Beliefs under Unawareness

Transcription

1 Beliefs under Unawareness Jing Li Department of Economics University of Pennsylvania 3718 Locust Walk Philadelphia, PA October 2007

2 Abstract I study how choice behavior under unawareness differs from choice behavior under zero probability beliefs or updating on zero probability events. Depending on different types of unawareness the decision maker suffers, behavior is either incomparable with the case of zero probability (in the case of pure unawareness), or sharply different from the behavior in a zero probability model (in the case of partial unawareness). I characterize beliefs under unawareness in both cases of unawareness. In terms of observed behavior, I show nontrivial partial unawareness is characterized by dynamically inconsistent behavior, while behavior in zero probability models exhibits dynamic consistency. Keywords: unawareness, zero probability, information, conditional probability system

3 1 Introduction It is a well-recognized fact that people may be unaware of some relevant uncertainties when making decisions. For example, most insurance companies in the 1970s were unaware of the harmful effects lead-based paint had on human body, which subsequently resulted in millions of dollars in compensations. The war against terrorism has been an extremely difficult endeavor precisely because we are unaware of many possible strategies the terrorists could employ. Understanding decision-making under unawareness thus is of great interest. 1 In particular, a frequently-raised question is that what is the observational difference between being unaware of an event and having zero probability belief of it. In this paper, I explore this issue. Conceptually, there is a clear distinction between the two cases: one is unaware of an event if one doesn t know it, and one doesn t know one doesn t know it, and so on; while assigning probability zero to an event requires being aware of it. Li (2007a, 2007b) characterize unawareness epistemically as a measurability constraint in single-agent environment: unawareness is equivalent to the absence of all knowledge concerning events that are too fine. For example, suppose there are two relevant uncertainties in the environment: whether there is a revolution, and whether the new product development is successful. The full state space specifying all uncertainties consists of four states, each of which describes how the two uncertainties are resolved. Now, suppose Alex is unaware of the possibility of a revolution. Then Alex s reasoning is contained in a subjective state space consisting of only two subjective states, one in which there is a revolution, and one in which there is no revolution. Fixing the algebra of events in the full state space and call it the objective algebra, the algebra of events Alex can reason about is a sub-algebra of the objective algebra. This suggests that, in a probabilistic model, agents with different awareness have different probability spaces, which is clearly different from having different null events. However, it is not clear how to differentiate the two cases in terms of observed behavior. There are two immediate behavioral implications following from the nature of unawareness. First, while one cannot bet on events of which one is unaware, one can certainly bet on events to which one assigns probability zero. The problem is, this is not testable: a rational decision-maker (DM) will never bet on events to which he assigns probability zero anyway. 2 Second, notice unawareness has a natural symmetry property: one is unaware of an event if and only if one is unaware of the negation of it. But one cannot assign probability zero to both an event and its negation. This implies that, while the DM can neither bet on nor against an event of which he is unaware, 1 For example, Modica, Rustichini and Tallon (1997) pioneers in this research by studying a general equilibrium model where agents can be unaware of some economic fundamentals. Recently, Ozbay (2006) and Filiz (2006) study games with unawareness, where one player has more awareness than the opponent and attempts to manipulate the opponent s beliefs. Heifetz, Meier and Schipper (2007) study Bayesian games where players may be unaware of relevant uncertainties. 2 In this context, rationality refers to expected utility maximization. 1

4 he would always want to bet against an event to which he assigns probability zero. For example, since Alex is unaware of both there is a revolution and there is no revolution, he can neither bet on the revolution nor bet against it; but he can either assign probability zero to the former event and hence would take on any bet against the revolution, or he can assign probability zero to the latter event, in which case he would take on any bet on revolution, and only one of these two scenarios can be true. But this observation, despite of the stark contrast between the two cases, only has minimal behavioral content: testing it requires asking Alex to rank bets involving the possibility of a revolution, but then Alex necessarily becomes aware of it. Thus I focus attention on bets concerning events of which Alex is aware, i.e., bets about the successful development of a new product. The first question I ask is whether Alex would rank these bets the same way when he is unaware of revolution and when he assigns probability zero to?. Here the question mark reflects the observation that unawareness is not directly comparable to assigning probability zero: it is impossible to have a probability measure defined on the set of all events where every event of which Alex is unaware has zero probability. Thus the first and foremost task in answering this question is to clarify what exactly the relevant comparison cases are when one compares unawareness with zero probability. A bet specifies what happens at each deterministic world. Call them objective bets. Since a DM who is unaware of some uncertainties is unaware of all deterministic worlds, he is necessarily unaware of all objective bets. For example, every (subjective) state Alex has in mind lacks the description about whether there is a revolution, and hence is not a deterministic world. On the other hand, arguably Alex can bet on events such as the new product development is successful. Call such bets subjective bets. Intuitively, Alex s ranking of these subjective bets reflects Alex s likelihood assessments of the involved events as as well as his perception of their consequences. Thus to compare unawareness and zero probability, the key issue is to relate Alex s perception of the consequences of the subjective bets to objective bets, in particular, objective bets involving events to which the DM assigns zero probability. There are two plausible scenarios. Perhaps when Alex bets on the new product development is successful, he has in mind a bet that pays off in all worlds where this event takes place. This is the case of pure unawareness, where the DM simply neglects the uncertainties of which he is unaware, and remains unbiased when reasoning about events of which he is aware. Under such perception, each subjective bet corresponds to one and only one objective bet. Consequently, the only implication of unawareness is the DM s inability to bet on events of which he is unaware. Pure unawareness and zero probability thus are incomparable. On the other hand, one can also imagine a natural asymmetry in some unawareness: in many situations, the DM s unawareness of an uncertainty seems to correspond to an unconscious certainty of a particular realization of the uncertainty. For example, one could argue that perhaps under his unawareness, Alex implicitly assumes that there is no revolution. When he bets on the event the new product development is 2

5 successful, Alex unconsciously takes it for granted that it is also true that there is no revolution. This case is particularly relevant when there is natural correlation between realizations of different possibilities. To see this, in the previous example, suppose the effort of developing a new product could produce three possible outcomes: successful development of the proposed product; successful development of a marketable product that is different from the proposed one; and complete failure, producing nothing marketable. Suppose Alex is unaware that the research effort could result in a product different from the proposed one yet marketable, and only aware of the other two outcomes. Since only one of the three outcomes can be true in any state, unawareness of the possibility of a different marketable product necessarily induces a bias in Alex s reasoning about the uncertain outcomes of the new product development: he unconsciously takes it for granted that there cannot exist a marketable product different from the proposed one. I refer to this case as partial unawareness. 3 Under partial unawareness, there is a sense in saying the DM is unaware of some objective states. For example, suppose Alex is partially unaware of the possibility of a revolution and that of a different marketable product. Then Alex s subjective state space can be viewed as consisting of those objective states where there is no revolution and the research team does not produce any other marketable product different from the proposed one, although he is unaware of this part of the description of the states. Alex s reasoning is confounded in this subjective state space, reasoning about every event of which he is aware relative to this universal event. In this sense Alex s subjective algebra of events is a relativization of the objective algebra. 4 It is this case of partial unawareness that begs a comparison with zero probability. I consider a two-period Anscomb-Aumann model, where the DM ranks acts mapping states to lotteries in each period (Anscombe and Aumann 1963). The (finite) full state space, S 2, and hence the set of all objective bets, are only revealed to the DM in the second period. In the first period, the DM has a subjective state space S 1 S 2, i.e., he is partially unaware of the states in S 2 \ S 1. There is no dynamic component in the situation: When ranking the subjective bets, the DM does not anticipate a second period where he updates his choice set. This is just a comparative statics exercise. 3 It is worth noting that the dichotomy of pure unawareness and partial unawareness arises from how the DM s subjective perceptions of consequences of actions in subjective states relate to the real consequences in objective states. This distinction is only meaningful in the context of decisionmaking. Epistemically, partial unawareness is not fundamentally different from pure unawareness. In cases involving correlations of realizations of different uncertainties, unawareness of an uncertainty could entail additional complication of unawareness of logical deductions. For example, not only Alex is unaware of both the research team develops a marketable product different from the proposed one and the research team may does not produce anything marketable, but also he is unaware that the research team may development the proposed product or it may produce nothing (of which he is aware) is equivalent to the research team does not produce anything marketable. See Galanis (2006) for a thorough discussion of such unawareness of logical deductions. 4 Technically, in this model a subjective state is a pair, consisting of the state itself and the subjective state space it belongs to. See Li (2007b) for details. 3

6 Fixing the set of lotteries, the DM s ranking of the subjective bets reflects how his partial unawareness affects his probabilistic assessments of those events of which he is aware. Absent of additional assumptions regarding the nature of unawareness, the only observation one can make is that the DM confounds those scenarios of which he is partially unaware when evaluating the subjective bets, and hence upon updating his state space, his ranking of the objective bets should reveal such confounding effects under the influence of partial unawareness in the first period. More specifically, I postulate his valuation of a subjective bet should fall between the value of a bestscenario completion of the subjective bet and a worst-scenario completion of it. I show that this axiom, combined with the standard Anscomb-Aumann axioms, is necessary and sufficient for the following characterization of subjective beliefs under partial unawareness: let µ i denote the DM s subjective beliefs over S i, then for every µ 1 -measurable event E, µ 1 (E) µ 2 (E). Thus, partial unawareness alone imposes little restriction on the DM s subjective beliefs as well as choice behavior. In particular, I show nontrivial partial unawareness is associated with dynamically inconsistent behavior: upon updating his awareness level, there exist bets such that the DM s preferences for them conditional on S 1 would reverse. In contrast, the case of zero probability not only imposes much more structure on the DM s beliefs, but also has the opposite implication for behavior. I explore three such models in the paper. The first model I explore is a static one in which the DM simply assigns zero probability to states in S 2 \ S 1. The second is the model of conditional probability systems proposed by Myerson (Myerson 1986b), in which the DM receives a zero probability signal and updates his beliefs. In both models the DM s behavior is characterized by dynamic consistency. This suggests partial unawareness is fundamentally different from zero probability, both conceptually and in terms of observed behavior. Finally, I show although it is always possible to match the behavior under partial unawareness in some model of zero probability, such practice sheds little light in understanding this environment. There has been little attempt in exploring beliefs under unawareness from decision-theoretic perspective. The closest work in the literature is Ahn and Ergin (2007). In their paper, the DM ranks objective bets in all periods, while the description of the bet becomes more and more detailed over time, which the authors interpret as different awareness levels. In contrast, the DM ranks different choice objects in different periods in this model. Consequently, while I compare beliefs of the same event under different awareness level, Ahn and Ergin compare how beliefs of different events relate to each other. The paper is organized as follows. Section 2 investigates how unawareness affect the DM s beliefs regarding events of which he is aware. I discuss both the benchmark case of pure unawareness and the main interest, the case of partial unawareness. Section 3 discusses three different models of zero probability and contrast them with the case of partial unawareness. Section 4 discusses caveats of the model and potential novel 4

7 issues a model of unawareness allows one to deal with. I conclude in Section 5. 2 Beliefs under Unawareness Let Z denote an arbitrary set of prizes and (Z) the set of simple lotteries over Z. 5 Let S i denote the state space for period i, i = 1, 2, and let S 2 be finite. Let i denote the DM s preference orderings over the period i choice set C i = ( (Z)) S i. Let i and i denote the asymmetric and symmetric parts of i, respectively. Let l denote a generic element in (Z). Slightly abusing notation, I also use l to denote the constant act that yields l in every state. As usual, convex combination of acts is defined by taking convex combination state-wise: for all α [0, 1] and any f, g C i, [αf + (1 α)g](s) = αf(s) + (1 α)g(s) for all s S i. First I postulate i satisfies the standard Anscomb-Aumann axioms. AA.1 (weak order): i is transitive and complete; AA.2 (continuity): for all g C i, the sets {f C i : g i f} and {f C i : f i g} are closed; AA.3 (independence): for all f, g, h C i, f i g, α (0, 1) implies αf + (1 α)h i αg + (1 α)h; AA.4 (non-triviality): there exist f, g C i such that f i g; To state the last axiom, some definitions are needed. Fixing f, g C i and E S i, I say the DM prefers f to g conditional on E, denoted by f i E g, if f i g where f (s) = g (s) for all s / E and f (s) = f(s) and g (s) = g(s) for all s E. An event E S i is Savage-null under i if f i E g for all f, g C i. A state s is said to be non-null if {s} is not Savage-null. AA.5 (state-independence): for all non-null s, t S i, l 1 i {s} l 2 if and only if l 1 i {t} l 2 for all constant acts l 1, l 2 (Z). Proposition 1 (Anscombe and Aumann (1963)): The axioms AA.1-5 are necessary and sufficient for i, i = 1, 2 to have the following representation: for all f C i, V i (f) = s S i µ i (s)u i (f(s)), (2.1) where u i : (Z) R is linear in probabilities and unique up to affine transformation, and µ i : 2 S i [0, 1] is the unique subjective probability on S i. 6 5 A simple lottery is a lottery that has finite support. 6 For notational ease, I omit the brackets when having a singleton set as the argument of µ i. 5

8 Next, I fix the DM s preferences for prizes across the two periods by requiring the DM to rank the constant acts in both periods in exactly the same way. B.1 (foreseen consequences): for all l 1, l 2 (Z), l 1 1 l 2 l 1 2 l 2. Adding axiom B.1 to AA.1-5 results in setting u 1 = u 2 = u in the above representations, which consequently allows one to separate the effects of unawareness on the DM s belief. The key axiom concerns how the DM s subjective bets relate to the objective ones, or, equivalently, how the DM s subjective state space relates to the full state space. First, I consider the case of pure unawareness as a benchmark. 2.1 Pure unawareness. Let S 2 = S 1 U, where U contains at least two elements. In this case, the DM is unaware of those uncertainties whose resolutions are described by U. When betting on a subjective event E S 1, the DM has in mind a bet that pays off whenever E is true, which, from the outside observer s perspective, simply means the objective event E U is true. Thus, a subjective bet on E under pure unawareness of U corresponds to an objective bet on E U. Let G : C 1 C 2 be the map that associates each subjective bet with its objective counterpart, i.e., for any f C 1, G(f)((s, u)) f(s) for all s S 1 and u U. Pure awareness amounts to the following axiom: B.2 (pure unawareness): for any f, g C 1, f 1 g G(f) 2 G(g). Proposition 2 Axioms AA.1-5 and B.1-2 are necessary and sufficient for i, i = 1, 2, to be represented as in (2.1), and in addition, 1. u 1 = u 2 ; 2. for all E S 1, µ 1 (E) = µ 2 (E U). The proof is straightforward and hence omitted. Proposition 2 says, pure unawareness is equivalent to a measurability constraint: the DM s preferences over the subjective bets are identified by a subset of his preferences over the objective bets. In terms of choice behavior, that the DM is unable to place bets on events of which he is unaware is the only implication of such unawareness; in terms of subjective beliefs, fixing µ 2 as the latent belief under full awareness, then the DM s beliefs under pure awareness is simply µ 2 restricted to those events of which he is aware. Pure unawareness is not comparable to assigning zero probability. 6

9 2.2 Partial unawareness. Let S 1 S 2. For notational ease, let U p S 2 \ S 1 denote the set of states of which the DM is partially unaware. The DM takes it for granted that S 1 is the universal event. Not only is the DM unaware of the uncertainty, but also he implicitly assumes it to be a particular certainty. Upon revelation of the full state space in the second period, the DM becomes aware of his own implicit assumption and reevaluates every event with respect to the expanded universal event. Let l 1 [E]l 2 [S i \E] denote the (subjective or objective) bet that yields the lottery l 1 on the event E and l 2 on its complement S i \ E. Given any f C i, let f denote its certainty equivalent under i (which exists by the Anscomb-Aumann axioms), i.e., f (Z) is the constant act such that f i f. Consider a subjective bet l 1 [E]l 2 [S 1 \ E]. Suppose l 1 1 l 2. From an outside observer s perspective, this subjective bet leaves states in S 2 \ S 1 unspecified; while from the DM s perspective in the first period, it specifies the payoff for every possible scenario. Intuitively, the DM implicitly confounds scenarios described in S 2 \ S 1 in his subjective states in S 1. Fixing a subjective bet f C 1, I say an objective bet g C 2 is an extension of f, or g extends f, if g(s) = f(s) for all s S 1 and g(s) {f(s) : s S 1 } for all s U p. Thus it seems a plausible assumption is, when betting on a subjective event in the first period under partial unawareness, the DM in fact implicitly has in mind some objective bet that extends it. Without additional assumptions about the nature of the DM s partial unawareness, one cannot pin down this objective counterpart as in the pure unawareness case. However, since preferences for prizes are state-independent, a natural assumption is that the objective counterpart of a subjective bet falls between its best scenario extension and worst scenario extension. B.3 (partial unawareness): for all E S 1, l 1 1 l 2, l 1 [E U p ]l 2 [S 1 \ E] 2 l 1 [E]l 2 [S 1 \ E] 2 l 1 [E]l 2 [S 2 \ E]. ( ) Axiom B.3 compares the DM s evaluation of a subjective bet l 1 [E]l 2 [S 1 \E] in the first period with his evaluation of the fully-specified objective bets in the second period, when he becomes fully aware, making use of the certainty equivalent l 1 [E]l 2 [S 1 \ E]. The objective bet l 1 [E U p ]l 2 [S 1 \ E] is the best-scenario extension of l 1 [E]l 2 [S 1 \ E], stipulating that the DM is to receive the better lottery in all states in U p ; while l 1 [E]l 2 [S 2 \ E] is the worst-scenario extension, where the DM receives the worse lottery in U p. B.3 asserts that, upon updating his awareness, the DM prefers the best-scenario extension to his original assessment of the subjective bet, and yet prefers the latter to the worst-scenario extension. Proposition 3 Axioms AA.1-5, B.1 and B.3 are necessary and sufficient for i, i = 1, 2, to be represented as in (2.1), and in addition, 7

10 1. u 1 = u 2 ; 2. for all E S 1, µ 2 (E) = s E α(s)µ 1(s), where α : S 1 [0, 1] is unique for all non-null s. Moreover, α(s) = 1 for all non-null s S 1 if and only if U p is Savage-null under 2. Proof: Only need to prove sufficiency for 2. Let V i (f) = s S i µ i (s)u(f(s)) represent i. Fix s S 1 and let l 1 1 l 2. Suppose {s} is Savage-null under 1 and hence µ 1 (s) = 0. Then l 1 [{s}]l 2 [S 1 \ {s}] 1 l 2, or l 1 [{s}]l 2 = l 2. By B.3, l 2 2 l 1 [{s}]l 2 [S 2 \ {s}]. By B.1, l 1 2 l 2, but then we must have µ 2 (s) = 0. Suppose {s} is not Savage-null. Then µ 1 (s) 0. By B.3, we have µ 1 (s)u(l 1 ) + (1 µ 1 (s))u(l 2 ) µ 2 (s)u(l 1 ) + (1 µ 2 (s)u(l 2 ). Set α(s) = µ 2(s) [0, 1]. Uniqueness of α(s) follows from the uniqueness of µ µ 1 (s) 1 and µ 2. Suppose S 2 \ S 1 is Savage-null under 2. Then for all E S 1, l 1 [E U p ]l 2 [S 1 \ E] 2 l 1 [E]l 2 [S 2 \ E]. By axiom B.3, we have l 1 [E]l 2 [S 1 \ E] 2 l 1 [E]l 2 [S 2 \ E], which then indicates µ 1 (E)u(l 1 )+(1 µ 1 (E))u(l 2 ) = µ 2 (E)u(l 1 )+(1 µ 2 (E))u(l 2 ). It follows µ 1 (E) = µ 2 (E). But then for all s S 1 such that {s} is not Savage-null, we must have α(s) = 1. For the converse, observe that α(s) = 1 implies µ 2 (s) = µ 1 (s). But then µ 2 (S 1 ) = µ 1 (S 1 ) = 1, and hence S \ S 1 is Savage-null under 2. The number α(s) can be interpreted as the DM s degree of partial unawareness of s. Intuitively, only α(s) portion of the scenario the DM associates with the subjective state s from the subjective state space S 1 under his partial unawareness corresponds to the objective state s from the full state space S 2. For example, consider the Alex example from Introduction. Arguably, Alex may associate all scenarios in which the research team fails to produce the proposed good with the subjective state the research team produces nothing. But in the full state space, only a subset of the described scenarios belong to the corresponding objective state, namely those in which the research team does not produce any other marketable product, too. Moreover, the DM s latent belief under full awareness assigns positive probability to those confounding scenarios. It is worth emphasizing that, absent of additional assumptions regarding the nature of unawareness, beliefs under partial unawareness are rather unrestrictive. That partial unawareness causes upward biases in the DM s probabilistic assessments with respect to his latent beliefs seems to be the only implication. 3 Models of Zero Probability. In this section, I explore whether the behavior under partial unawareness is consistent with a standard model, possibly involving Savage-null events. 8

11 The natural contrasting point is that the DM is fully aware of S 2 in the first period, but simply assigns zero probability to U p. Z.1 (zero probability): U p is Savage-null under 2. Requiring Z.1 in addition to axioms AA.1-5 and B.1 leads to identifying 1 as the conditional preferences 2 S 1 : for all f, g C 1, f 1 g if and only if f 2 S 1 g, where f and g extend f and g, respectively. Notice axiom Z.1 is consistent with B.3: if both preferences in ( ) are 2, then B.3 reduces to Z.1. In other words, axiom B.3 allows for the zero probability interpretation, as well as the interpretation that although the DM is partially unaware of U p, his latent beliefs assign zero probabilities to it anyway. This is true for individual states as well: if α(s) = 1, then either the DM has complete understanding of s despite of his partial unawareness, or the DM s latent belief assigns zero probability to the confounding scenarios the DM implicitly considers when reasoning about s. This model does not distinguish them. On the other hand, when the DM has nontrivial partial unawareness, i.e., his latent belief assigns a positive probability to U p under full awareness, then the behavior sharply differs from that under assigning probability zero. Corollary 4 Given axioms AA.1-5, B.1 and B.3, the set U p is not Savage-null under 2 if and only if there exists f, g C 1, f, g C 2, f and g are extensions of f and g respectively, such that f 1 g, while g 2 S 1 f. Proof: Only need to show sufficiency. Let µ 2 (U p ) > 0. Then there exists non-null s S 1 such that α(s) < 1. Let E = {s S 1 : α(s) < 1}. The set E is non-empty. It follows µ 2 (S 1 \ E) = µ 1 (S 1 \ E). Let l 1 1 l 2 and consider the subjective bet f = l 1 [E]l 2 [S 1 \ E] and let f C 2 be any extension of it. By proposition 3, V 1 (f) = u(l 1 )µ 1 (E) + u(l 2 )µ 1 (S 1 \ E). By standard result on conditional preferences (see, for example, Savage (1954)), V 2(g) = µ 2 (s) s S u(g(s)) represents 1 µ 2 (S 1 ) 2 S 1. Thus, V 2 (f s E ) = u(l 1 ) α(s)µ 1(s) µ 2 (S 1 + u(l ) 2 ) µ 1(S 1 \E) µ 2 (S 1. ) Since µ 2 (S 1 ) < 1, V 1 (f) < V 2 (f). By the continuity axiom (AA.2), there exists l (Z) such that f 1 l while l 2 S 1 f. Corollary 4 says nontrivial partial unawareness contrasts sharply with the case of assigning zero probability. Behavior under partial unawareness has a preference reversal feature: upon expanding his awareness level, the DM revises his choice. Thus, the DM is dynamically inconsistent. While the naive comparison of partial unawareness with assigning zero probability seems to suggest a sharp distinction between the two cases, one can also argue that the relevant comparison is not with a static model involving null events, but rather, a dynamic model involving updating on a zero probability event. A plausible story along 9

12 this line is as follows. Suppose the DM is always fully aware. In the first period, the DM assigns zero probability to the set U p. Upon being informed of S 2, the DM interprets it as the signal that U p actually occurs with positive probability, and hence updates his belief to give U p positive probability. However, in this interpretation, the signal S 2 expands the support of the belief rather than narrows it, and hence its informational content regarding the event U p is not well-defined. An alternative perspective is to take a spin on the suggested timing structure, and think about these preferences as reflecting how the DM takes into account the information he receives. More specifically, one can interpret 1 as the DM s preferences over the set of objective events conditional on receiving the information S 1, while 2 as the DM s preferences conditional on having the information S 2, in which case U p could be assigned positive probability. Let the primitives be { S } S 2 S 2, where S is interpreted as the DM s preferences over C 2 conditional on receiving information S. Since S 1, S 2 are taken to be information regarding which states are relevant in making a decision, the following axioms seem natural: Z.2 (event consistency): for any f, g C 2 and disjoint S, T S 2, f S g and f 2 T g imply f S T g; Z.3 (interest): for any s S 2, there exists f, g C 2 such that f {s} g. Myerson (1986b) proves the above axioms, combined with Anscomb-Aumann axioms, are necessary and sufficient for S, S S 2 to be represented by V S (f) = s S 2 µ(s S)u(f(s)), where µ( S) is unique, and the collection {µ( S)} S S2 consists of a conditional probability system (CPS) in the following sense: µ : 2 S 2 2 S 2 [0, 1] is such that, for all S, T, V S 2, V, 1. S T = µ(s T V ) = µ(s V ) + µ(t V ); 2. µ(s 2 V ) = µ(v V ) = 1; 3. S T V, T, µ(s V ) = µ(s T )µ(t V ). 7 Notice axiom Z.2 captures a form of dynamic consistency. To see it more clearly, let S = S 1 and T = U p. The axiom says, what happens in U p should not affect the DM s assessments regarding what happens in S 1. Clearly, this axiom rules out the 7 Blume, Brandenburger and Dekel (1991a) proves another version of this result in terms of lexicographic probability system (LPS). 10

13 type of dynamic inconsistency axiom B.3 mandates in the case of nontrivial partial unawareness. How to understand this? Intuitively, there is a fundamental conceptual difference between updating on new factual information and updating on new awareness information. Given state-independent preferences for lotteries, such separability is natural when information is about which states are relevant when making a decision. However, when there is partial unawareness, the problem is precisely the DM s inability to separate states due to his unawareness in the first period. In that context, the new information is not about revealing relevant facts, but rather, is about revealing the hidden correlation between events or states, which is not separable by nature. Consequently, while dynamic consistency characterizes the choice behavior under different factual information, even when some information can take the form of null events, dynamic inconsistency characterizes non-trivial partial unawareness. Thus, there remains one last question to be asked: although neither Savage-null events nor updating on null event generates the same behavior as in the case of partial unawareness, is there some model of zero probability that can do so? The answer is yes. It is always possible to construct a zero probability model to match behavior under partial unawareness. The idea is, one can always view the subjective state space S 1 and the full state space S 2 as two disjoint events and consider an expanded state space. More specifically, given the primitives i over C i, i = 1, 2, I can define another model à la Myerson (1986b) as follows. Let S be the disjoint union of S 1 and S 2. Let Si over C = (Z) S be defined by: f Si g if and only if f Si i g Si, and S = Si. Intuitively, this model describes the following behavior: on the expanded state space, the DM s initial signal is given by S 1, and hence assigns zero probability to the full state space S 2. In the second period, the DM receives the information S 2, updates his beliefs, and ranks objective bets according to S2, which matches 2 in the original model. However, interpretation of such a model is subtle. The state space S is an artificial construct, consisting of both subjective states and full states. There is no obvious interpretation of what a bet means in this context. In particular, the connection between subjective state and full states is lost, which is presumably the main interests in studying this environment. Although theoretically one can use this model to match behavior under partial unawareness, one still needs to go back to a model of unawareness to draw plausible assumptions about such behavior, which seems in turn to entirely invalidate such an approach. 11

14 4 Discussion 5 Conclusion References Ahn, David S. and Haluk Ergin, Framing Contingencies, Working paper. Anscombe, Frank J. and Robert J. Aumann, A Definition of Subjective Probability, Annals of Mathematical Statistics, 1963, 34, Blume, Lawrence, Adam Brandenburger, and Eddie Dekel, Lexicographic Probabilities and Choice Under Uncertainty, Econometricca, 1991a, 59, Filiz, Emel, Incorporating Unawareness into Contract Theory, Working paper. Galanis, Spyros, Unawareness of Theorems, Working paper, University of Rochester. Heifetz, Aviad, Martin Meier, and Burkhard C. Schipper, Unawareness, Beliefs, and Games, Working paper. Li, Jing, Information Structures with Unawareness, 2007a. Working paper., Modeling Unawareness in Arbitrary State Spaces, 2007b. Working paper. Modica, Salvatore, Aldo Rustichini, and Jean-Marc Tallon, Unawareness and Bankruptcy: A General Equilibrium Model, Economic Theory, 1997, 12, Myerson, Roger, Axiomatic Foundations of Bayesian Decision Theory, 1986b. Discussion Paper No. 671, the Center for Mathematical Studies in Economics and Management Science, Northwestern University. Ozbay, Erkut Y., Unawareness and Strategic Announcements in Games with Uncertainty, Working Paper. Savage, Leonard J., The Foundations of Statistics, Dover Publications,