Interactive epistemology in simple dynamic games with a continuum of strategies

Abstract

We extend the epistemic analysis of dynamic games of Battigalli and Siniscalchi (J Econ Theory 88:188–230, 1999, J Econ Theory 106:356–391, 2002, Res Econ 61:165–184, 2007) from finite dynamic games to all simple games, that is, finite and infinite-horizon multistage games with finite action sets at nonterminal stages and compact action sets at terminal stages. We prove a generalization of Lubin’s (Proc Am Math Soc 43:118–122, 1974) extension result to deal with conditional probability systems and strong belief. With this, we can provide a short proof of the following result: in every simple dynamic game, strong rationalizability characterizes the behavioral implications of rationality and common strong belief in rationality.

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Notes

  1. 1.

    In finite games of complete information without chance moves, strong rationalizability coincides with (the correlated version of) “extensive-form rationalizability,” a solution concept put forward by Pearce (1984). Battigalli (2003) coined the term “strong rationalizability” to distinguish it from other legitimate, but weaker notions of rationalizability for dynamic games (e.g., Ben Porath 1997). For more on rationalizability in dynamic games, see the references therein.

  2. 2.

    Of course, the general epistemic framework of BS allows for the analysis of different epistemic assumptions not involving forward-induction reasoning, as illustrated in Battigalli and Siniscalchi (1999).

  3. 3.

    Actually, the extension result of Lubin (1974) follows from Lemma 2.2 of Varadarajan (1963).

  4. 4.

    See Battigalli (2003, 2006) and Battigalli and Siniscalchi (2003). Battigalli and Siniscalchi (2007), Battigalli and Friedenberg (2012), and Battigalli and Prestipino (2013) relate strong \(\Delta \)-rationalizability to the epistemic analysis of forward-induction reasoning.

  5. 5.

    Note that if C is clopen, then so is \(C\times T\).

  6. 6.

    Information types may be different from types in the sense of Harsanyi (1967–1968), that implicitly determine hierarchies of exogenous initial beliefs, and from the epistemic types introduced later, that represent hierarchies of conditional probability systems.

  7. 7.

    Note that \(A^{n}\) is the set of all sequences of elements of A with length n. In the rest of the paper, it will be clear from the context whether a superscript stands for a product (as in this case) or as a mere index (as, for example, in the statement of Lemma 3).

  8. 8.

    Such features can be implicitly modeled within the present framework by letting \(0\in I\) be an indifferent “fictitious” player. Then, \(\theta _{0}\) parameterizes residual uncertainty, about which no nonfictitious player has private information, and \(\mathcal {A}_{0}(h)\) is the set of realizations of a chance move. Knowledge about the objective probabilities of chance moves can be modeled by restrictions on players’ beliefs.

  9. 9.

    Battigalli and Prestipino (2013) allow for imperfect monitoring, a dimension of common uncertainty, and chance moves, but they maintain that the game structure is finite.

  10. 10.

    More generally, in all games with perfect recall, \(S(h_{i})={\text {proj}}_{S_{i}}S(h_{i})\times {\text {proj}} _{S_{-i}}S(h_{i})\) for each information set \(h_{i}\) of each player i, which is all we really need.

  11. 11.

    The intermediate step from game structure to game form could be skipped, defining utility/payoff functions over terminal nodes. We keep this step because it adds conceptual clarity at little cost.

  12. 12.

    Roughly, this means that everybody’s lower-order beliefs are the marginals of higher-order beliefs and there is common belief of this conditional on each history.

  13. 13.

    Furthermore, Battigalli and Siniscalchi (1999) prove that the canonical type structure is also universal, i.e., that every other type structure can be universally (hence uniquely up to isomorphism) mapped into it preserving beliefs hierarchies. On such belief-preserving morphisms, see Heifetz and Samet (1998) and Friedenberg and Meier (2011).

  14. 14.

    Each type \(t_{i}\) is also a type in the sense of Harsanyi (1967–1968), because it determines an initial belief about the information and epistemic types of the co-players: \(t_{i}\mapsto {\text {mrg}}_{\varTheta _{-i}\times T_{-i}}\beta _{i,\varnothing }(t_{i})\). Therefore, \(\varGamma \) and \(\langle I,\left( T_{i},\beta _{i}\right) _{i\in I}\rangle \) together yield a Bayesian game.

  15. 15.

    See Battigalli and Friedenberg (2012), and Battigalli and Prestipino (2013).

  16. 16.

    The formal definition of best reply is given below.

  17. 17.

    Battigalli and Friedenberg (2012) analyze forward-induction reasoning when contextual assumptions rule out completeness.

  18. 18.

    Catonini (2017b), and Harrington (2017). See Sect. 5.

  19. 19.

    Artemov et al. (2013) analyze robust virtual implementation with respect to \(\Delta \)-rationalizability in static mechanisms. Mueller (2016) applies strong rationalizability to dynamic mechanisms. Bergemann and Morris (2016 and references therein) apply to static mechanisms the solution concept without belief restrictions and call it “belief-free rationalizability.”

  20. 20.

    Battigalli and Siniscalchi (2002, 2003) correctly rely on such simplified algorithm.

  21. 21.

    In this case, 2’s belief in the rationality of 1 after any Stackelberg history \(h^{*}=\left( \left( a_{1}^{*},a_{2} ^{1}\right) , \left( a_{1}^{*},a_{2}^{2}\right) ,\ldots \right) \) is implied by \(\varepsilon \)-possibility of \(\theta ^{c}\).

  22. 22.

    If \(\theta ^{c}\) is a “commitment type” whose only feasible action is the Stackelberg actions, then the result also holds for “weak,” or “initial” \(\Delta \)-rationalizability. See also Battigalli and Watson (1997), who consider a countable sequence a short-run players in the role of player 2.

  23. 23.

    For an epistemic analysis of finite, but otherwise general dynamic games of incomplete information, see Battigalli and Prestipino (2013).

  24. 24.

    See the discussion in Battigalli and Friedenberg (2012) and Battigalli and Prestipino (2013).

  25. 25.

    See, for example, Hart et al. (2017) and the references therein.

  26. 26.

    We thank Gabriele Beneduci for proving this generalization. See the working paper version: http://www.igier.unibocconi.it/folder.php?vedi=6337&tbn=albero&id_folder=4878.

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Correspondence to Pierpaolo Battigalli.

Additional information

We are very grateful to Gabriele Beneduci, for excellent research assistance and for proving a generalization of the main technical lemma of this paper. We thank Federico Bobbio, Roberto Corrao, Enrico De Magistris and Giacomo Lanzani for careful proof reading and Emiliano Catonini, Nicodemo De Vito, Michael Greinecker, the Editor, and an anonymous referee for helpful suggestions. Pierpaolo Battigalli gratefully acknowledges financial support from the European Research Council (Grant 324219).

Appendix: Proofs

Appendix: Proofs

Proof of Lemma 3

Fix some \(\nu \in \Delta ^{\mathcal {C}}(X)\) that strongly believes \(({\text {proj}}_{X}E_{1},\ldots ,{\text {proj}}_{X}E_{n})\). Define recursively a partition \(\{ \mathcal {C}_{0},\mathcal {C}_{1},\ldots \}\) of \(\mathcal {C}\) as follows: let \(\mathcal {C}_{0}=\{C\in \mathcal {C}:\nu _{X}(C)>0\}\). Suppose we have defined \(\mathcal {C}_{0},\ldots ,\mathcal {C}_{m}\), and let

$$\begin{aligned} \mathcal {{\bar{C}}}_{m+1}=\{ {\bar{C}}\in \mathcal {C}\setminus \mathcal {(C}_{0} \cup \dots \cup \mathcal {C}_{m}):\forall C\in \mathcal {C},\,C\supset {\bar{C}}\Rightarrow C\in \mathcal {C}_{0}\cup \dots \cup \mathcal {C}_{m}\} \end{aligned}$$

be the collection of conditioning events \({\bar{C}}\) not in \(\mathcal {C}_{0} \cup \dots \cup \mathcal {C}_{m}\) whose strict predecessors C are all in \(\mathcal {C}_{0}\cup \dots \cup \mathcal {C}_{m}\); then, let

$$\begin{aligned} \mathcal {C}_{m+1}=\{D\in \mathcal {C}\setminus \mathcal {(C}_{0}\cup \dots \cup \mathcal {C}_{m}):\exists {\bar{C}}\in \mathcal {{\bar{C}}}_{m+1},{\bar{C}}\supseteq D,\nu _{{\bar{C}}}(D)>0\}. \end{aligned}$$

For convenience, let \(\bar{\mathcal {C}}_{0}=\{X\}\). Note that, by definition of \(\mathcal {C}_{m}\) and since \(\nu \) is a CPS, we have \(\mathcal {{\bar{C}}} _{m}\subseteq \mathcal {C}_{m}\) for every \(m=0,1,\ldots \) (in particular, \(\bar{\mathcal {C}}_{0}=\left\{ X\right\} \subseteq \mathcal {C}_{0}\)).

Claim 1

Every element of \(\mathcal {C}_{m}\)has a unique predecessor in\(\bar{\mathcal {C}}_{m}\).

First note that it is true by definition that each element of \(\mathcal {C} _{m}\) has at least one predecessor in \(\mathcal {{\bar{C}}}_{m}\); we only have to prove uniqueness. The claim is obvious for \(m=0\). For \(m=1,2,\ldots \), let \(C\in \mathcal {C}_{m}\) and consider any two distinct predecessors \(C^{\prime },C^{\prime \prime }\supset C\) (if they exist); then—by the tree-like property of \(\mathcal {C}\)—either \(C^{\prime \prime }\supset C^{\prime }\) or \(C^{\prime }\supset C^{\prime \prime }\). Thus, by definition of \(\bar{\mathcal {C}}_{m}\), at most one of \(C^{\prime }\) and \(C^{\prime \prime }\) can belong to \(\bar{\mathcal {C}}_{m}\). \(\square \)

Claim 2

\(\{ \mathcal {C}_{0},\mathcal {C}_{1},\ldots \}\)is a partition of\(\mathcal {C}\).

By the construction above, it is clear that the collections \(\mathcal {C}_{m}\) are pairwise disjoint. Let us show that \(\mathcal {C}= {\displaystyle \bigcup \nolimits _{n\ge 0}} \mathcal {C}_{n}\). Pick any C in \(\mathcal {C}\). By definition of simple conditional space, the collection of predecessors (supersets) of C in \(\mathcal {C}\), denoted by \(\mathcal {C}^{\supseteq }(C)\), is finite and totally ordered by \(\supseteq \). Hence, \(\mathcal {C}^{\supseteq }(C)\) can be written as \(\{C_{0},\ldots ,C_{m}\}\) where all \(C_{k}\)s are distinct (\(k=0,\ldots ,m\)) and \(X=C_{0}\supset \dots \supset C_{m-1}\supset C_{m}=C\). Remark that, for every k less than m, \(\{C_{0},\ldots ,C_{k}\}\) is the set of strict predecessors of \(C_{k+1}\). We show inductively that

$$\begin{aligned} C_{k}\in \mathcal {C}_{0}\cup \dots \cup \mathcal {C}_{k} \end{aligned}$$

for all \(k=0,\ldots ,m\), so that, in particular, \(C=C_{m}\in \mathcal {C}_{0} \cup \dots \cup \mathcal {C}_{m}\subseteq {\displaystyle \bigcup \nolimits _{n\ge 0}} \mathcal {C}_{n}\). Indeed, \(C_{0}=X\in \mathcal {C}_{0}\) is trivially true. Suppose that \(C_{k}\in \mathcal {C}_{0}\cup \dots \cup \mathcal {C}_{k}\) for all \(k=0,\ldots ,\ell \le m-1\): if \(C_{\ell +1}\in \mathcal {C}_{0}\cup \dots \cup \mathcal {C}_{\ell }\) we are done; if not, every strict predecessor of \(C_{\ell +1}\) belongs to \(\mathcal {C}_{0}\cup \dots \cup \mathcal {C}_{\ell }\) by the inductive hypothesis, which implies—by definition of \(\mathcal {{\bar{C}} }_{\ell +1}\)—that \(C_{\ell +1}\in \mathcal {{\bar{C}}}_{\ell +1}\subseteq \mathcal {C}_{\ell +1}\).

By Claims 12, we can define a map \(e:\mathcal {C}\rightarrow \mathcal {C}\) that associates each \(C\in \mathcal {C}\) with the earliest predecessor \({\bar{C}}\) of C such that \(\nu _{{\bar{C}}}(C)>0\): let \(\mathcal {C}_{m}\) be the cell of partition \(\{ \mathcal {C}_{0},\mathcal {C}_{1},\ldots \}\) that contains C, then \({\bar{C}}=e(C)\) is the unique predecessor of C in \(\mathcal {{\bar{C}}}_{m}\).

Note that \(C\cap {\text {proj}}_{X}E_{m}\ne \emptyset \) if and only if \(E_{m}\cap (C\times T)\ne \emptyset \), for all \(m=0,\ldots ,n\) and C in \(\mathcal {C}\). Since \(\nu \) strongly believes each \({\text {proj}} _{X}E_{m}\), by Lemma 2 we can find an array of probability measures \({\bar{\mu }}=({\bar{\mu }}_{C})_{C\in \mathcal {C}}\) in \([\Delta (X\times T)]^{\mathcal {C}}\) (possibly not a CPS) such that \({\text {mrg}}_{X} {\bar{\mu }}=\nu \) and \({\bar{\mu }}\) strongly believes \(\mathcal {E}\). More explicitly: for all C in \(\mathcal {C}\) and for all \(m=0,\ldots ,n\),

$$\begin{aligned} {\text {mrg}}_{X}{\bar{\mu }}_{C}=\nu _{C}, \end{aligned}$$

and

$$\begin{aligned} E_{m}\cap (C\times T)\ne \emptyset \Rightarrow {\bar{\mu }}_{C}(E_{m}\cap (C\times T))=1. \end{aligned}$$

We use some of these measures to construct the desired CPS \(\mu \):

For all \(m=0,1,\ldots \), for all C in \(\mathcal {C}_{m}\), and for all E in \(\mathcal {B}(X\times T)\), let

$$\begin{aligned} \mu _{C}(E)=\frac{{\bar{\mu }}_{e(C)}(E\cap (C\times T))}{{\bar{\mu }}_{e(C)}(C\times T)}, \end{aligned}$$

where the denominator is positive because \({\bar{\mu }}_{e(C)}(C\times T)={\text {mrg}}_{X}{\bar{\mu }}_{e(C)}(C)=\nu _{e(C)}(C)>0\) by definition of \(e\left( C\right) \). Since \(\{ \mathcal {C}_{0},\mathcal {C}_{1},\ldots \}\) is a partition of \(\mathcal {C}\), we obtain an array \((\mu _{C})_{C\in \mathcal {C}}\) in \([\Delta (X\times T)]^{\mathcal {C}}\). Clearly, \(\mu _{C}(C\times T)=1\) for all C in \(\mathcal {C}\).

Claim 3

\(\nu \)is the marginal of \(\mu \).

For each C in \(\mathcal {C}\) and each event of the form \(E\times T\) for E in \(\mathcal {B}(X)\),

$$\begin{aligned} \mu _{C}(E\times T)&=\frac{{\bar{\mu }}_{e(C)}((E\cap C)\times T))}{\bar{\mu }_{e(C)}(C\times T)}=\frac{\nu _{e(C)}(E\cap C)}{\nu _{e(C)}(C)}\\&=\frac{\nu _{C}(E\cap C)\, \nu _{e(C)}(C)}{\nu _{e(C)}(C)}=\nu _{C}(E), \end{aligned}$$

where we used the definition of e(C) and the fact that \(\nu \) satisfies the chain rule (1). \(\square \)

Claim 4

\(\mu \)strongly believes\(\mathcal {E}\).

Let \(m\in \{1,\ldots ,n\}\) and \(C\in \mathcal {C}\). We must show that \(E_{m} \cap (C\times T)\ne \emptyset \) implies \(\mu _{C}(E_{m})=1\). If \(E_{m} \cap (C\times T)\ne \emptyset \), then \(C\cap {\text {proj}}_{X}E_{m} \ne \emptyset \), and \(e(C)\cap {\text {proj}}_{X}E_{m}\ne \emptyset \) because \(C\subseteq e(C)\) by definition. Since \({\bar{\mu }}\) strongly believes \(\mathcal {E}\), \({\bar{\mu }}_{e(C)}(E_{m})=1\), which implies \({\bar{\mu }} _{e(C)}(E_{m}\cap (C\times T))={\bar{\mu }}_{e(C)}(C\times T)\) and thus

$$\begin{aligned} \mu _{C}(E_{m})=\frac{{\bar{\mu }}_{e(C)}(E_{m}\cap (C\times T))}{{\bar{\mu }} _{e(C)}(C\times T)}=\frac{{\bar{\mu }}_{e(C)}(C\times T)}{{\bar{\mu }} _{e(C)}(C\times T)}=1. \end{aligned}$$

\(\square \)

Claim 5

\(\mu \)satisfies the chain rule (1).

Fix an event E and two conditioning events C and D in \(\mathcal {C}\) with \(E\subseteq D\times T\subseteq C\times T\). We must show that \(\mu _{C} (E)=\mu _{D}(E)\, \mu _{C}(D\times T)\). There are two cases: (i) If \(e(C)=e(D)\), then

$$\begin{aligned} \mu _{C}(E)&=\frac{{\bar{\mu }}_{e(C)}(E)}{{\bar{\mu }}_{e(C)}(C\times T)} =\frac{{\bar{\mu }}_{e(D)}(E)}{{\bar{\mu }}_{e(D)}(C\times T)}\\&=\frac{{\bar{\mu }}_{e(D)}(E)}{{\bar{\mu }}_{e(D)}(D\times T)}\frac{\bar{\mu }_{e(D)}(D\times T)}{{\bar{\mu }}_{e(D)}(C\times T)}\\&=\mu _{D}(E)\, \mu _{C}(D\times T). \end{aligned}$$

(ii) If \(e(C)\ne e(D)\), then \(D\subseteq e(D)\subset C\subseteq e(C)\) and \(\nu _{e(C)}(e(D))=0\). Therefore \({\bar{\mu }}_{e(C)}(E)={\bar{\mu }}_{e(C)}(D\times T)=0\), which implies that (1) holds:

$$\begin{aligned} \mu _{C}(E)&=\frac{{\bar{\mu }}_{e(C)}(E)}{{\bar{\mu }}_{e(C)}(C\times T)}\\&=0\\&=\frac{{\bar{\mu }}_{e(D)}(E)}{{\bar{\mu }}_{e(D)}(D\times T)}\frac{\bar{\mu }_{e(C)}(D\times T)}{{\bar{\mu }}_{e(C)}(C\times T)}\\&=\mu _{D}(E)\, \mu _{C}(D\times T). \end{aligned}$$

\(\square \)

Since \(\nu \) is the marginal of \(\mu \), which strongly believes \(\mathcal {E}\) and satisfies all the properties of a CPS, the theorem is proved. \(\square \)

Other proofs

Proof of Lemma 4

Let \(H_{n}=H\cap A^{n}\) be the set of nonterminal histories of length n. By assumption, \(H_{n}\) is finite for each \(n=0,1,\ldots \); therefore, \(H=\bigcup _{n\in \mathbb {N}_{0}}H_{n}\) is countable. Each set \(\mathcal {A} _{j}(h)\) is a compact metrizable space; hence, the countable Cartesian product \(S_{j}=\prod _{h\in H}\mathcal {A}_{j}(h)\) is compact and metrizable as well. The set \(\varTheta _{j}\) is compact metrizable and \(I\setminus \{i\}\) is countable, hence \(\varSigma _{-i}=\prod _{j\ne i}(\varTheta _{j}\times S_{j})\) is a compact metrizable space.

We have to show that \(\mathcal {C}_{i}\) is a countable collection of clopen subsets of \(\varSigma _{-i}\) with a tree-like structure. The collection \(\mathcal {C}_{i}\) is countable because H is countable, and it inherits its tree-like structure from the countable tree H: indeed, \(\varSigma _{-i} =\varSigma _{-i}(\varnothing )\in \mathcal {C}_{i}\), and the set of strict predecessors of any \(\varSigma _{-i}(h)\) in \(\mathcal {C}_{i}\) is the finite collection \(\mathcal {C}_{i}^{\prec }(h)=\{ \varSigma _{-i}(h^{\prime }):h^{\prime }\prec h\}\); pick any pair of distinct predecessors \(\varSigma _{i}(h^{\prime })\) and \(\varSigma _{i}(h^{\prime \prime })\) in \(\mathcal {C}_{i}^{\prec }(h)\). Since H is a tree, either \(h^{\prime }\prec h^{\prime \prime }\) and \(\varSigma _{-i}(h^{\prime \prime })\subset \varSigma _{-i}(h^{\prime })\) or \(h^{\prime \prime }\prec h^{\prime }\) and \(\varSigma _{-i}(h^{\prime })\subset \varSigma _{-i} (h^{\prime \prime })\).

Obviously, \(\varSigma _{-i}(\varnothing )=\varSigma _{-i}\) is closed and open in \(\varSigma _{-i}\). To see that \(\varSigma _{-i}(h)\) is closed for each h in \(H\setminus \{ \varnothing \}\), let \(h=(a^{1},\ldots ,a^{n})\) and note that

$$\begin{aligned} \varSigma _{-i}(h)=\prod _{j\ne i}\left( \varTheta _{j}\times \{a_{j}^{1}\} \times \dots \times \{a_{j}^{n}\} \times \prod _{h^{\prime }\in H:h^{\prime }\nprec h}\mathcal {A}_{j}(h^{\prime })\right) \end{aligned}$$

where each set in the product is closed. To see that \(\varSigma _{-i}(h)\) is also open in \(\varSigma _{-i}\), we first show that \(\varSigma \left( h\right) \) is open in \(\varSigma \). Note that \(\{ \varSigma (h^{\prime }):h^{\prime }\in H_{n}\}\) is the finite partition of subsets of \(\varSigma \) obtained from the preimages through map \(\left( \theta ,s\right) \mapsto \zeta \left( s\right) \) of the elements of the finite partition \(\{Z(h^{\prime }):h^{\prime }\in H_{n}\}\) of Z. Each \(\varSigma \left( h^{\prime }\right) \) is a product of closed subsets (see above) and it is closed as well; therefore, the finite union \(\bigcup _{h^{\prime }\in H_{n}\setminus \{h\}}\varSigma (h^{\prime })\) is closed and \(\varSigma (h)=\varSigma \setminus \bigcup _{h^{\prime }\in H_{n}\setminus \{h\}}\varSigma (h^{\prime })\) is open. Since a Cartesian product \(C=\prod _{j\in I}C_{j}\subset \varSigma \) is open if and only if each \(C_{j}\) is open, it follows that each \(\varSigma _{j}\left( h\right) \) is open and \(\varSigma _{-i}(h)=\prod _{j\ne i}\varSigma _{j}(h)\) is open. \(\square \)

Proof of Lemma 5

The result follows from standard compactness-continuity arguments and is therefore omitted (see Battigalli 2003). \(\square \)

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Battigalli, P., Tebaldi, P. Interactive epistemology in simple dynamic games with a continuum of strategies. Econ Theory 68, 737–763 (2019). https://doi.org/10.1007/s00199-018-1142-8

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Keywords

  • Epistemic game theory
  • Simple infinite dynamic game
  • Strong belief
  • Strong rationalizability

JEL Classification

  • C72
  • C73
  • D82