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Inductive reasoning about unawareness

Abstract

We develop a model of games with awareness that allows for differential levels of awareness. We show that, for the standard modal-logical interpretations of belief and awareness, a player cannot believe there exist propositions of which he is unaware. Nevertheless, we argue that a boundedly rational individual may regard the possibility that there exist propositions of which she is unaware as being supported by inductive reasoning, based on past experience and consideration of the limited awareness of others. In this paper, we provide a formal representation of inductive reasoning in the context of a dynamic game with differential awareness. We show that, given differential awareness over time and between players, individuals can derive inductive support for propositions expressing their own unawareness. We consider the ecological rationality of heuristics to guide decisions in problems involving differential awareness.

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Notes

  1. 1.

    Osborne and Rubinstein use the term ‘belief system’. Our terminology has been chosen to avoid confusion with the belief operator to be defined below.

  2. 2.

    This includes the action of nature governed by the probabilities \(f^{c}\) and the resolution of any randomization over actions by the players.

  3. 3.

    As one referee points out, this is not an innocuous existential assumption as it rules out what some may view as naturally conceivable situations of unawareness in strategic settings. However, by imposing it, we ensure the decision problem facing any individual and the decision problems they impute for any other player (including themselves at a later stage of the game) are always well defined. We acknowledge the limitations inherent with such an existential requirement, but as we shall see below, the simplification it affords us in characterizing “equilibrium” behavior allows us to turn our attention to what we view is the main focus of the paper, namely the inductive reasoning that these players may engage in when contemplating their own past and possibly future limited awareness.

  4. 4.

    In particular, it may require her to add histories of which she has become newly aware to the information sets she had encountered in the play of the game. However, given that \(\Gamma \) is a game of perfect recall and hence any restriction of that game she perceives herself to be playing is also a game of perfect recall, she will never forget the action choice she made from any information set she had previously encountered.

  5. 5.

    If the players are all aware of the game \(\Gamma \), then the behavioral rule corresponds to what we referred to in Sect. 2.1 as a behavioral strategy profile. In the setting we consider below, however, one or more of the players may not be fully aware of the game \(\Gamma \) that they are actually playing. Although a less than fully aware player might adopt a “rule of play” that will determine her choice at ever information set she may encounter in the game \(\Gamma \), in general she will not have access to the behavioral rule generated by the way she and her opponents decide on their choice of actions during the course of play. Hence, we feel it inappropriate to refer to such a behavioral rule as a strategy profile.

  6. 6.

    Note that, at a price of 1, both parties strictly prefer to trade, and each imputes to the other a game in which they are indifferent between trading and no-trading. Further, all of this is common knowledge. This example does not, however, allow for common knowledge of a strict preference for trade. Heifetz et al. (2012) show that, in general, unawareness cannot produce common knowledge of mutual strict preference for speculative trade.

  7. 7.

    We owe this characterization of the result to an anonymous referee.

  8. 8.

    A closely related argument is prominent in philosophical debates over “realism,” namely the view that the success of science reflects its correspondence to objective truth. Critics such as Laudan (1981) argue on the basis of historical experience that, since successful theories have been proven false in the past, the success of a theory cannot be regarded as evidence for its truth. Similarly, in our analysis, the fact that models used with some success in decision making have nonetheless been discovered to be incomplete in the past supports the view that the model currently held by any given decision maker is also unlikely to be complete.

  9. 9.

    A referee points out that the string \(q_{\forall }a_{i}q\) is simply a propositional constant and could be replaced with an arbitrary string, such as \(helloworld\), without any effect on the logical validity of the axiomatizations. This point is entirely consistent with the observation that, within a game of differential awareness, individuals must always believe the proposition \(q_{\forall }a_{i}q\) to be true, at least as regards themselves.

  10. 10.

    Recall in the specification of the game that for party 1 the buyer \(\alpha ^{1}\) corresponds to the action “make an offer of 1” (action \(\alpha _{1}\) in Figs. 1, 2, 3, 4) and for party 2, the owner \(\alpha ^{2}\) is the action “accept the offer of 1” (action \(\alpha _{A}\) in Figs. 1, 2, 3, 4).

  11. 11.

    One common way of meeting this requirement is to seek funding for research that is effectively complete, but has not yet been published.

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Correspondence to John Quiggin.

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Financial support from the Australian Research Council Discovery Grant DP120102463 is gratefully acknowledged.

Appendix A

Appendix A

Proof of Proposition 1

By standard arguments (e.g., Osborne & Rubinstein [1994, p 227]), it follows that an assessment in a game of perfect recall is sequentially rational \(\left( {\varvec{\beta }, \varvec{\mu }}\right) \) if and only if it satisfies the one-shot deviation property. That is, at each \(h\), for the behavioral strategy \(\beta _{h}^{P\left( h\right) }\) of player \( P\left( h\right) \) in the continuation of the game \(\Gamma _{\tilde{Z}\left( h\right) }\), there is no subsequent information set \(\mathcal I _{\tilde{Z} \left( h\right) }\left( h^{\prime }\right) \), with \(P\left( h^{\prime }\right) =P\left( h\right) \) in the continuation of the game at which a change in \(\beta _{h}^{P\left( h\right) }\left( h^{\prime }\right) \) increases his payoff conditional on reaching \(\mathcal I _{\tilde{Z}\left( h\right) }\left( h^{\prime }\right) \).

Therefore, we will first establish the existence of a trembling-hand equilibrium for the agent-normal form of \(\mathcal G \), which by the one-shot deviation property also constitutes a trembling-hand equilibrium of \(\mathcal G \). It will then suffice to show that for any trembling-hand equilibrium strategy profile \({\varvec{\beta }}\), there exists a subjective probability system \({\varvec{\mu }}\) such that \(\left( {\varvec{\beta }, \varvec{\mu }}\right) \) is a sequential equilibrium.

We take the agent-normal form of the game with awareness \(\left( \Gamma ,\tilde{Z}\left( \cdot \right) \right) \), to be the game with awareness \(\left( \Gamma ^{an},\tilde{Z}\left( \cdot \right) \right) \), where \(\Gamma ^{an}\) is the agent-normal form \(\Gamma \), in which there is one player for each information set in the extensive-form game and where player \(h\) is imputed to be playing \(\Gamma _{\tilde{Z}\left( h\right) }^{an} \) the agent-normal form of \(\Gamma _{\tilde{Z}\left( h\right) }\). For each \(h \), denote the perturbation of the game \(\Gamma _{\tilde{Z}\left( h\right) }^{an}\) by first fixing the strategies of all players \(h^{\prime \prime }\), such that there exists \(h^{\prime }\in \mathcal I _{\tilde{Z} \left( h\right) }\left( h\right) \) and \(h^{\prime \prime }\preceq h^{\prime } \) and \(h^{\prime \prime }\ne h^{\prime }\) (i.e., player \(h^{\prime \prime } \) is a player has already moved by the time the game reaches the information set \(\mathcal I _{\tilde{Z}\left( h\right) }\left( h\right) \)) and then letting the set of actions of each player \(h^{\prime \prime }\), such that there exists history \(h^{\prime }\in \mathcal I _{\tilde{Z}\left( h\right) }\left( h\right) \) and \(h^{\prime }\preceq h^{\prime \prime }\) (i.e., player \(h^{\prime \prime }\) is a player who moves in the continuation of the game \(\Gamma _{\tilde{Z}\left( h\right) }^{an}\) after information set \(\mathcal I _{\tilde{Z}\left( h\right) }\left( h\right) \)) be the set of mixed strategies in \(A_{\tilde{Z}\left( h\right) }\left( h^{\prime \prime }\right) \cap A_{\tilde{Z}\left( h^{\prime \prime }\right) }\left( h^{\prime \prime }\right) \) that assign probability of at least \(\varepsilon _{h^{\prime \prime }}^{a}\left( h\right) \) to each action that player \(h\) (at \(h\)) imputes to player \(h^{\prime \prime }\) at her information set \( \mathcal I _{\tilde{Z}\left( h\right) }\left( h^{\prime \prime }\right) \). That is, this constrains \(h\) and every player who follows \(h\) to use every strategy \(h\) imputes that they have available with some minimal probability. Consider a sequence of such perturbed games in which \(\varepsilon _{h^{\prime \prime }}^{a}\left( h\right) \rightarrow 0\), for all \(h\), for all \(h^{\prime \prime }\) and \(a\); by the compactness of the set of strategy profiles, some sequence of selections \(\left( {\varvec{\beta }}^{k}\right) \) from the sets of strategy profiles that are sequentially rational for all \(i\), of the games \(\{\Gamma _{\tilde{Z}\left( h\right) }^{an}:h\in H\}\) converges to say \({\varvec{\beta }}\). It is straightforward to show from its construction that \({\varvec{\beta }}\) corresponds to a trembling-hand perfect equilibrium of the game with awareness.

Now, take the sequence \(\left( {\varvec{\beta }}^{k}\right) \). At each information set \(\mathcal I \left( h\right) \) define the belief \(\mu _{h^{\prime }}\) for each \(h^{\prime }\) in \(\mathcal I \left( h\right) \), to be the limit of the beliefs defined from \(\left( {\varvec{\beta }}^{k}\right) \) using Bayes rule. The assessment \(\left({\varvec{\beta },{ \mu } }\right) \) is then by construction consistent. Since the strategies are completely mixed as can be done consistently, each information set consistent with each player’s level of awareness is reached with positive probability and each agent’s strategy is a best response when the beliefs at each information set are defined by \({\varvec{\mu }}\). Thus, \(\left( {\varvec{\beta },\varvec{\mu }}\right) \) is a sequential equilibrium. \(\square \)

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Grant, S., Quiggin, J. Inductive reasoning about unawareness. Econ Theory 54, 717–755 (2013). https://doi.org/10.1007/s00199-012-0734-y

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Keywords

  • Unawareness
  • Bounded rationality
  • Induction

JEL Classification

  • D80
  • D82