GAMES WITH OR WITHOUT UNAWARENESS RYAN Y. FANG Abstract. We show that the behavioral predictions of the unawareness model developed by Heifetz, Meier, and Schipper (2006, 2007) (the HMS model) can be replicated by standard models with properly defined state spaces and belief structures. This is achieved essentially by allowing individuals to hold false beliefs (yet is more involved than simply letting the individuals assign zero probability to events of which they are unaware in the HMS model). However, no other assumptions made in the standard rational model needs to be relaxed. Moreover, our model is truly standard and admits a standard, unawareness-free interpretation.

1. Introduction In the past two decades, much attention has been devoted to developing models for “unawareness”, the situation where an individual does not “think about” certain possibilities when faced with a decision problem (Dekel, Lipman, and Rustichini, 1998a). In epistemic terms, unawareness can be thought of as when an individual does not know something (an event), does not know that she does not know, and ad infinitum. That is, the individual has no positive knowledge of the event (Dekel, Lipman, and Rustichini, 1998b; Li, 2009). In contrast to the standard notion of uncertainty, which describes a lack of knowledge, unawareness amounts to a lack of conception (Heifetz, Meier, and Schipper, 2006). Modica and Rustichini (1994) and Dekel, Lipman, and Rustichini (1998b) illustrate how standard state-space models coupled with non-partitional information correspondences can not model unawareness. Richer structures are developed by Modica and Rustichini (1999) and Halpern (2001) in order to model unawareness in a meaningful manner. The rest of Date: November 27, 2011. This paper was inspired by discussions with Philip Reny, to whom we owe a great debt of gratitude. We would also like to thank Eddie Dekel, Bart Lipman, Roger Myerson and Marciano Siniscalchi for their valuable comments on earlier drafts of the note. 1

our discussion will focus on later developments by Heifetz, Meier, and Schipper (2006, 2007) and Li (2008a, 2009), whose treatments of unawareness are purely semantic (do not rely on explicit usage of the modal syntax) and are rich enough for analyzing multi-person unawareness. Consequently, these models easily lend themselves to game theoretical analyses, which are developed in Heifetz, Meier, and Schipper (2007). Although not the focus of our discussion, Dekel, Lipman, and Rustichini (1998a) also survey some earlier work on decision theoretical approaches to unforeseen contingencies. In light of these developments in the unawareness literature, one may want to ask the question: how much has this new literature added to our understanding of individuals’ behavior? More specifically, do the new models generate novel behavioral patterns that cannot be captured by the more conventional approach? As we shall argue below, the answer to the second question seems to be “no”. In what follows, we show that, the behavioral predictions of the unawareness model by Heifetz, Meier, and Schipper (henceforth HMS model or HMS Bayesian game) can be replicated by standard uncertainty models with properly defined state spaces and belief structures. We focus on the HMS model since there is a readily available game theoretical formulation. However, given the obvious similarity between the HMS model and the Li (2008a, 2009) model1, we believe that the same logic will extend to any similar game theoretical formulation based on the later. What we mean by “standard” will be clarified later. Among other things, the model encompasses a standard state space in the Dekel, Lipman, and Rustichini (1998b) sense and is given an “unawareness-free” interpretation. We do not claim that we can model unawareness using a standard model, which has been proved impossible. Nor do we assert that individuals behave identically under unawareness and uncertainty in general. What we show is that to the extent that behaviors under unawareness can be generated by the

1

The similarity stems from the fact that in Li (2008a, 2009), each individual’s awareness function effectively gives rise to a lattice of subjective state spaces similar to that in the HMS model and, like in the HMS model, the individual’s reasoning is confined within a sub-space in the lattice. This, as will become clear in what follows, is the key fact that drives our results. 2

HMS model (and models that are similar), they can also be generated by more conventional approaches. There are two possible explanations for this behavioral equivalence between the HMS model and the more conventional model. First, it could be that behaviors under unawareness are indeed the same as under uncertainty in general. Second, there could be key features of unawareness that would give rise to novel behavioral patterns but are not captured by the HMS model. In light of the clear conceptual differences between unawareness and uncertainty stressed by Dekel, Lipman, and Rustichini (1998b) and other authors, we are sympathetic of the view that there can be novel behavioral consequences of unawareness. However, one can not be sure until there is a model that formally demonstrates this. We discuss a couple of important features of unawareness that are not captured in the HMS model towards the end of the paper. Such features may or may not give rise to behaviors unique to unawareness. Again, this can only be proved once we have a formal model. The rest of the paper is organized as follows: the next section gives a brief summary of the HMS unawareness model. Section 3 makes a first attempt to replicate the behavioral predictions of the HMS model by a standard model and highlights that the key deviation from the rational model is that individuals are now allowed to hold false beliefs. Section 4 proves the result with a new state space that affords the model an unawareness-free interpretation. The new construction also allows us to give a rather intuitive explanation of the behavioral equivalence. Section 5 concludes with further remarks.

2. The HMS Unawareness Model In this section we briefly summarize the HMS model. While the HMS model is purely semantic, we will from time to time refer to syntactic examples as it greatly assists the exposition. Readers familiar with the HMS model can skip this section and proceed to our discussion in Section 3. Let S = {Sα }α∈A be a complete lattice of disjoint “state-spaces”, with the partial order  on S, where Sα  Sβ means “Sα is more expressive than Sβ ”. The term “more expressive” 3

warrants a little more explanation. Roughly speaking, it means that “states of Sα describe situations with a richer vocabulary than states of Sβ ” (Heifetz, Meier, and Schipper, 2006). Heifetz, Meier, and Schipper (2006) give the following syntactic example. Let the relevant facts (aspects of the problem) be {p, q}. For α ⊆ {p, q}, let Sα = {true, f alse}α . Then the space S{p,q} is more expressive than S{p} . A state of S{p,q} is (p, ¬q), that is, a state where p is true and q is false. Clearly, a state of S{p} , (p), can only describe partially the expressed details of (p, ¬q). In the HMS model, when the “true state” is (p, ¬q), an individual unaware of the fact q may think the state is (p). Each S ∈ S is assumed to be measurable, with σ-field FS . Let Ω = ∪α∈A Sα be the (overall) state space. 0

For every S and S 0 such that S 0  S, there is a measurable surjective projection rSS : S 0 → 0

S, where rSS is the identity. For any ω ∈ S 0 , rSS (ω) is the restriction of the description ω to the S

S

{p,q} {p,q} less expressive vocabulary S. In the above example, rS{p} ((p, ¬q)) = (p) = rS{p} ((p, q)).

The events, negation of an event, conjunction and disjunction of events can then be defined 0

using the projections rSS . We omit the definitions here. The reader is referred to the authors’ original papers. However, we will highlight that, these definitions differ significantly from those in the standard model. For example, the negation of an event does not in general equal to its complement. Let I be the non-empty set of individuals. For each individual i ∈ I, there is a possibility correspondence Πi : Ω → 2Ω \{∅}. So, when in the state ω, individual i thinks states in Πi (ω) are possible. The possibility correspondences are assumed to satisfy a number of properties, including the “confinement” condition: if ω ∈ S then Πi (ω) ⊆ S 0 for some S 0  S; and the “stationarity” condition: ω 0 ∈ Πi (ω) implies Πi (ω 0 ) = Πi (ω). Moreover, let 4 (S) denote the set of probability measures on (S, FS ). For each individual i ∈ I, there is a type mapping ti : Ω → ∪α∈A 4 (Sα ), where ti (ω), as usual, is interpreted as individual i’s belief in the state ω. The type mapping is assumed to satisfy a number of properties that guarantee the coherence of belief and awareness of the individuals down the lattice structure. Note that, for each i ∈ I and each ω ∈ Ω, ti (ω) belongs to ∪α∈A 4 (Sα ), i.e. a probability measure over some S ∈ S rather than Ω. As an example, if the true state is (p, q) ∈ S{p,q} , an individual 4

i that is unaware of the fact q will think the state lies in the space S{p} (Πi ((p, q)) ⊆ S{p} ) and his belief ti ((p, q)) will be given by a probability measure over the space S{p} . We are now ready to define HMS Bayesian games with unawareness and the corresponding equilibrium concept. For notational convenience, the authors restrict attention to finite I, Ω (where for each S ∈ S, FS is assumed to be 2S ) and finite sets of actions. Formally,

Definition 1. A HMS Bayesian game with unawareness of events consists of an unawareness 0
belief structure Ω =< S, rSS S 0 S , (ti )i∈I > that is augmented by a tuple < (Mi )i∈I , (ui )i∈I > defined as follows: For each player i ∈ I, there is (i) (ii)

a finite nonempty set of actions Mi , and ! Y a utility function ui : Mi × Ω → R i∈I

A strategy of player i is a function σi : Ω → 4 (Mi ) such that for all ω ∈ Ω, ti (ω) = ti (ω 0 ) implies σi (ω) = σi (ω 0 ). Let GHM S denote the set of HMS Bayesian games with unawareness.

In state ω, player i only thinks states in Sti (ω) possible, where Sti (ω) is the space of which ti (ω) is a probability measure. Therefore the expected utility of player-type (i, ti (ω)) from   0 the strategic profile σSti (ω) ≡ (σj (ω ))j∈I , where σj (ω 0 ), a member of ∆ (Mj ), is ω 0 ∈Sti (ω)

0

the strategy of the player-type (j, tj (ω )) in the normal form of the game (defined below), can be written as: 

U(i,ti (ω)) σSti (ω)



ˆ X

≡ ω 0 ∈S

ti (ω)

m∈

Q

j∈I

Y

σj (ω 0 ) (mj ) · ui (m, ω 0 ) dti (ω)

Mj j∈I

The equilibrium concept can then be defined as follows: 5

(2.1)

Definition 2. An equilibrium of a HMS Bayesian game with unawareness of events < Ω, (Mi )i∈I , (ui )i∈I > is a Nash equilibrium of the strategic game defined by: (i)

{(i, ti (ω)) | ω ∈ Ω and i ∈ I} is the set of players, and for each player (i, ti (ω)) ,

(ii) (iii)

the set of mixed strategies is 4 (Mi ) , and the utility function is given by equation (2.1)

For expositional purposes, we call the strategic game defined as such the normal form of the HMS Bayesian game with unawareness.

We have thus reproduced the HMS formulation of a Bayesian game with unawareness. In the next section we will argue that the behavioral predictions of the HMS model can be replicated by a standard model.

3. The Standard Approach - First Attempt We first define what we mean by a standard Bayesian game or a standard model. For simplicity, we continue to work with finite set of players, finite state space and finite action spaces for each player. 6

Definition 3. A standard Bayesian game consists of: (i) (ii)

a non-empty set of players I, a measurable state-space Ω, with FΩ = 2Ω , and for each player i ∈ I,

(iii) (iv)

a finite nonempty set of actions Mi , ! Y a utility function ui : Mi × Ω → R, i∈I

(v)

a strictly positive prior belief ρi ∈ 4Ω, so that ρi (E) > 0 for all E ∈ FΩ , and,

(vi)

a possibility correspondence Πi : Ω → 2Ω \ {∅}.

For each ω ∈ Ω and each i, Πi (ω) is called i’s information set in state ω. A strategy of player i is a function σi : Ω → 4 (Mi ) such that for all ω ∈ Ω, Πi (ω) = Πi (ω 0 ) implies σi (ω) = σi (ω 0 ). Let G denote the set of standard Bayesian games.

In state ω, player i with prior ρi forms a posterior belief conditioning on her information set Πi (ω) in the standard Bayesian manner. We can summarize this by a function ti : Ω → ∆Ω, such that ti (ω) is player i’s posterior belief in state ω. We call this function i’s type mapping and each distinct ti (ω) a type of i2. In a rational model, the possibility correspondences are assumed to be partitional, that is, the range of each possibility correspondence is a partition of the state space. In epistemic terms, this follows from the familiar assumptions that individuals know only the truth, know what they know and know what they do not know (Samet, 1990). In what follows, we will allow for some of these assumptions to be relaxed. Our formulation also accommodates the possibility of heterogeneous priors.

2Hence

ti (Ω) is the set of i’s types. 7

Let σ denote a strategic profile. The expected utility of player-type (i, ti (ω)) from σ with respect to i’s posterior ti (ω) is given by: ˆ U(i,ti (ω)) (σ) ≡

X ω 0 ∈Ω

m∈

Q

Y

σj (ω 0 ) (mj ) · ui (m, ω 0 ) dti (ω)

(3.1)

j∈I j∈I Mj

Definition 4. An equilibrium of a standard Bayesian game is a Nash equilibrium of the strategic game defined by: (i)

{(i, ti (ω)) | ω ∈ Ω and i ∈ I} is the set of players, and for each player (i, ti (ω)) ,

(ii) (iii)

the set of mixed strategies is 4 (Mi ) , and the utility function is given by equation (3.1)

We call the strategic game defined as such the normal form of the Bayesian game.

Before proceeding to stating and proving our main propositions, we state the following assumption:

(A1) For every HMS game GHM S =< Ω, (Mi )i∈I , (ui )i∈I >, there exist possibility correspondences (Πi )i∈I , such that each Πi satisfies the conditions imposed by Heifetz, Meier, and Schipper (2006) and, for every player i and every ω, ω 0 ∈ Ω, (i) (ii)

The support of ti (ω) is equal to Πi (ω) ; and, Πi (ω) = Πi (ω 0 ) implies ti (ω) = ti (ω 0 ) .

That is, the players’ beliefs are consistent with their information: beliefs assign positive probability only to events they think are possible and players have the same belief in different states if they also have same information in those states. 8

Although our assumption is not explicitly imposed in (Heifetz, Meier, and Schipper, 2007), we believe it is implied by the internal coherence of the HMS construction. We will not need this assumption to prove our result, however, if we make the type mappings primitive in our definition instead of being derived from a prior. However, if we do so, the resulting definition will be less standard and, more importantly, we might not be able to speak of prior beliefs in the natural sense. For every HMS game GHM S =< Ω, (Mi )i∈I , (ui )i∈I >, let the σ-algebra on Ω, FΩ , be generated by the union ∪S∈S FS 3. Let 4Ω denote the set of probability measures on Ω. We have the following Lemma: Lemma 5. For every GHM S =< Ω, (Mi )i∈I , (ui )i∈I >∈ GHM S , if (A1) holds, then there exist functions (ρi )i∈I , such that, for each i, ρi ∈ ∆Ω, ρi (E) > 0 for all E ∈ FΩ , and ρi conditioning on Πi (ω) agrees with ti (ω) on Sti (ω) (and assigns zero probability to Ω \ Sti (ω) ), where (Πi )i∈I are the possibility correspondences given in (A1) and Sti (ω) ∈ S is the space of which ti (ω) is a probability distribution. Proof. For each i and each ω ∈ Ω, let t0i (ω) ∈ ∆Ω be the extension of ti (ω) to FΩ by assigning zero probability to Ω \ Sti (ω) . Recall the “stationarity” condition: ω 0 ∈ Πi (ω) implies Πi (ω 0 ) = Πi (ω). It follows that for any ω, ω 0 ∈ Ω, if Πi (ω 0 ) ∩ Πi (ω) 6= ∅, then Πi (ω 0 ) = Πi (ω). In other words, Πi (Ω) is a partition of ∪Πi (Ω). (ii) of (A1) implies that there is a bijection between Πi (Ω) and ti (Ω) and, consequently, between Πi (Ω) and t0i (Ω). (i) of (A1) implies that the support of t0i (ω) is equal to Πi (ω). If ∪Πi (Ω) = Ω, Let ρi be a convex combination of the members of t0i (Ω), with strictly positive weights assigned to each of them. If ∪Πi (Ω) 6= Ω, then Πi (Ω) ∪ {Ω \ ∪Πi (Ω)} partitions Ω. In such cases, let τi ∈ ∆Ω have support Ω \ ∪Πi (Ω) and let ρi be a (strictly) convex combination of the members of t0i (Ω) and τi . It is straightforward to check that ρi constructed as such has the properties described in Lemma 5 and t0i (ω) will correspond to i’s posterior belief in state ω, where her information set is Πi (ω)4. 3In



the finite case, this is simply 2Ω . is similar to the observation by Samet (1998).

4This

9

We are now ready to show that the behavioral predictions of any HMS Bayesian game with unawareness can be replicated by a standard Bayesian game, or, formally: Proposition 6. If (A1) holds, then for every GHM S ∈ GHM S , there exists G ∈ G, such that, the equilibria of GHM S are the same as those of G5. Proof. Our proof is by construction. Fix any GHM S =< Ω, (Mi )i∈I , (ui )i∈I >∈ GHM S . We can construct a Bayesian game G as follows: (i)

Let G have the same state space, Ω, same set of players, I, as GHM S ; and, for every i ∈ I,

(ii) (iii)

Let her action space and utility function be the same as in GHM S ; Let i have the corresponding possibility correspondence Πi , as described in (A1) ; and,

(iv)

Let i have prior belief ρi as constructed in Lemma 5 and let t0i denote the resulting type mapping.

It then follows from Lemma 5 that the normal forms of the games G and GHM S have the same set of players (up to renaming (i, ti (ω)) as (i, t0i (ω))). Moreover, the utility functions in the normal forms are in the following sense the same: for any strategy profile σ, we have: 

U(i,ti (ω)) σSti (ω)



ˆ X

≡

Y

σj (ω 0 ) (mj ) · ui (m, ω 0 ) dti (ω)

ω 0 ∈Sti (ω) m∈Q j∈I j∈I Mj

ˆ =

X

Y

ω 0 ∈Ω m∈Q j∈I j∈I Mj

σj (ω 0 ) (mj ) · ui (m, ω 0 ) dt0i (ω)

≡ U(i,t0 (ω)) (σ) i where σSti (ω) , as in equation 2.1, is the restriction of σ to Sti (ω) ⊆ Ω and, from the first to the second line, we use the fact that t0i (ω) agrees with ti (ω) on Sti (ω) . This then implies that 5Strictly

speaking, the sets of equilibria from the two games are isomorphic, since we “rename” the players from one normal form to the other. 10

every equilibrium to the normal form of GHM S will be an equilibrium to the normal form of G and vice versa.



Thus, we have shown that, the behavioral predictions of a HMS Bayesian game with unawareness are the same as those of a properly defined standard Bayesian game. The two games are essentially identical, except that in the standard game, the players are “aware” of all possible events. Note that, an HMS model with non-trivial unawareness will have ti (ω) ∈ ∆S 0 for some i, some ω ∈ S ∈ S and some S 0 ≺ S. Consequently, in our standard model G, we must have ω ∈ / Πi (ω). That is, in state ω, i has false information and hence holds false belief. In epistemic terms, individuals can “know” something that is not true and we no longer have a rational model. Note, however, as shown in the proof to Lemma 5, Πi (Ω) is still a partition of ∪Πi (Ω). It follows that it remains true that individuals will know what they know and what they do not know. Therefore, our formulation makes it clear that, in order to replicate behavioral consequences of the HMS unawareness model, all we need is to allow individuals to hold false beliefs. No other assumptions made in the standard rational model needs to be relaxed. Our result echoes Schipper (2011), who shows that, from a decision theoretical perspective, unawareness can be modeled by zero probability beliefs under what he terms a “impersonal expected utility representation”. However, he argues that this representation is problematic as it takes as primitive an unawareness structure and the beliefs cannot be reasonably interpreted as subjective probabilities. Interpretation indeed represents a challenge to us. For G to be a truly standard model, it needs to also admit a standard interpretation. In particular, we need to be able to understand the belief structures without appealing to the notion of unawareness. However, this is not possible, since, without appealing to unawareness, the members of Ω, can no longer be interpreted as states of the world6. 6Clearly,

G cannot be given an interpretation under the HMS framework either, since then probabilistic beliefs over Ω would not be meaningful. 11

In Savage’s(1972) original framework for decision-making under uncertainty, the states have to “specify all relevant aspects of the problem” and “exhaust all possibilities”. Take the earlier example where the relevant aspects are p and q and consider the state (p). When an individual is unaware of the aspect q, the state (p) then specifies all aspects of the problem of which she is aware. Therefore, as a subjective state, (p) then satisfies Savage’s requirement. However, when the individual is aware of the aspect q, as in our standard Bayesian setting, the state (p) only provides a partial description of the relevant facts. Indeed, in conventional treatments, a state space “exhausting all possibilities” is simply given by S{p,q} , a proper subset of Ω. Thus, in order to argue that the behavioral predictions of the HMS model can be captured by a standard model, we need to construct games with “real states” that can be given an “unawareness-free” interpretation. This will be done in the next section, where we achieve this by introducing another dimension of uncertainty.

4. The Standard Approach with Real States The standard model for decision-making under uncertainty is very general and can be used to model uncertainties arise from a wide range of situations. In particular, it allows for the individuals to be uncertain about how different aspects of the world (facts) will affect their payoffs or whether a particular aspect of the world (fact) will affect their payoffs at all. This additional dimension of uncertainty clearly introduces another aspect of the problem that needs to be considered by the individual when making her decision and, consequently, specified in any state of the state space. For concreteness, consider the situation where an individual draws one from two lottery tickets at random. One of the two lotteries involves betting against the realization of both aspects p and q of the world, while the other one involves betting only against the realization of aspect p. Clearly, a state where p is true, q is true and the first lottery is drawn should be distinguished from a state where p and q are both true and the second lottery is drawn. With this kind of interpretation in mind, we construct a state space that admits a standard 12

interpretation, which will allow us to derive behavioral equivalence between the HMS model and a standard model. Formally, for every HMS model GHM S =< Ω, (Mi )i∈I , (ui )i∈I >, we define a new state e = Sα × A, where α ∈ A is the index of the most expressive vocabulary, that is, space Ω e We interpret the Sα  Sα for all α ∈ A7. Further, for each α ∈ A, let Seα = Sα × {α} ⊆ Ω. e as specifying not only all the basic facts for the problem but also which facts affect states in Ω payoff. For each state in Seα , the facts that actually affect payoff are those expressible in the e constructed as (HMS) states in Sα . For every HMS state space Ω, we call the state space Ω, such, the “corresponding real state space”. Take the syntactic example again. A = {{p, q}, {p}, {q}, φ} and the richest vocabulary is e is given by (s, α) where s ∈ S{p,q} and α ∈ A. A state in given by {p, q}. A generic state in Ω Se{p} is ((p, ¬q) , {p}), that is, p is true, q is false and only the aspect p affects payoff. As far as the individual is concerned, the states ((p, q) , {p}) and ((p, ¬q) , {p}) are essentially the same, since even though the aspect q obtains different value in the two states, that aspect of the problem does not affect the individual’s well-being8. Note that, the two (HMS) states (p, q) and (p, ¬q) have the same projection onto the space e are “essentially the same”, if S{p} . Formally, we say that two states (s, α) and (s0 , α0 ) in Ω Sα Sα α = α0 and rSα (s) = rSα (s0 ).

e Call The sets of essentially the same states then form equivalence classes and partition Ω. e Each equivalence class in Λ e can be identified with a state the set of equivalence classes Λ. e → Ω such that for all λ ∈ Λ, Γ (λ) = rSα (s) for some in Ω. Formally, define a function Γ : Λ Sα (s, α) ∈ λ. Since the states in λ are all essentially the same, Γ is well-defined and it is easy to verify that it is a bijection9. 7Heifetz,

Meier, and Schipper (2006, 2007) call this space, Sα , the objective state space. We refrain from e will be the objective state space. using this term for Sα , since, in our formulation, Ω 8Our state space, Ω, e can be constructed from the syntax in the Samet-Aumann approach by introducing propositions p0 and q 0 , which specify that p and, respectively, q, is payoff-relevant. Thus the states in Se{p} are those in which p0 is true and q 0 is false. 9While we need to refer to the unawareness structure in the HMS model when defining Ω, e Λ e and Γ, it is only to identify our model with the HMS one for the purpose of proving behavioral equivalence. The e is “unawareness free” and completely standard. interpretation of our model, in particular, the state space Ω, 13

We can now state our second result: Proposition 7. If (A1) holds, then for every GHM S ∈ GHM S , there exists G0 ∈ G ,such that the equilibria of G0 are the same as the equilibria of GHM S 10. Furthermore, the state space of G0 is the corresponding real state space to the state space of GHM S . Proof. Our proof is by construction. Fix any GHM S =< Ω, (Mi )i∈I , (ui )i∈I >∈ GHM S . We can construct a Bayesian game G0 as follows: (i)

Take the corresponding Bayesian game G we constructed in the proof to e Proposition 6. Let G0 has the corresponding real state space, Ω, same set of players, I, same action space, Mi , for every i ∈ I, as G;

(ii)

Let the utility function of every i ∈ I in G0 be defined as follows: u0i (m, ω e ) = ui (m, ω) for all m ∈ × Mi , ω ∈ Ω, and ω e ∈ Γ−1 (ω) ; i∈I

where ui is i’s utility function in G; (iii)

Define the possibility correspondence of every i, Π0i , as follows: Π0i (e ω ) = Γ−1 (Πi (ω)) for all ω ∈ Ω and ω e ∈ Γ−1 (ω) , where Πi is player i’s possibility correspondence in G; and,

(iv) (a)

Let the prior belief of every i, ρ0i , satisfy the following conditions:
ρ0i Γ−1 (ω) = ρi (ω) for all ω ∈ Ω, where ρi is i’s prior in G (Recall that, Ω is assumed to be finite, and FΩ = 2Ω ); and,

(b)

ρ0i (E) > 0 for all E ∈ FΩe = 2Ω . e

Step (iii) above ensures that all states that are essentially the same give rise to the same type for each player. Consequently, the normal forms of G0 and G (and hence GHM S ) have the same (isomorphic) sets of players. Step (ii) to (iv) imply that the players in the normal 10This

is, again, up to renaming the players. 14

form of G0 have utility functions that are the same, in the same sense as in the proof to Proposition 6, to the utility functions of the players in the normal form of GHM S . It then follows that every equilibrium to the normal form of GHM S will be an equilibrium to the normal form of G0 and vice versa.



Thus we have shown that the behavioral predictions of a HMS model can be replicated by a standard model with properly defined state space and belief structures. The model is truly standard, as it admits a standard interpretation. As discussed in Section 3, this is achieved essentially by allowing players to hold false beliefs and no other assumptions made in the standard rational model needs to be relaxed. As a consequence of holding false beliefs, the beliefs of the players in our model assign zero probability to events that are in fact true. The debate on whether HMS unawareness can be modeled by zero probability beliefs is not new (see Schipper (2011) and Li (2008b)). In the HMS model, when individuals are unaware of certain events, they simply do (can) not take them into consideration. Therefore, they may be expected to behave identically to an individual who is aware of such events yet does not believe they will obtain. Thus, one may be tempted to conclude that the (choice) behavior of an individual unaware of certain events will be identical to that of someone whose belief assigns zero probability to such events. This reasoning is, however, flawed. As pointed out by Schipper (2011), when an individual is unaware of an event, she must also be unaware of the negation of the event. On the other hand, in the standard models, if an individual’s belief assigns zero probability to an event, she must believe that the negation of the event will obtain with certainty. Thus, individuals can be expected to have different preferences over lotteries in these situations. Our approach is immune from the above critique. Since, in our model, the individuals do not simply assign zero probability to events of which her HMS counterpart is unaware. Rather, she believes that the aspect of the problem, of which her counterpart is unaware, is (payoff) irrelevant. The intuition of our result can be better illustrated by considering the simple syntactic example. An individual in the HMS model with awareness level S{p} , that is, she is aware of p 15

but unaware of q, is simply an expected utility maximizer with a subjective state space S{p} . This individual corresponds to an individual in our model whose belief assigns probability one to the event Se{p} , where only the aspect p is payoff relevant. This is indeed true from her perspective, since her payoff function and her opponents’ behavior do not depend on the specification of the aspect q. In other words, her actions amount to no more or less than betting against the realization of the aspect p. It is no surprise then, with the same marginal beliefs over the events “p is true” and “p is false” and facing the same choices and (utility) consequences, the two individuals should behave identically. An expected utility maximizer undertakes the same calculations when she is unaware of certain aspects of the problem and, consequently, fail to take them into consideration as when she chooses to ignore these aspects since she considers them irrelevant. Consequently, she should behave identically in both situations. We have two final comments on the standard model constructed above. First, the description “q is payoff irrelevant” is merely a choice of abstraction. Concrete circumstances resulting in such irrelevance should be found in specific application. An example of such circumstances is given at the beginning of this section. Second, we constructed the standard model with the purpose of generating the same behavior as the HMS model. However, the framework allows for more flexibility in specifying beliefs of the individuals. For example, while it would not be reasonable to let an individual to have belief in ∆Sα in a state in Sβ with Sα strictly more expressive than Sβ , it is legitimate for an individual to have subjective belief in ∆Seα in a state in Seβ . That is, the HMS model also imposes a number of restrictions on the players’ belief structures that are not captured in our standard model. To fully appreciate the implications of the HMS model, one should also examine the implications of imposing these restrictions. This is, however, beyond the scope of this paper.

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5. Conclusion In this paper, we show that the behavioral predictions of a HMS unawareness model can be replicated by a standard model. This is achieved essentially by allowing players to hold false beliefs and no other assumptions made in the standard rational model needs to be relaxed. The reason for this behavioral equivalence is that, in the HMS model, an individual with awareness level Sα is simply an standard expected utility maximizer with subjective state space Sα , who should behave identically to someone who is aware of all aspects of the problem but believes that she only needs to consider a subset of them, specifically, those describable in Sα , since the rest are believed to be irrelevant. As stated in the introduction, we do not claim that standard models are sufficient to model unawareness. Our results only apply to the HMS formulation of unawareness and other models in which unawareness is reduced to expected utility maximization with a similarly restricted state space. It should be pointed out that, a key feature of unawareness that motivated much of the thinking in the literature is not captured in the HMS model. While it is not reasonable for an individual to be aware of exactly which event she is unaware of, it is plausible for an individual to be aware of the fact that there may be “some” event of which she is unaware. This kind of awareness of unawareness is not captured in the HMS model. Indeed, to model this, one may have to go beyond the Samet (1990) and Aumann (1999a,b) constructions of the state space, since they do not afford us the ability to formalize notions such as “there exists some propositions (events) that...”. Moreover, the HMS model does not help us understand how, in a dynamic setting, an individual will behave when she becomes aware of something. We are not aware of any model that successfully captures these aspects of unawareness11. Perhaps future research on these features will also afford us better understanding of the behavioral consequences of unawareness in contrast to those of uncertainty.

11Li

(2008b) provides some insight into the dynamic problem. 17

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