Volume 21, Issue 2 - Special Issue on Unawareness; Guest Editor: Burkhard C. Schipper

Decision theory and game theory are extended to allow for information processing errors. This extended theory is then used to reexamine market speculation and consensus, both when all actions (opinions) are common knowledge and when they may not be. Five axioms of information processing are shown to be especially important to speculation and consensus. They are called nondelusion, knowing that you know (KTYK), nested, balanced, and positively balanced. We show that it is necessary and sufficient that each agent's information processing errors be (1) nondeluded and balanced so that the agents cannot agree to disagree, (2) nondeluded and positively balanced so that it cannot be common knowledge that they are speculating, and (3) nondeluded and KTYK and nested so that agents cannot speculate in equilibrium. Each condition is strictly weaker than the next one, and the last is strictly weaker than partition information.

In modeling game and decision theory situations, it has been usual to start by considering Ω, the set of conceivable states of the world. I wish to propose a more fundamental view. I do not assume that the agent knows Ω. Instead the agent is assumed to derive for herself a representation of the universe. Given her knowledge and her ability to reason about it the agent deduces a set of conceivable states of the world and a set of possible states of the world. The epistemic model considered in this paper uses a propositional framework. The model distinguishes between the knowledge of the existence of a proposition, which I call awareness, and the knowledge of the truth or the falsity of the proposition. Depending upon whether one assumes that the agent is aware or not of all the propositions, she will or will not have a “complete model” of the world. When the agent is not aware of all the propositions, the states of the world and the possibility correspondence imaginable by her are coarser than the modeler's. The agent has an incomplete knowledge of both the states of the world and the information structure. In addition, I extend the model with “incompleteness” to a dynamic setting. Under the assumption that the agent's knowledge is non-decreasing over time, I show that the set of states of the world conceivable by the agent and her possibility correspondence get finer over time.

We provide a tool to model and solve strategic situations where players’ perceptions are limited, as well as situations where players realize that other players’ perceptions may be limited and so on. We define normal, repeated, incomplete information, and extensive form games with unawareness using a unified methodology. A game with unawareness is defined as a collection of standard games (of the corresponding form). The collection specifies how each player views the game, how she views the other players’ perceptions of the game and so on. The modeler’s description of perceptions, the players’ description of other players’ perceptions, etc. are shown to have consistent representations. We extend solution concepts such as rationalizability and Nash equilibrium to these games and study their properties. It is shown that while unawareness in normal form games can be mapped to incomplete information games, the extended Nash equilibrium solution is not mapped to a known solution concept in the equivalent incomplete information games, implying that games with unawareness generate novel types of behavior.

Bayesian game theory investigates strategic interaction of players with full awareness but incomplete information about their environment. We extend the analysis to players with incomplete awareness, who might not be able to reason about all contingencies in the first place. We develop three logical systems for knowledge, probabilistic beliefs and awareness, and characterize their axiom systems. Bayesian equilibrium is extended to games with incomplete awareness and we show that it is consistent with common prior and speculative trade, when common knowledge of rationality is violated.

We define a cautious version of extensive-form rationalizability for generalized extensive-form games with unawareness that we call prudent rationalizability. It is an extensive-form analog of iterated admissibility. In each round of the procedure, for each tree and each information set of a player a surviving strategy of hers is required to be rational vis-a-vis a belief system with a full-support belief on the opponents' previously surviving strategies that reach that information set. We demonstrate the applicability of prudent rationalizability. In games of disclosure of verifiable information, we show that prudent rationalizability yields unraveling under full awareness but unraveling might fail under unawareness. We compare prudent rationalizability to extensive-form rationalizability. We show that prudent rationalizability may not refine extensive-form rationalizability strategies but conjecture that the paths induced by prudent rationalizable strategy profiles (weakly) refine the set of paths induced by extensive-form rationalizable strategies.

This paper studies an environment in which a decision maker choosing between acts may initially be unaware of certain consequences. We follow the approach of Karni and Vierø (2013) to modeling increasing awareness, which allows for the decision maker's state space to expand as she becomes aware of new possible consequences. We generalize the main result in Karni and Vierø (2013) by allowing the discovery of new consequences to nullify some states that were non-null before the discovery. We also provide alternative assumptions which strengthen the predictions of the belief updating model.

Ambiguity in the ordinary language sense means that available information is open to multiple interpretations. We model this by assuming that individuals are unaware of some possibilities relevant to the outcome 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 corresponds to the usual unique (Bayesian) prior but with less than full awareness generates multiple priors. 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 awareness, however, leads to an expansion of the individual’s subjective state and (in general) a contraction in the set of priors under consideration.

The ability of markets to aggregate information through prices is examined in a dynamic environment with unawareness. We find that if all traders are able to minimally update their awareness when they observe a price that is counterfactual to their private information, they will eventually reach an agreement, thus generalising the result of Geanakoplos and Polemarchakis (1982). Moreover, if the traded security is separable, then agreement is on the correct price and there is information aggregation, thus generalizing the result of Ostrovsky (2012) for non-strategic traders. We find that a trader increases her awareness if and only if she is able to become aware of something that other traders are already aware of and, under a mild condition, never becomes aware of anything more. In other words, agreement is more the result of understanding each other, rather than being unboundedly sophisticated.

We incorporate unawareness into the delegation problem between a financial expert and an investor, and study their pre-delegation communication. The expert has superior awareness of the possible states of the world, and decides whether to reveal some of them to the investor. We find that the expert reveals all the possible states to the investor if the investor is initially aware of a large set of possible states, but reveals partially or nothing otherwise. An investor with a higher degree of unawareness tends to delegate a larger set of projects to the expert, giving rise to a higher incentive for the expert to keep her unaware.