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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.
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 ∩ FE ∩ F ⊆, 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) .
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.