, Jan

2 Issues per year

IMPACT FACTOR 2016: 0.229
5-year IMPACT FACTOR: 0.271

CiteScore 2016: 0.30

SCImago Journal Rank (SJR) 2016: 0.398
Source Normalized Impact per Paper (SNIP) 2016: 0.232

Mathematical Citation Quotient (MCQ) 2016: 0.08

See all formats and pricing
More options …

Transparent Restrictions on Beliefs and Forward-Induction Reasoning in Games with Asymmetric Information

Pierpaolo Battigalli / Andrea Prestipino
Published Online: 2013-05-08 | DOI: https://doi.org/10.1515/bejte-2012-0005

Abstract: We analyze forward-induction reasoning in games with asymmetric information assuming some commonly understood restrictions on beliefs. Specifically, we assume that some given restrictions Δ on players’ initial or conditional first-order beliefs are transparent, that is, not only do the restrictions Δ hold but there is also common belief in Δ at every node. Most applied models of asymmetric information are covered as special cases whereby Δ pins down the probabilities initially assigned to states of nature. But the abstract analysis also allows for transparent restrictions on beliefs about behavior, e.g. independence restrictions or restrictions induced by the context behind the game. Our contribution is twofold. First, we use dynamic interactive epistemology to formalize assumptions that capture foward-induction reasoning given the transparency of Δ, and show that the behavioral implications of these assumptions are characterized by the Δ-rationalizability solution procedure of Battigalli (1999, 2003). Second, we study the differences and similarities between this solution concept and a simpler solution procedure put forward by Battigalli and Siniscalchi (2003). We show that the two procedures are equivalent if Δ is “closed under compositions” a property that holds in all the applications considered by Battigalli and Siniscalchi (2003). We also show that when Δ is not closed under compositions, the simpler solution procedure of Battigalli and Siniscalchi (2003) may fail to characterize the behavioral implications of forward-induction reasoning.

Keywords: epistemic game theory; rationalizability; forward induction; transparent restrictions on beliefs

JEL Classification codes: C72; C73; D82


  • Aradillas-Lopez, A., and E. Tamer. 2008. “The Identification Power of Equilibrium in Simple Games.” Journal of Business & Economics Statistics 26:261–83.Web of ScienceCrossrefGoogle Scholar

  • Aumann, R. J. 1987. “Correlated Equilibrium as an Expression of Bayesian Rationality.” Econometrica 55:1–18.CrossrefGoogle Scholar

  • Battigalli, P. 1996. “Strategic Rationality Orderings and the Best Rationalization Principle.” Games and Economic Behavior 13:178–200.CrossrefGoogle Scholar

  • Battigalli, P. 1997. “On Rationalizability in Extensive Form Games.” Journal of Economic Theory 74:40–61.Web of ScienceCrossrefGoogle Scholar

  • Battigalli, P. 1991. “Rationalizability in Incomplete Information Games.” EUI working papers, ECO 99/17.Google Scholar

  • Battigalli, P. 2003. “Rationalizability in Infinite Dynamic Games with Incomplete Information.” Research in Economics 57:1–38.Google Scholar

  • Battigalli, P. 2006. “Rationalization in Signaling Games: Theory and Applications.” International Game Theory Review 8:1–27.Google Scholar

  • Battigalli, P., and A. Friedenberg. 2009. “Context-Dependent Forward Induction Reasoning.” IGIER working paper 351, Bocconi University.Google Scholar

  • Battigalli, P., and A. Friedenberg. 2012. “Forward Induction Reasoning Revisited.” Theoretical Economics 7:57–98.CrossrefWeb of ScienceGoogle Scholar

  • Battigalli, P., and M. Siniscalchi. 1999. “Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games.” Journal of Economic Theory 88:188–230.CrossrefGoogle Scholar

  • Battigalli, P., and M. Siniscalchi. 2002. “Strong Belief and Forward Induction Reasoning.” Journal of Economic Theory 106:356–91.CrossrefGoogle Scholar

  • Battigalli, P., and M. Siniscalchi. 2003. “Rationalization and Incomplete Information.” Advances in Theoretical Economics 3(1), Article 3, http://www.bepress.con/bejte/advances/vol3/iss1/art3.

  • Battigalli, P., and M. Siniscalchi. 2007. “Interactive Epistemology in Games with Payoff Uncertainty.” Research in Economics 61:165–84.Google Scholar

  • Battigalli, P., and P. Tebaldi. 2012. “Interactive Epistemology in Simple Dynamic Games with a Continuum of Strategies.” typescript.Google Scholar

  • Battigalli, P., A. Di Tillio, and D. Samet. 2011. “Strategies and Interactive Beliefs in Dynamic Games,” IGIER w.p. n. 375 (2011), forthcoming in Advances in Economics and Econometrics (D. Acemoglu, M. Arellano, E. Dekel Eds.)Google Scholar

  • Battigalli, P., A. Di Tillio, E. Grillo, and A. Penta. 2011. “Interactive Epistemology and Solution Concepts for Games with Asymmetric Information.” The B.E. Journal of Theoretical Economics 11(1)(Advances), Article 6.Google Scholar

  • Bergemann, D., and S. Morris. 2012. “Robust Mechanism Design.” The Role of Private Information and Higher Order Beliefs. Singapore: World Scientific Publishing.Google Scholar

  • Brandenburger, A., and E. Dekel. 1987. “Rationalizability and Correlated Equilibria.” Econometrica 55:1391–402.CrossrefGoogle Scholar

  • Brandenburger, A., and E. Dekel. 1993. “Hierarchies of Beliefs and Common Knowledge.” Journal of Economic Theory 59:189–98.CrossrefGoogle Scholar

  • Brandenburger, A., and A. Friedenberg. 2010. “Self-Admissible Sets.” Journal of Economic Theory 145:785–811.CrossrefWeb of ScienceGoogle Scholar

  • Brandenburger, A., A. Friedenberg, and H. J. Keisler. 2008. “Admissibility in Games.” Econometrica 76:307–52.Web of ScienceGoogle Scholar

  • Cappelletti, G. 2010. “A Note on Rationalizability and Restrictions on Beliefs.” B.E. Journals in Theoretical Economics 10(1) (Topics), Article 40.Google Scholar

  • Catonini, E. 2012. “Non-Binding Agreements and Forward Induction Reasoning.” typescript (2012), Bocconi University.Google Scholar

  • Cho, I. K., and D. Kreps. 1987. “Signaling Games and Stable Equilibria.” Quarterly Journal of Economics 102:179–222.CrossrefGoogle Scholar

  • Dekel, E., D. Fudenberg, and S. Morris. 2007. “Interim Correlated Rationalizability.” Theoretical Economics 2:15–40.Google Scholar

  • De Vito, N. 2012. “Complete Conditional Type Structures.” typescript, Bocconi University.Google Scholar

  • Friedenberg. 2010. “When Do Type Structures Contain All Hierarchies of Beliefs?” Games and Economic Behavior 68:108–29.Web of ScienceCrossrefGoogle Scholar

  • Harsanyi, J. 1967–68. “Games of Incomplete Information Played by Bayesian Players.” Parts I, II, III, Management Science 14:159–82, 320–34, 486–502.CrossrefGoogle Scholar

  • Mertens, J.-F., and S. Zamir. 1985. “Formulation of Bayesian Analysis for Games with Incomplete Information.” International Journal of Game Theory 14:1–29.CrossrefGoogle Scholar

  • Mueller, C. 2012. “Robust Virtual Implementation Under Common Strong Belief in Rationality.” typescript, Carnegie Mellon University.Google Scholar

  • Osborne, M., and A. Rubinstein. 1994. A Course in Game Theory. Cambridge, MA: MIT Press.Google Scholar

  • Pearce, D. 1984. “Rationalizable Strategic Behavior and the Problem of Perfection.” Econometrica 52:1029–50.CrossrefGoogle Scholar

  • Renyi, A. 1955. “On a New Axiomatic Theory of Probability.” Acta Mathematica Academiae Scientiarum Hungaricae 6:285–335.CrossrefGoogle Scholar

  • Schweinzer, P., and M. Shimoji. 2012. “Implementation without Incentive Compatibility: Two Stories with Partially Informed Planners.” typescript, University of York, 2012.Google Scholar

  • Tan, T., and S. Werlang. 1988. “The Bayesian Foundation of Solution Concepts of Games.” Journal of Economic Theory 45:370–91.CrossrefGoogle Scholar

  • Tebaldi, P. 2011. “Non-Binding Agreements in Dynamic games.” MSc Thesis, Bocconi University.Google Scholar

About the article

Published Online: 2013-05-08

Published in Print: 2013-01-01

For a result of this sort see, for example, Battigalli et al. (2011, Theorem 3): absent independent restrictions, rationality and common belief in rationality have equivalent behavioral implications in the two scenarios.

The strategies and conditional beliefs pinned down by this argument concide with the sequential equilibrium selected by the Intuitive Criterion of Cho and Kreps (1987).

We refrain from saying that such restrictions are “common knowledge”. We find the use of the “common knowledge” terminology much too casual in economic theory, as there is often either a terminological or even a conceptual conflation of (common) knowledge and (common, probability one) belief. We find it semantically and conceptually useful to reserve “knowledge” for the justified true belief that comes from observation and logical deduction.

We distinguish between states of nature, that parametrize payoff functions, and states of the world that describe every relevant aspect of the situation of strategic interaction.

On extensive-form rationalizability see Pearce (1984) and Battigalli (1996, 1997).

Battigalli (1999, 2003) and Battigalli and Siniscalchi (2007) also analyze a less demanding solution concept that only requires initial common belief in rationality and the restrictions Δ. To stress the difference between these two solution concepts, these papers call “weak Δ-rationalizability” the one based on assumptions about initial beliefs, and “strong Δ-rationalizability” the one capturing forward-induction reasoning. Like other papers that only analyze the latter solution concept (e.g. Battigalli and Siniscalchi 2003; Battigalli and Friedenberg 2012), here we simply call it “Δ-rationalizability.”

For applications of the Δ-rationalizability approach to economic models see, e.g., Battigalli (2003, 2006), Battigalli and Siniscalchi (2003) and references therein; for applications to robust mechanism design see Schweinzer and Shimoji (2012) and Mueller (2012) and the survey by Bergemann and Morris (2012), for applications to non-binding agreeements see Catonini (2012) and Tebaldi (2011); for empirical applications see Aradillas-Lopez and Tamer (2008). Attention is restricted to first-order beliefs for the sake of simplicity, as Δ-rationalizability is meant to be a relatively simple reduction procedure whose implementation does not involve type structures and beliefs about beliefs. But restrictions on higher-order beliefs (not implied by the transparency of first-order beliefs restrictions) may well be appropriate in some applications; taking higher-order belief restrictions into account does not change the essential features of the approach. See, for example, the rationalizability analysis of Spence’s model by Battigalli (2006). Also, Theorem 4 (a straightforward extension of results due to Battigalli and Friedenberg 2012) provides a kind of equivalence between forward-induction reasoning under transparency of first-order and of higher-order beliefs.

Informally, we call “expressible” an assumption that can be stated using primitive terms and terms derived from primitive terms or other derived terms (see Battigalli et al. 2011).

More formally, we say that a collection of expressible assumptions justifies a solution concept, or equivalently that characterizes the behavioral implications of A, if for each player i and each piece of private information θi, the set of strategies allowed by for θi coincides with the set of strategies allowed by for θi.

On the other hand, the conjunction of strong belief in E and strong belief in F implies strong belief in.

Indeed, considering the same events and informational setup as above,E ∩ FEF ⊆, but we have just shown that strong belief in E ∩ F does not imply strong belief in both E and F. Thus, monotonicity does not hold.

Cf. Osborne-Rubinstein (1994), chapters 6, 11.

For any given set Y, Y<N denotes the set of finite sequences of elements of Y, including the empty sequence ∅, that is, with .

No information set contains two ordered histories; furthermore, whenever and , there is a history such that and .

This is not to be confused with “complete information,” which means that all the rules of the game and players’ preferences over consequences are common knowledge. Indeed, we allow for the opposite, if there is payoff uncertainty, there is incomplete information.

Battigalli (2003) allows for type-dependent actions sets.

Our framework allows for the possibility that players do not have common beliefs about the probabilities of chance moves.

The construction of a canonical type structure à la Battigalli and Siniscalchi applies to this more general setting (see Battigalli and Siniscalchi 1999; Battigalli 2003). The extension of the main epistemic characterization result of this paper involves, directly or indirectly, a measurable selection argument (see Battigalli and Tebaldi 2012).

Г exhibits complete information if the payoff map is constant (which is trivially true when Θ is a singleton), otherwise Г has incomplete information.

Compactness of the relevant spaces is assumed for simplicity, it can be relaxed with some additional technical complications.

is called conditional probability space in Renyi (1955).

That is, .

If the information set of i containing also contained another node, then it would contain two nodes on the same path, thus violating perfect recall.

If two information sets differ only because of moves of i, then . Thus, the cardinality of Hi may be smaller than the cardinality of . This redundancy is innocuous in our analysis.

History h is inconsistent with (or counterfactual at) .

Battigalli and Siniscalchi (1999) uses a slightly different definition of type structure. But all the arguments in Battigalli and Siniscalchi (1999) can be easily adapted to the present framework.

The representation of a type structure as a belief-closed substructure of the canonical one eliminates redundant types, i.e. types that yield the same hierarchy of CPS’s. Redundant types do not play any role in our analysis.

A result by Friedenberg (2010) implies that in static games (games where for each i) every compact-continuous complete structure contains all the “conceivable” hierarchies of beliefs and is in a precise sense equivalent to the canonical structure. It can be shown that the same holds more generally for all the dynamic games considered here (De Vito 2012).

See the discussion section in Battigalli, Di Tillio, and Samet (2011).

For any measurable (closed) subset is measurable (closed).

We can prove our main results without assuming compactness of [Δ], but we are not able to do it without complicating the analysis. Clearly, compactness of [Δ] may not hold in interesting applications. In the incomplete-information scenario of the Beer–Quiche example analyzed in Section 5, Δ2 is not compact. But exactly the same analysis goes through with any compact subset of Δ2 sufficiently close to Δ2 (that is, sufficiently close in the Hausdorff topology to the closure of Δ2).

See Lemma 1 in Section 5.

Battigalli and Siniscalchi (2007) also put forward an incorrectly stated conjecture, the correct version of which is Theorem 1 above.

With this notation, .

We conjecture that if Δ is regular and each Δi,h is closed and convex, then the characterization of Δ-rationalizability as iterated Δ-dominance (Cappelletti 2010) can be extended to the present extensive-form setting.

If the reader is wondering why a complete information game corresponds to a Bayesian game, he should remember that in our terminology (which we claim to be the correct one) “complete information” is a substantive assumption, i.e. common knowledge of the payoff functions. On the other hand, Bayesian games are just mathematical structures that may be used to analyze both games with incomplete information and games with asymmetric, imperfect information about an initial chance move, such as poker. The interpretation of such mathematical structures is immaterial for Harsanyi’s equilibrium analysis, but not for rationalizability analysis. The reason is that standard notions of rationalizability for Bayesian games implicitly incorporate independence restrictions, and different restrictions are relevant under different interpretations (see Battigalli et al. 2011).

Our original example showing the difference between Δ-rationalizability and naive Δ-rationalizability did not have the latter feature. We thank Amanda Friedenberg for providing this example.

We use the following notation: O is the class of realization-equivalent strategies choosing Out, otherwise strategies are denoted by lists of action labels in an obvious way.

Note that in games with complete information Σ = S.

Battigalli and Friedenberg (2012) provide examples of complete information games where the inclusion also fails for the corresponding sets of paths, that is, for some game and some restrictions Δ.

Battigalli and Friedenberg (2012) is the abridged published version of Battigalli and Friedenberg (2009). The latter elab-orates more on the context interpretation of incomplete type structures and how they are related to transparent restrictions on beliefs. Battigalli and Friedenberg build on previous work on admissibility by Brandenburger, Friedenberg, and Keisler (2008) and Brandenburger and Friedenberg (2010).

The equivalences stated in the Theorem 4 adapt results of Battigalli and Friedenberg as follows: adapts Proposition 1 of Battigalli and Friedenberg (2012); adapts Theorem 1 of Battigalli and Friedenberg (2012); adapts Proposition A1 of Battigalli and Friedenberg (2009) applied to the E-restriction TE of a belief-complete type structure T; finally follows from Theorem 2 and (the adaptation of) Proposition 1 in Battigalli and Siniscalchi (2012) (the set Δ constructed in the proof is finite, hence compact), follows from Theorem 1 and .

The coalition formed by the chance player and Player 1 has perfect recall. Therefore, a correlated strategy of the coalition is realization equivalent to a behavioral strategy of the coalition, which is in turn equivalent to a product measure.

As observed by Battigalli et al. (2011), the application of solution concepts (such as rationalizability, or iterated dominance) to the ex ante strategic form implicitly relies on the independence assumption described above.

We write X.Y for the strategy of Player 1 that selects X in the surly state and Y in the wimp state; similarly we write x.y for the strategy of Player 2 that selects x if B and y if Q.

Since and we are abusing notation here. This should cause no confusion.

Lemma 7 of Battigalli and Siniscalchi (2007), which is a special case of our Proposition 1.

Citation Information: The B.E. Journal of Theoretical Economics, ISSN (Online) 1935-1704, ISSN (Print) 2194-6124, DOI: https://doi.org/10.1515/bejte-2012-0005.

Export Citation

©2013 by Walter de Gruyter Berlin / Boston. Copyright Clearance Center

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Emiliano Catonini
Electronic Proceedings in Theoretical Computer Science, 2017, Volume 251, Page 102
Joseph E. Harrington
Research in Economics, 2017, Volume 71, Number 1, Page 140
Makoto Shimoji and Paul Schweinzer
Games and Economic Behavior, 2015, Volume 91, Page 258

Comments (0)

Please log in or register to comment.
Log in