Ambiguity and awareness: a coherent multiple priors model. Simon Grant Ani Guerdjikova y John Quigginz February 20, 2019 Abstract Ambiguity in the ordinary language sense is that available informa- tion is open to multiple interpretations. We model this by assuming that individuals are unaware of some possiblities relevant to the out- come of their decisions and that multiple probabilities may arise over an individual’s subjective state space depending on which of these possibilities are realized. We formalize a notion of coherent multiple priors and derive a representation result that with full awareness cor- responds to the usual unique (Bayesian) prior but with less than full awareness generates multiple priors. We show when information is received with no change in awareness, each element of the set of priors is updated in the standard Bayesian fashion (that is, full Bayesian updating). An increase in awarenss, however, leads to an expansion of the individual’s subjective state and (in general) a contraction in the set of priors under consideration. Keywords ambiguity, unawareness, multiple priors. JEL Classification: D81 Research School of Economics, Australian National University, AUSTRALIA Email: [email protected]. yGrenoble Applied Economics Lab, University of Grenoble Alpes, FRANCE. Email: [email protected]. zSchool of Economics, University of Queensland, AUSTRALIA. Email: [email protected]. 1 Introduction The idea that choices under uncertainty are subject to problems arising from ambiguity was first put forward by Ellsberg (1961), drawing on the earlier work of Knight (1921, 2006). Like Knight, Ellsberg argued that, in many cases, decisionmakers do not, and could not be expected to, act as if they assigned well-defined numerical probabilities to all the possible outcomes of a given choice. His well-known thought experiments illustrating this argu- ment formed the basis of a large subsequent literature both theoretical and empirical. In most of this literature, the term ‘ambiguity’ has been treated as a synonym for what Knight called ‘uncertainty’namely the fact that relative likelihoods are not characterized by well-defined numerical probabilities. The standard method of dealing with ambiguity in decision theory is to endow the decisionmaker with multiple priors as in Gilboa and Schmeidler (1989). This approach may be combined with a variety of preference models, notably including the maxmin model of Gilboa and Schmeidler (1989) and the smooth model of Klibanoff, Marinacci, and Mukerji (2005). For a non-specialist this is puzzling; there is no obvious link to the ordi- nary meaning (or meanings1) of ambiguity as a characteristic of propositions with more than one interpretation. In its normal usage, ambiguity is a lin- guistic concept, but in decision theory it is typically treated as a property of preferences. The now-standard usage is quite different from that in Ellsberg’soriginal 1 Empson (1930) famously distinguished seven types of ambiguity. 1 article. Ellsberg treated ambiguity, not as a property of preferences or relative likelihoods, but as a property of the information on which judgements of relative likelihoods might be based Responses from confessed violators [of the EU axioms] indicate that the difference is not to be found in terms of the two factors commonly used to determine a choice situation, the relative de- sirability of the possible pay-offs and the relative likelihood of the events affecting them, but in a third dimension of the problem of choice: the nature of one’sinformation concerning the relative likelihood of events. What is at issue might be called the am- biguity of this information, a quality depending on the amount, type, reliability and “unanimity”of information, and giving rise to one’s degree of “confidence” in an estimate of relative likeli- hoods. In this paper, we argue that informational ambiguity, in the ordinary lan- guage sense that the available information is open to multiple interpretation, may be modelled using concepts from the literature on unawareness. When individuals are unaware of some possibilities relevant to the outcome of their decisions, there are multiple probability distributions that may be applicable, depending on whether or not these possibilities are realised. To represent this idea, we adopt a syntactic representation, in which the state of the world is characterized by the truth values of a finite set of elementary propositions P . The state space is given by the set of all logically possible combinations of truth values, that is, by the truth table for 2 P . Any sentence ` in the language generated by P corresponds to an event L E in . In particular, any information that arrives at time t can be described by a sentence in and the corresponding event (that is, subset) of . L An unboundedly rational decision-maker is aware of all the propositions in P and sentences in , and therefore has access to a complete and unambigious L description of the state space . Assuming the Savage axioms are satisfied, the decision-maker can assign a unique subjective probability to any event E, and update that probability in line with Bayes rule as new information is received. We represent a boundedly rational decisionmaker as one who is unaware of at least some propositions in P . For simplicity, consider the case when there is only one such proposition p. Now consider any sentence ` that does not include p, but assigns a truth value to every other p. Such a sentence is potentially ambiguous, since it might correspond to the state associated with ` p or alternatively to ` p. ^ ^ : The idea may be illustrated by the case of the Ellsberg one-urn problem, as discussed by Billot and Vergopoulos (2018), though the interpretation given here differs from theirs. In the Billot and Vergorpoulos example, the color of the balls in the ‘ambiguous’urn is determined by the values of two parameters i and j, where j 0, 1 . In particular, if i = 3, a ball added to 2 f g the urn will be black if j = 0 and white if j = 1. However, the decisionmaker is unaware of the role of j. Thus, the information contained in the proposition i = 3 is ambiguous and incomplete, in a way that the decisionmaker cannot fully understand. A multiple priors representation of probabilities reflects this ambiguity. 3 Any complete probability distribution over the pair (i, j) gives rise to two conditional distributions 0 corresponding to (i j = 0) and 1 correspond- j ing to (i j = 1) . Although the decision-maker, being unaware of j, cannot j formulate the full probability distribution , she may entertain both 0 and 1, as well as any convex combination of the two, as prior beliefs about i. Further diffi culties arise when we consider updating in the multiple prior framework. As is well known from the works of Epstein and Breton (1993) and Ghirardato (2002), in a dynamic setting, deviations from additive beliefs require the relaxation of either consequentialism or dynamic consistency, or alternatively, a restriction on the class of preferences. This has lead to three different approaches to updating of multiple priors in the literature. The first approach restricts preferences to be recursive (and thus, dynamically- consistent and consequentialist) as in the case of recursive smooth ambiguity aversion in Klibanoff, Marinacci, and Mukerji (2009), rectangular max-min preferences in Epstein and Schneider (2003), and the generalization of their approach to -max-min preferences in Beissner, Lin, and Riedel (2016). The second class of models preserves consequentialism, but relaxes dynamic con- sistency, such as the full-bayesian updating axiomatized by Pires (2002), the maximum-likelihood rule of Gilboa and Schmeidler (1993) and the general- ized Bayesian updating rule for the -max-min preferences by Ghirardato, Maccheroni, and Marinacci (2008). Finally, a third approach consists in preserving dynamic consistency, but relaxing consequentialism as in the up- dating rule proposed by Hanany and Klibanoff (2007) for the smooth model of ambiguity formulated by Klibanoff, Marinacci, and Mukerji (2005). In this paper, we formalize this idea to derive a coherent multiple priors 4 (CMP) model. Our goals are twofold. First, we derive a representation theo- rem for the CMP models and show that, with full awareness, it corresponds to the usual Bayesian model. Second, we consider the problem of updat- ing beliefs, which has proved problematic in the multiple-priors setting. In our setting, updating may arise in response to the receipt of new informa- tion or to increased awareness, represented as awareness of new elementary propositions p. We show that when information is received with no change in awareness, each element of the set of priors is updated in the standard Bayesian fashion as in Ghirardato, Maccheroni, and Marinacci (2008). An increase in awareness is represented by an expansion of the state space to which the decision maker has access, and by a corresponding contraction in the set of priors under consideration. We show that, as the decisionmaker approaches full awareness, the set of priors contracts to a singleton . Rel- f g ative to the set of priors at any time t is made of conditional probabilities, depending on the truth values of propositions of which the decisionmaker is unaware The paper is organized as follows. We first set up the description of the decisionmaking problem in both propositional (syntactic) and state-space (semantic) terms. Awareness, information and acts are defined. Next we consider preferences and ambiguity in a timeless setting. We re- state the Ghirardato, Maccheroni, and Marinacci (2004) axioms. The crucial result of this section is to show that preferences satisfying the Ghirardato, Maccheroni, and Marinacci (2004) axioms may be derived from the prefer- ences of a fully aware EU-maximizer, by introducing unawareness. The key idea is that any set of truth values for the propositions of which the in- 5 dividual is unaware, induces a conditional probability distribution over the truth values of the propositions of which she is aware, and therefore over the associated (awareness-constrained) state space.