Home Transparent Restrictions on Beliefs and Forward-Induction Reasoning in Games with Asymmetric Information
Article
Licensed
Unlicensed Requires Authentication

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

  • Pierpaolo Battigalli EMAIL logo and Andrea Prestipino
Published/Copyright: May 8, 2013
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 13 Issue 1

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

5 Appendix

Subsection 5.1 provides a complete forward-induction analysis of the Beer–Quiche game under different scenarios and related restrictions on beliefs; 5.2 derives the epistemic justifications of Δ-rationalizability; 5.3 proves the equivalence between Δ-rationalizability and the naïve reduction algorithm when Δ is closed under compositions.

5.1 Two scenarios for Δ-rationalizability in the Beer–Quiche game

Complete information scenario We formalize the informal argument provided in Section 1 for the Beer–Quiche game of Figure 1 and Example 1 (1). As formally described in Example 2, assume the following restrictions Δ are transparent:

  • Players ex ante assign 90% (10%) probability to the surly (wimp) state,

  • According to the ex ante beliefs of Player 1, the chance move and the strategy of Player 2 are independent.

The behavioral implications of (correct) common strong belief of rationality and transparency of Δ can be derived with the (naïve) Δ-rationalizability algorithm (Theorem 1, Corollary 1). To apply the latter, we first note that we may add wlog further restriction that, according to the beliefs of Player 2, the strategy of Player 1 and the chance move are independent.43 With this, Δ-rationalizability is a refinement of rationalizability on the ex ante strategic form of the Bayesian game defined by the first restriction.44 In the Table 4, we multiply all the expected payoffs by a factor of 10:

Table 4

Ex ante strategic form of the Beer and Quiche game.

9, 19, 129, 929, 9
10, 112, 028, 1030, 9
0, 118, 102, 020, 9
1, 121, 91, 121, 9

Iterated weak dominance on this strategic form deletes f.f and Q.B in step 1, f.d in step 2, Q.Q in step 3, d.d in step 4, and B.Q in step 5.45 This is equivalent (for this game) to iterated conditional dominance on the strategic form, which gives the Δ-rationalizability solution. We summarize the procedure in Table 5. For intuitive explanations based on the epistemic analysis see Section 1.

Table 5

Solution of the Beer and Quiche game for the complete information scenario.

StepsPlayer 1Player 2
1
2
3
4
5

Incomplete-information scenario In this scenario, there is no chance move, the game begins at the “interim” stage, when Player 1 knows the true state; see Example 1 (2). Now there are two conceivable information types of Player 1, s and w, each with strategy space . Thus,

In this scenario, Player 1 has no belief on {s, w}. Intuitively, the reason is that is just an attribute of Player 1 known to him, not something that he learns. Formally, his primitive uncertainty space is . With this, the independence assumption of the complete-information scenario cannot be expressed as a property of his first-order beliefs. As we mentioned in Section 1, there is a property of the second-order beliefs of Player 2 that could replace the independence property expressible in the complete-information scenario: according to the second-order beliefs of Player 2, the information type of Player 1 and his first-order belief about the strategy of Player 2 are independent. We could provide an epistemic analysis of the game based on the transparency of first and second-order restrictions and derive results similar to those obtained for the complete-information scenario. But in this paper we restricted our attention to transparency of first-order restrictions. Therefore, as an illustration of our approach we assume the following first-order restrictions Δ to be transparent:

  1. Player 2 initially assigns 90% probability to θ = s,

  2. conditional on B, Player 2 believes that s is more likely than w.

Formally,

where we used obvious abbreviations for unconditional, conditional, and marginal probabilities. Note, this is just meant to be an example; we do not claim that the assumption in the second bullet is more plausible, or “nicer” than the assumption about the second-order beliefs of Player 2 described above.

The behavioral implications of (correct) common strong belief in rationality and transparency of the restrictions are characterized by a solution procedure that deletes profiles from the set . We summarize the steps in Table 6 and then comment.

Table 6

Solution of the Beer and Quiche game for the incomplete information scenario.

StepsPlayer 1Player 2
1
2
3
4

Step 1, Player 2: We delete f.f (as in the complete-information scenario) and also f.d (fight if and only if Beer); the reason for the latter is that it is rational to fight if and only if the conditional probability of the surly type is no more that 50%, therefore, given the restrictions on beliefs, if it is rational to fight after B then it is a fortiori rational to fight after Q. To sum up, Player 2 does not fight after B.

Step 2, Player 1: Given the above, the surly type has his preferred meal, i.e., type/strategy pair (s,Q) is deleted.

Step 3, Player 2: By a forward-induction argument similar to Step 4 of the complete-information scenario (see Section 1), d.d is deleted.

Step 4: Each type of Player 1 is certain of d.f, therefore both types choose Beer.

5.2 Proofs of the characterization results

The proof of the main result (Theorem 1) is adapted from Battigalli and Siniscalchi (2007). First, we need some preliminaries. Fix a belief-complete type structure and a collection of compact subsets Δ. is a belief-closed subset of this structure that can be constructed as follows (recall we introduced the singleton for notational convenience): let

for each and

Clearly, ,46 thus . Note that we had to define as a subset of rather than , because the restrictions on first-order beliefs may depend on the information type θi. Elements of will be simply called “types”.

We begin with a few preliminary results. Let and denote the θi-section of, respectively, and . Clearly, .

Lemma 1For alland, the sets, , and are closed.

Proof For every is closed because fi is continuous and Δθi is closed. Therefore, is closed because it is the union of finitely many closed sets.

Now suppose by way of induction that is closed for every i and θi, so that is closed for every i. Then the set of probability measures

is closed. This implies that the set of conditional beliefs that fully believe , that is,

is closed as well, because is closed. Since gi is continuous,

is closed. This implies that, for every is closed, because is closed (inductive hypothesis) and

Hence, is closed as well. ■

Lemma 2For every, projΘi=Θi, therefore there exists a functionsuch that.

Proof We prove below by induction that for every . By Lemma 1, is a decreasing sequence of non-empty closed subsets of a compact space. By the finite intersection property, . Hence, for every i

Then, we can define as follows: for each and let .

Now we prove by induction that, for every , and .

Basis step First, for every and arbitrarily. Also, let . Now define an array of probability measures as follows: for every measurable subset and

It can be checked that is a CPS, that is, . Since is onto (belief-completeness), there is some such that . By construction, thus, .

As a matter of notation, let for every and .

Induction step Now suppose that, for every i and and pick for each a profile . For every and θi, define an array of probability measures as follows: for every measurable subset and

As in the basis step, it can be checked that . Since gi is onto, there is some such that . By construction, . Furthermore, for every and

(the equalities hold by construction for , then they also hold for because ). Since

it follows that . ■

Lemma 3Fix compact restrictions Δ and mapssuch that, where. Fix a playerand, for eachθi, a first-order. Then, for eachθi, there exists an epistemic typesuch thatand, for each, has finite support and

[7]
[7]

Proof. Define a candidate by setting

for every and and , extending the assignments by additivity. Axioms 1 and 2 follow immediately from the observation that the map yields an embedding of supp (a finite sub setset of ∑‒i) in , so that, for every is indeed a probability measure on . By the same argument, νi must also satisfy Axiom 3, i.e. it must be a CPS; of course, each has finite support by construction. Since gi is onto, there exists a type such that

for every and . To see that note that by construction and for each h, which implies for each h and n. ■

We can now prove our main result.

Proof of Theorem 1: To prove the first part, we rely on Lemma 2 and Lemma 3 to recursively define, for each , a profile of functions such that, for each , and whenever .

For the reader’s convenience, we report below the conditions for surviving the (n+1)-th step of Δ-rationalizability:

For every and , if and only if there exists a CPS such that

[8]
[8]
[9]
[9]

The maps for Player 0 are trivial: for every and n. As for the real players, let be any profile of functions such that such functions exist by Lemma 2. Next, assuming that has been defined and satisfies for every m = 0, … n, define as follows: for each i and let for each there is first-order such that eqs. [8] and [9] hold and, by Lemma 3, there is an epistemic type with such that eq. [7] holds (for each ) with , so that – in particular then let . Clearly, .

Claim 1For every and ,

[10]
[10]
[11]
[11]

Equation [10] implies

Equation [11] implies

Therefore, the claim implies that

for every m.

Proof of the claim Recall that is a Cartesian product. To ease notation, for each and , we write

Basis step Fix (θ, s) arbitrarily. Suppose that . By construction of 1, for each and . Therefore, . Since , this proves eq. [10] for m= 0. Next, fix any such that . Then, for each and ; therefore . This proves eq. [11] for m = 0.

Induction step Assume that eqs. [10] and [11] hold for each and m = 0, …, n‒1. Then, for each m = 0, …, n – 1,,

[12]
[12]

Fix (θ, s) arbitrarily. Suppose that . By construction of , for every and ; therefore . By the inductive hypothesis and the construction of , for every , and

Hence, by construction of and eq. [12], for every and ,

Next, note that

It follows that , showing that eq. [10] holds for m = n.

Now fix any such that . Then, for every , and (by the inductive hypothesis) for every and

By eq. [12], the formula above is equivalent to

for every and . Therefore, , showing that eq. [11] holds with m = n. □

Next, we prove the second part of the thesis: . Pick any . Since, for every , we conclude that for every so . Hence, . Now pick any and consider the sequence of sets is closed and (by standard arguments) R is closed as well. For every closed event E, i, and is closed, therefore is also closed. is obviously closed. Therefore, each is non-empty (because and closed subset of the compact space ; also, the sequence of sets is decreasing, and hence has the finite intersection property. Then and . Therefore, . ■

Proof of Theorem First observe that the set of states of player i in is

where is the set of information/epistemic types defined at the beginning of this subsection.

Proof of eq. [3] Fix i, θi. By definition of , the right-hand-side of eq. [3] is contained in the left-hand-side. To see that also the converse holds, fix ; by Lemma 3 there is a type such that and . By construction, , for each (the second equality holds because and is the restriction of on ). Thus, for some such that for every si.

Proof of eq. [4]Basis step. Each first-order belief map is the restriction of on and ; therefore

Induction step. Suppose that . Then

where the second equality follows from the induction hypothesis, the fourth holds because and is the restriction of gi on , and the other equalities hold by definition. □

Proof of eqs. [5] and [6]. Given eqs. [4], [5] and [6] follow from Theorem 1. ■

Proof of Theorem 3. To shorten the proof, we take advantage of the result due to Battigalli and Siniscalchi (2007) mentioned in Section 1:

Theorem 5Fix a collection of compact subsets of first-order CPS’s and a belief-complete type structure . Then, for every n ≥ 0,

and

Sketch of proof of Theorem 5. By inspection of the proof of the main characterization result in Battigalli and Siniscalchi (2007), it is clear that a separate lemma47 shows the equivalence of naïve Δ-rationalizability and Δ-rationalizability under the assumption that Δ is regular, whereas the rest of the proof shows that Theorem 5 holds. This is done within the observable-actions framework, but it is clear that the arguments of the rest of the proof are unaffected by considering the more general framework of this paper. □

Theorems 1 and 5 imply

for each , including . An induction argument shows that all the claimed inclusions hold. Indeed,

Suppose that . Recall from subsection 3.2 that, for each and , we let ,

is similarly defined. Since and have the same projection on , they are consistent with the same information sets of Player i, even if . Let denote the collection of such information sets; then, by the inductive hypothesis and monotonicity of the Bi, h operators,

Recall that, for each . Therefore, given the above definitions and inclusions,

■

5.3 Equivalence of Δ-rationalizability and naïve Δ-rationalizability

We begin with a preliminary result about closedness under compositions.

Definition 11A finite sequenceis admissible ifis a decreasing sequence of product sets withand, for everystrongly believes (for each.

Definition 12Fix an admissible sequenceand let. A system of beliefsis the composition ofif

Lemma 4The compositionμof an admissible sequenceis a CPSsuch thatfor eachwith.

Proof. We only have to prove that μ satisfies the chain rule, the rest follows immediately by inspection of Definition 12. Fix , and E‒i so that (g is a prefix of h); then . If then

If then, by definition of . By admissibility of the sequence, . Therefore, and

■

Lemma 5Letbe closed under compositions and fix an admissible sequencesuch thatfor each m. Then also the composition ofgives a CPS in.

Proof. The statement is true by definition for admissible sequences of length 2. Assume by way of induction that the result holds for admissible sequences of length and consider an admissible sequence of length n, viz. . Let μ be the composition of the (admissible) prefix . By the inductive hypothesis, . The pair is admissible since and μ is a composition of so that, in particular, μ is a CPS such that implies for each (Lemma 4). Now let be the composition of . Since is closed under compositions, . By construction, is the composition of . ■

Proof of Proposition 1. The statement is obvious for . Now pick and assume by way of induction that the statement it is true for each positive integer up to n. We have to show that .

If then there is some satisfying Eqs. [8] and [9]; since , then by the inductive hypothesis ; moreover, since (again by the inductive hypothesis), eq. [9] implies 1. We conclude that .

In the other direction, suppose . Then also for , so we can find CPS’s , such that, for each m, and, implies and, . Therefore, the sequence is admissible. Now let be the composition of this sequence. Clearly, and satisfies eq. [9]. Moreover, by Lemma 5 . Therefore . ■

Acknowledgments

The authors thank an anonymous referee for his or her thoughtful and constructive review of an earlier draft of this paper. The authors gratefully acknowledge useful comments by Emiliano Catonini, Nicodemo De Vito, Amanda Friedenberg, Elena Manzoni, Sara Negrelli, Jacopo Perego, and Pietro Tebaldi. All errors are their own. Pierpaolo Battigalli thanks NYU-Stern for its hospitality and gratefully acknowledges financial support from ERC grant 324219.

References

Aradillas-Lopez, A., and E.Tamer. 2008. “The Identification Power of Equilibrium in Simple Games.” Journal of Business & Economics Statistics26:261–83.10.1198/073500108000000105Search in Google Scholar

Aumann, R. J.1987. “Correlated Equilibrium as an Expression of Bayesian Rationality.” Econometrica55:1–18.10.2307/1911154Search in Google Scholar

Battigalli, P.1996. “Strategic Rationality Orderings and the Best Rationalization Principle.” Games and Economic Behavior13:178–200.10.1006/game.1996.0033Search in Google Scholar

Battigalli, P.1997. “On Rationalizability in Extensive Form Games.” Journal of Economic Theory74:40–61.10.1006/jeth.1996.2252Search in Google Scholar

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

Battigalli, P.2003. “Rationalizability in Infinite Dynamic Games with Incomplete Information.” Research in Economics57:1–38.10.1016/S1090-9443(02)00054-6Search in Google Scholar

Battigalli, P.2006. “Rationalization in Signaling Games: Theory and Applications.” International Game Theory Review8:1–27.10.1142/S0219198906000795Search in Google Scholar

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

Battigalli, P., and A.Friedenberg. 2012. “Forward Induction Reasoning Revisited.” Theoretical Economics7:57–98.10.3982/TE598Search in Google Scholar

Battigalli, P., and M.Siniscalchi. 1999. “Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games.” Journal of Economic Theory88:188–230.10.1006/jeth.1999.2555Search in Google Scholar

Battigalli, P., and M.Siniscalchi. 2002. “Strong Belief and Forward Induction Reasoning.” Journal of Economic Theory106:356–91.10.1006/jeth.2001.2942Search in Google Scholar

Battigalli, P., and M.Siniscalchi. 2003. “Rationalization and Incomplete Information.” Advances in Theoretical Economics3(1), Article 3, http://www.bepress.con/bejte/advances/vol3/iss1/art3.Search in Google Scholar

Battigalli, P., and M.Siniscalchi. 2007. “Interactive Epistemology in Games with Payoff Uncertainty.” Research in Economics61:165–84.10.1016/j.rie.2007.10.003Search in Google Scholar

Battigalli, P., and P.Tebaldi. 2012. “Interactive Epistemology in Simple Dynamic Games with a Continuum of Strategies.” typescript.Search in 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.)Search in 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 Economics11(1)(Advances), Article 6.Search in Google Scholar

Bergemann, D., and S.Morris. 2012. “Robust Mechanism Design.” The Role of Private Information and Higher Order Beliefs. Singapore: World Scientific Publishing.10.1142/8318Search in Google Scholar

Brandenburger, A., and E.Dekel. 1987. “Rationalizability and Correlated Equilibria.” Econometrica55:1391–402.10.2307/1913562Search in Google Scholar

Brandenburger, A., and E.Dekel. 1993. “Hierarchies of Beliefs and Common Knowledge.” Journal of Economic Theory59:189–98.10.1006/jeth.1993.1012Search in Google Scholar

Brandenburger, A., and A.Friedenberg. 2010. “Self-Admissible Sets.” Journal of Economic Theory145:785–811.10.1016/j.jet.2009.11.003Search in Google Scholar

Brandenburger, A., A.Friedenberg, and H. J.Keisler. 2008. “Admissibility in Games.” Econometrica76:307–52.10.1111/j.1468-0262.2008.00835.xSearch in Google Scholar

Cappelletti, G.2010. “A Note on Rationalizability and Restrictions on Beliefs.” B.E. Journals in Theoretical Economics10(1) (Topics), Article 40.Search in Google Scholar

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

Cho, I. K., and D.Kreps. 1987. “Signaling Games and Stable Equilibria.” Quarterly Journal of Economics102:179–222.10.2307/1885060Search in Google Scholar

Dekel, E., D.Fudenberg, and S.Morris. 2007. “Interim Correlated Rationalizability.” Theoretical Economics2:15–40.Search in Google Scholar

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

Friedenberg. 2010. “When Do Type Structures Contain All Hierarchies of Beliefs?” Games and Economic Behavior68:108–29.10.1016/j.geb.2009.05.005Search in Google Scholar

Harsanyi, J.1967–68. “Games of Incomplete Information Played by Bayesian Players.” Parts I, II, III, Management Science14:159–82, 320–34, 486–502.Search in Google Scholar

Mertens, J.-F., and S.Zamir. 1985. “Formulation of Bayesian Analysis for Games with Incomplete Information.” International Journal of Game Theory14:1–29.10.1007/BF01770224Search in Google Scholar

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

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

Pearce, D.1984. “Rationalizable Strategic Behavior and the Problem of Perfection.” Econometrica52:1029–50.10.2307/1911197Search in Google Scholar

Renyi, A.1955. “On a New Axiomatic Theory of Probability.” Acta Mathematica Academiae Scientiarum Hungaricae6:285–335.10.1007/BF02024393Search in Google Scholar

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

Tan, T., and S.Werlang. 1988. “The Bayesian Foundation of Solution Concepts of Games.” Journal of Economic Theory45:370–91.10.1016/0022-0531(88)90276-1Search in Google Scholar

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

  1. 1

    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.

  2. 2

    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).

  3. 3

    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.

  4. 4

    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.

  5. 5

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

  6. 6

    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.”

  7. 7

    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.

  8. 8

    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).

  9. 9

    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.

  10. 10

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

  11. 11

    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.

  12. 12

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

  13. 13

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

  14. 14

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

  15. 15

    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.

  16. 16

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

  17. 17

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

  18. 18

    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).

  19. 19

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

  20. 20

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

  21. 21

    is called conditional probability space in Renyi (1955).

  22. 22

    That is, .

  23. 23

    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.

  24. 24

    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.

  25. 25

    History h is inconsistent with (or counterfactual at) .

  26. 26

    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.

  27. 27

    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.

  28. 28

    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).

  29. 29

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

  30. 30

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

  31. 31

    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).

  32. 32

    See Lemma 1 in Section 5.

  33. 33

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

  34. 34

    With this notation, .

  35. 35

    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.

  36. 36

    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).

  37. 37

    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.

  38. 38

    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.

  39. 39

    Note that in games with complete information Σ = S.

  40. 40

    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.

  41. 41

    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).

  42. 42

    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 .

  43. 43

    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.

  44. 44

    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.

  45. 45

    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.

  46. 46

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

  47. 47

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

Published Online: 2013-5-8
Published in Print: 2013-1-1

©2013 by Walter de Gruyter Berlin / Boston

Articles in the same Issue

  1. Masthead
  2. Masthead
  3. Advances
  4. Dependence and Uniqueness in Bayesian Games
  5. Monopolistic Signal Provision†
  6. Multi-task Research and Research Joint Ventures
  7. Transparent Restrictions on Beliefs and Forward-Induction Reasoning in Games with Asymmetric Information
  8. A Simple Bargaining Procedure for the Myerson Value
  9. On the Difference between Social and Private Goods
  10. Optimal Use of Rewards as Commitment Device When Bidding Is Costly
  11. Labor Market and Search through Personal Contacts
  12. Contributions
  13. Learning, Words and Actions: Experimental Evidence on Coordination-Improving Information
  14. Are Trust and Reciprocity Related within Individuals?
  15. Optimal Contracting Model in a Social Environment and Trust-Related Psychological Costs
  16. Contract Bargaining with a Risk-Averse Agent
  17. Academia or the Private Sector? Sorting of Agents into Institutions and an Outside Sector
  18. Topics
  19. Poverty Orderings with Asymmetric Attributes
  20. Dictatorial Mechanisms in Constrained Combinatorial Auctions
  21. When Should a Monopolist Improve Quality in a Network Industry?
  22. On Partially Honest Nash Implementation in Private Good Economies with Restricted Domains: A Sufficient Condition
  23. Revenue Comparison in Asymmetric Auctions with Discrete Valuations
https://doi.org/10.1515/bejte-2012-0005
Keywords for this article
epistemic game theory; rationalizability; forward induction; transparent restrictions on beliefs

Articles in the same Issue

  1. Masthead
  2. Masthead
  3. Advances
  4. Dependence and Uniqueness in Bayesian Games
  5. Monopolistic Signal Provision†
  6. Multi-task Research and Research Joint Ventures
  7. Transparent Restrictions on Beliefs and Forward-Induction Reasoning in Games with Asymmetric Information
  8. A Simple Bargaining Procedure for the Myerson Value
  9. On the Difference between Social and Private Goods
  10. Optimal Use of Rewards as Commitment Device When Bidding Is Costly
  11. Labor Market and Search through Personal Contacts
  12. Contributions
  13. Learning, Words and Actions: Experimental Evidence on Coordination-Improving Information
  14. Are Trust and Reciprocity Related within Individuals?
  15. Optimal Contracting Model in a Social Environment and Trust-Related Psychological Costs
  16. Contract Bargaining with a Risk-Averse Agent
  17. Academia or the Private Sector? Sorting of Agents into Institutions and an Outside Sector
  18. Topics
  19. Poverty Orderings with Asymmetric Attributes
  20. Dictatorial Mechanisms in Constrained Combinatorial Auctions
  21. When Should a Monopolist Improve Quality in a Network Industry?
  22. On Partially Honest Nash Implementation in Private Good Economies with Restricted Domains: A Sufficient Condition
  23. Revenue Comparison in Asymmetric Auctions with Discrete Valuations
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 18.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/bejte-2012-0005/html
Scroll to top button