Vagueness of Language: Indeterminacy under Two-Dimensional State-Uncertainty

Saori Chiba

doi:10.1515/bejte-2018-0093

Article

Vagueness of Language: Indeterminacy under Two-Dimensional State-Uncertainty

Saori Chiba

Published/Copyright: October 22, 2019

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From the journal The B.E. Journal of Theoretical Economics Volume 20 Issue 1

Abstract

We study indeterminacy of indicative meanings (disagreements about meanings of messages among players), a kind of language vagueness examined in Blume and Board (2013. “Language Barriers.” Econometrica 81 (2): 781–812). They, using a cheap talk model in which the state-distribution and the players’ language competence were ex-ante uncertain, demonstrated that this vagueness occurs as an equilibrium language. We expand the work of Blume and Board by using a model between an uninformed decision maker and an informed agent in which the state-distribution and the state are both ex-ante uncertain. We show that this two-dimensional uncertainty also leads to indeterminacy of indicative meanings, that is, to a set of conditions in which an agent with different perceptions of the state-distribution intentionally uses the same symbol for the different extents of information on the state. The vagueness, contrary to common expectations, can actually lead to welfare improvement.

Keywords: information; language; state-uncertainty; vagueness

JEL Classification: D82; D83; M14

Acknowledgements

I benefited from suggestions made by Sambuddha Ghosh, Kohei Kawamura, Ming Li, Marco LiCalzi, Barton Lipman, Tadashi Sekiguchi, Nora Wegner, and two anonymous reviewers as well as seminar participants at XXIX Jornadas de Economía Industrial, XXXIX Simposio de la Asociación Española de Economía, Hitotsubashi University, and Tsukuba University. I appreciate the editorial support from Jose de Jesus Herrera Velasquez and Rahel O’More. I am grateful to Dipartimento di Management, Università Ca’ Foscari Venezia for grants (project SSD SECS-S/06, 536/2013 and 569/2014) and Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (C) (no. 16K03549). This paper is a substantially revised version of an earlier paper entitled “Extensions and Vagueness of Language under Two-Dimensional State Uncertainty” (Chiba 2014). All remaining errors are my own.

Appendix

A.1 Proofs for Section 3

A.1.1 Definitions

Define a^l for l∈{R,S}:

aS(θ;b):=arg⁡maxa∈RUS(a,θ,b) for ∀θ, ∀b

a^R is characterized similarly. From assumptions, a^l is well defined.

Define Ul¯ for l∈{R,S}:

US¯(a,s′,b;i)=E[US(a,θ,b)|s=s′,t=i]=(p+(1−p)qi)US(a,Fi−(s′),b)+∑j∈{1,2,...,n}∖{i}(1−p)qjUS(a,Fj−(s′),b).

UR¯ is expressed similarly. Ul¯ is a convex combination of U^l and hence preserves the following properties: Ul¯ is continuously twice differentiable; UR¯(a,s;i)=US¯(a,s,0;i) for ∀a, ∀s; U11S¯<0<U12S¯ for ∀a, ∀s; and 0<U13S¯ for ∀a, ∀b.

Thus, there also exists a well-defined A^l for l∈{R,S}:

AS(s,i,b):=arg⁡maxa∈RUS¯(a,s,b;i)

Then, it follows from 0<U13S¯ that AS(s,i,b)=AR(s,i) if b = 0 for ∀s,∀i and AS(s,i,b)−AR(s,i) increases with b for any b and θ. A^l is bounded below and above by Al(0,i) and Al(1,i), respectively, because the support ∪j∈{1,2,...,n}Tj is bounded.

A.1.2 Proof for Lemma 1

Under one-dimensional state-uncertainty, there are n subgames, one game per type of S. Lemma 1 and Theorem 1 in CS directly applies to each subgame since S perfectly knows that θ and the payoff functions satisfy the required conditions. Thus, each type uses finite messages and induces finite actions. Last, even if different types commonly use some message, R infers differently for different types. Thus, the claim holds.

A.1.3 Proof for Lemma 2

It follows from Lemma 1 and Theorem 1 in CS. Thus, if the set of actions induced in equilibrium is finite, the equilibrium should have a partitional form because U12S¯>0. Thus, it suffices to show that if multiple actions are induced by one type, for every two distinct actions a and a' induced by one type, there is ε > 0 such that |a′−a|≥ϵ.

Consider two cases: first, no message is commonly used by multiple types; second, at least one message is used by at least two types.

In the first case, the analysis is equivalent to the proof of Lemma 1 because a message reveals the type and payoff functions Ul¯ satisfy the required conditions for Lemma 1 and Theorem 1.

In the second case, suppose a is induced by type 1 and type 2. Suppose a' > a is also induced by both types. Let a be induced by type 1 given s₁ and type 2 given s₂, respectively. Let a' be induced by type 1 given s1′ and type 2 given s2′ , respectively. Hence, by weakly revealed preferences and continuity there exists a unique si¯ such that US¯(a,si¯,b;i)=US¯(a′,si¯,b;i). Thus:

a<AS(si¯,i,b)<a′for i∈{1,2}.

Type k, where k∈{1,2}, does not induce a for any s>si¯ nor a' for any s>si¯. This and U12R¯>0 imply that:

a≤γAR(s1¯,1)+(1−γ)AR(s2¯,2)≤a′ for some γ∈(0,1).

Thus, there is an ε > 0 such that a' – a ≥ ε.

Last, suppose a is induced by type 1 and type 2. Suppose a' > a is induced only by type 1. For type 1, there exists a unique s1¯ such that US¯(a,s1¯,b;1)=US¯(a′,s1¯,b;1), but for type 2, US¯(a,s,b;2)>US¯(a′,s,b;2) holds for any s. Thus:

a<AS(s1¯,1,b)<a′,a<AS(1,2,b)<a′.

Both types induce a, but type 1 does not induce it for any s>si¯. Only type 1 induces a' and he does not induce it for s<s1¯:

AR(s1¯,1)≤a′,a≤γAR(s1¯,1)+(1−γ)AR(1,2)for some γ∈(0,1).

Thus, there is an ε > 0 such that a' – a ≥ ε.

A.1.4 Proof for Proposition 1

Fix b∈(0,∞) and p∈[0,1]. Consider any equilibrium under two-dimensional state-uncertainty (i.e. t is S’s private information).

Consider a set of types N¯ such that ∩i∈N¯⁡Ti has a positive measure and condition (1) (i.e. Fi(θ)≠Fj(θ) almost everywhere over Ti∩Tj) holds for every pair of i,j∈N¯, where i≠j.

Let Fi− represent an inverse function of F_i supported on [0,1].

The next claim suffices to show that all types belonging to N¯ should commonly use at least one message.

Claim 1. Take θ′∈∩i∈N¯⁡Ti such that Fi(θ′)≠Fj(θ′) for every pair of i,j∈N¯, where i≠j. Let si=Fi(θ′). Then, for any i∈N¯, an action a ' induced by type i given s = s_i (and hence in its neighborhood) should be induced by type j given some s′∈[min{sj,si},max{sj,si}] (and hence in its neighborhood) for every j∈N¯.

This claim implies that all types belonging to N¯ should induce at least one same action. Hence, all types belonging to N¯ should commonly use at least one message.

Proof.

Without loss of generality, consider 1,2∈N¯ and s1>s2. Then:

F1−(s2)<F1−(s1)=θ′=F2−(s2)<F2−(s1)⇔aS(F1−(s2))<aS(θ′)<aS(F2−(s1)).

Besides, type 1 (i.e. S observing t = 1) infers:

F={F1withprobabilityp+(1−p)q1F2withprobability(1−p)q2,Fjwherej∈N∖{1,2}otherwise

and type 2 (i.e. S observing t = 2) infers:

F={F1withprobability(1−p)q1F2withprobabilityp+(1−p)q2.Fjwherej∈N∖{1,2}otherwise

Hence:

US¯(a,s1,b;1)=(p+(1−p)q1)US(a,θ′,b)+(1−p)q2US(a,F2−(s1),b)+∑∀j∈N∖{1,2}(1−p)qjUS(a,Fj−(s1),b),US¯(a,s1,b;2)=(1−p)q1US(a,θ′,b)+(p+(1−p)q2)US(a,F2−(s1),b)+∑∀j∈N∖{1,2}(1−p)qjUS(a,Fj−(s1),b),US¯(a,s2,b;1)=(p+(1−p)q1)US(a,F1−(s2),b)+(1−p)q2US(a,θ′,b)+∑∀j∈N∖{1,2}(1−p)qjUS(a,Fj−(s2),b),US¯(a,s2,b;2)=(1−p)q1US(a,F1−(s2),b)+(p+(1−p)q2)US(a,θ′,b)+∑∀j∈N∖{1,2}(1−p)qjUS(a,Fj−(s2),b).

Then,

AS(s2,b;1)<AS(s2,b;2)<AS(s1,b;1)<AS(s1,b;2)

due to U11S<0, Fj−(s2)<Fj−(s1), F1−(s2)<θ′<F2−(s1) because:

U1S¯(a,s2,b;1)>US¯1(a,s2,b;2)>US¯1(a,s1,b;1)>US¯1(a,s1,b;2)for any a.

Also, for any a > a'

US¯(a,s2,b;1)−US¯(a′,s2,b;1)<US¯(a,s2,b;2)−US¯(a′,s2,b;2)<US¯(a,s1,b;1)−US¯(a′,s1,b;1)<US¯(a,s1,b;2)−US¯(a′,s1,b;2)

due to U12S>0, Fj−(s2)<Fj−(s1), F1−(s2)<θ′<F2−(s1).

Take an action a' induced by type 1 given s = s₁ (and hence in its neighborhood). Then, a' should be induced by type 2 given some s′∈[s2,s1] (and hence in its neighborhood) as follows.

If a′∈(AS(s2,b;2),AS(s1,b;2)), by continuity, there must be s′∈(s2,s1) such that AS(s′,b;2)=a′. The action a' should be induced by type 2 given s in the neighborhood of s'.

Consider a′≤AS(s2,b;2). If some action a~>a′ is inducible on the equilibrium path, due to the revealed preference:

0≥US¯(a~,s1,b;1)−US¯(a′,s1,b;1),

and

US¯(a~,s1,b;1)−US¯(a′,s1,b;1)>US¯(a~,s2,b;2)−US¯(a′,s2,b;2).

In this case, because type 1 given s₁ weakly prefers a' to a~, type 2 given s = s₂ strictly prefers a' to a~. If a' is the highest inducible action, type 2 given s = s₂ strictly prefers a' to any action a < a' because US¯(a,s2,b;2) is increasing with a for a < a'. The action a' should be induced by type 2 given s in the neighborhood of s₂.

Consider a′≥AS(s1,b;2). If some action a~<a′ is inducible on the equilibrium path, due to the revealed preference:

0≤US¯(a′,s1,b;2)−US¯(a~,s1,b;2),

and:

US¯(a′,s1,b;1)−US¯(a~,s1,b;1)<US¯(a′,s1,b;2)−US¯(a~,s1,b;2).

In this case, because type 1 given s₁ weakly prefers a' to a~, type 2 given s = s₁ strictly prefers a' to a~. If a' is the lowest inducible action, type 2 given s = s₁ strictly prefers a' to any action a > a' because US¯(a,s1,b;2) is decreasing with a for a > a'. The action a' should be induced by type 2 given s in the neighborhood of s₁.

Similarly, an action induced by type 2 given s = s₂ is also induced by type 1. (Take a' induced by type 2 given s = s₂. If a′∈(AS(s2,b;1),AS(s1,b;1)), a' should be induced by type 1 given s in the neighborhood of s' where AS(s′,b;1)=a′. If a′≤AS(s2,b;1), a' should be induced by type 1 given s in the neighborhood of s₂. If a′≥AS(s1,b;1), a' should be induced by type 1 given s in the neighborhood of s₁).

The above proof applies to any types i,j∈N¯. ■

The next claim suffices to show that if p∈(0,1), all types belonging to N¯ should commonly use at least one message for different extensions.

Claim 2. Fix p∈(0,1). When two types commonly use some message and at least one type use multiple messages, then, the two types use the common message for different extensions.

Proof.

If three actions a < a' < a" are induced by type 1, and a and a" are induced by type 2 as well, then, a' should be induced by type 2.

It suffices to consider two cases: first, two actions are induced by type 1 and type 2; second, only one action is induced by both types.

In the first case, let a and a' be induced by type 1 and type 2, where a < a' . Type 1 and type 2 are indifferent between the two actions at s = x and s = y, respectively. Let m be used to induce a. The underlying states (given which each sends m) include the neighborhoods of θ=F1−(x) and θ=F2−(x) for type 1 and the neighborhoods of θ=F1−(y) and θ=F2−(y) for type 2. Thus, it suffices to show x≠y. This inequality holds because S’s indifference conditions require:

US¯(a,x,b;1)=US¯(a′,x,b;1)⇔p+(1−p)q1(1−p)q2=US(a′,F2−(x),b)−US(a,F2−(x),b)US(a,F1−(x),b)−US(a′,F1−(x),b)

and:

US¯(a,y,b;2)=US¯(a′,y,b;2)⇔(1−p)q1p+(1−p)q2=US(a′,F2−(y),b)−US(a,F2−(y),b)US(a,F1−(y),b)−US(a′,F1−(y),b),

but p+(1−p)q1(1−p)q2≠(1−p)q1p+(1−p)q2.

In the second case, suppose type 1 induces a for s∈(x,x′) and a' for s∈(x′,x"), where a < a'; type 2 induces a for s∈(y,y′) and a" for s∈(y′,y"), where a < a"≠ a'. Single crossing properties imply x′≠y′ . Suppose not, x′=y′. Then, min{a′,a"}>max{AS(x′,1,b),AS(x′,2,b)}. US¯(a,x′,b;1) and US¯(a′,y,b;2) both decrease with a over a≥max{AS(x′,1,b),AS(x′,2,b)}, and type 1’s choice implies that a' is the highest inducible action over a≥max{AS(x′,1,b),AS(x′,2,b)}, i.e. a' ≥ a". Type 2’s choice implies a' ≤ a". There is a contradiction. ■

If p = 1 or p = 0, x = y should hold. If p = 1, multiple types belonging to N¯ can commonly use a message (messages) for the same extension. If p = 0, the sender does not have type information.

A.2 Section 4 (the Uniform-Quadratic Case)

We present results to support the arguments in Section 4, including Remark 1.

A.2.1 One-Dimensional State-Uncertainty

In A.2.1, we consider a game under one-dimensional state-uncertainty (i.e. p = 1 and type signal t is public information).

Lemma.

Fix b∈(0,∞). Consider a partition equilibrium with N₁–N₂intervals. In a subgame given type t, where t∈{1,2}:

(1) Type t’s partition satisfies Δθi=Δθi−1+4b for i∈{2,...,Nt}.

(2) Type t sends m = m_igiven θ∈(θi−1,θi) so that mi≠mi′ for any i′≠i.

(3) R selects a = a_igiven m = m_isuch that ai=θi−1+θi2 for i∈{1,...,Nt}.

Proof.

For each subgame, the outcome is defined by S’s indifference conditions (S’s ICs) and R’s best responses (R’s BRs).

At any boundary point θ = θ_i for i∈{1,...,Ni−1}, any type of S should be indifferent between two actions, a = a_i and a = a_i + 1:

−(θi+b−ai)2=−(ai+1−θi−b)2⇔ai+1+ai=2θi+2b(S's ICs)

Given m = m_i for i∈{1,...,Nt}, R updates his belief on the state using Bayes’ rule and selects the optimal action a=a_i:

ai=E[θ|θ∈(θi−1,θi)]=θi−1+θi2.(R's BRs)

Hence, regardless of S’s type, S’s ICs and R’s BRs define the partition such that Δθi=Δθi−1+4b for i∈{2,...,Nt}. ■

Lemma 4.

Fix b∈(0,∞). Consider a subgame given type t, where t∈{1,2}. Then, there is a positive integer NtCS(b) such that there is an equilibrium where type t partitions the state space into N intervals for N∈{1,2,...,NtCS(b)}. Further, NtCS(b) decreases with b.

Proof.

The previous lemma implies Δθi=θ1+4(i−1)b for i∈{1,...,N} for any type. Hence, the existence of an equilibrium with N intervals requires θ₁ > 0 and:

θN=∑i=1NΔθi=Nθ1+2N(N−1)b=i.

For any b > 0, NtCS(b) is well defined such that N1CS(b)=n for b∈[bn+1,bn) for n ≥ 1 and N2CS(b)=N for b∈[LbN+1,LbN) for N ≥ 1. ■

A.2.2 Two-Dimensional State-Uncertainty

In A.2.2, we consider a game under two-dimensional state-uncertainty (i.e. type signal t is S’s private information).

Lemma 5.

Fix p∈(0,1) and b∈(0,∞). Any PBE is a partition equilibrium. Further, in a partition equilibrium with N₁–N₂intervals:

There is a unique N₁for N₂, and N1≤N2.
Gb,p(N1)=ΔxN1ΔxN1+ΔyN1((L(1+p)+1−p)(yN1−1+yN1)4−(1+p+L(1−p))(xN1−1+xN1)4)≥0.
Type 1’s partition x(N1) satisfies:

Δ x i = Δ x i − 1 + 8 b 1 + p + L ( 1 − p ) f o r i ∈ { 2 , . . . , N 1 − 1 } . Δ x i ≤ Δ x i − 1 + 8 b 1 + p + L ( 1 − p ) f o r i = N 1 .
Type 2’s partition y(N2) satisfies:

Δ y i = { Δ y i − 1 + 8 b L ( 1 + p ) + 1 − p f o r i ∈ { 2 , . . . , N 2 } ∖ { N 1 , N 1 + 1 } , Δ y i − 1 + 8 b L ( 1 + p ) + 1 − p + 4 L ( 1 + p ) + 1 − p G b , p ( N 1 ) f o r i ∈ { N 1 , N 1 + 1 } .
Relationship between partitions of both types is:
x i = { L ( 1 + p ) + 1 − p 1 + p + L ( 1 − p ) y i f o r i ∈ { 1 , . . . , N 1 − 1 } , 1 ( ≤ L ( 1 + p ) + 1 − p 1 + p + L ( 1 − p ) y N 1 ) f o r i = N 1 .
Both types send m = m_igiven s∈(xi−1,xi) (if this interval exists) and s∈(yi−1,yi), respectively, for i∈{1,...,N2}, where mi≠mj for any j≠i.
R selects a = a_igiven m = m_isuch that:

a i = { ( L ( 1 + p ) + 1 − p ) ( y i − 1 + y i ) 4 \textit{for} i ∈ { 1 , . . . , N 2 } ∖ { N 1 } , ( L ( 1 + p ) + 1 − p ) ( y i − 1 + y i ) 4 − G b , p ( N 1 ) \textit{for} i = N 1 ,

where a₁, ..., aN1 is also described as follows:

a i = { ( 1 + p + L ( 1 − p ) ) ( x i − 1 + x i ) 4 \textit{for} i ∈ { 1 , . . . , N 2 } ∖ { N 1 } , ( 1 + p + L ( 1 − p ) ) ( x i − 1 + x i ) 4 + Δ y N 1 Δ x N 1 G b , p ( N 2 ) \textit{for} i = N 1 .

Proof.

To define a partition, we consider indifference conditions for each type of S (S’s ICs) and R’s best responses (R’s BRs) for the game under two-dimensional state-uncertainty.

Suppose a_i and a_i+1 are induced in the i-th and (i+1)-th intervals, respectively, by type 1. At the boundary point s = x_i, type 1’s ICs require:

ai+1+ai=(1+p+L(1−p))xi+2b.(S's ICs (t = 1))

Similarly, suppose a_i and a_i+1 are induced in the i-th and (i+1)-th intervals, respectively, by type 2. At the boundary point s = y_i, type 2’s ICs require:

ai+1+ai=(L(1+p)+1−p)yi+2b.(S's ICs (t = 2))

Thus, if a_i and a_i + 1 are induced by both types of S, then xi=L(1+p)+1−p1+p+L(1−p)yi. Type 1’s ideal action given signal s = x_i is equivalent to type 2’s ideal action given signal s = y_i for i∈{1,2,...,N1−1} (because type 1’s ideal action given s is a=1+p+L(1−p)2s+b while type 2’ ideal action given s is a=L(1+p)+1−p2s+b).

R’s BRs require that given m = m_i for any i∈{1,2,...,N1} (i. e. both types use m_i):

given m = m_i for i∈{N1+1,...,N2} (i. e. only type 2 use m_i) is

ai=E[θ|mi,t=2],(R's BRs)

where E[θ|mi,t=1]=(1+p+L(1−p))(xi−1+xi)4 and E[θ|mi,t=2]=(L(1+p)+1−p)(yi−1+yi)4. S’s ICs imply E[θ|mi,t=1]=E[θ|mi,t=2] for i∈{1,...,N1−1} while E[θ|mN1,t=1]≤E[θ|mN1,t=2]. Thus, a_i is well defined as claimed. □

Lemma 6.

Fix p∈(0,1) and b∈(0,∞). Consider m_t in a partition equilibrium with N₁–N₂ intervals, where t∈{1,2,...,N1−1}. Information provided by type 2 second order stochastically dominates (SOSD) information provided by type 1.

Proof.

Type 1 uses m_i given s∈(xi−1,xi) while type 2 uses m_i given signal s∈(yi−1,yi). The two conditional distributions have the same mean

(L(1+p)+1−p)(yi−1+yi)4=(1+p+L(1−p))(xi−1+xi)4,

but different supports; and the former has larger variances than the latter as follows.

Table 1:

Type	Support	Variances
t=1	( x i − 1 , x i ) ∪ ( L x i − 1 , L x i )	( 1 + p + L 2 ( 1 − p ) ) ( x i − 1 2 + x i − 1 x i + x i 2 ) 6 − ( 1 + p + L ( 1 − p ) ) ( x i − 1 + x i ) 2 4 2
t=2	( y i − 1 , y i ) ∪ ( L y i − 1 , L y i )	( L 2 ( 1 + p ) + 1 − p ) ( y i − 1 2 + y i − 1 y i + y i 2 ) 6 − ( L ( 1 + p ) + 1 − p ) 2 ( y i − 1 + y i ) 2 4 2

This table compares two conditional distributions: one for information which type 1 transmits when sending message mi (see the second row) and another for information which type 2 transmits when sending mi (see the third row).

□

A.3 Proofs for Section 5.1 (Discrete Model)

A.3.1 Proof for Remark 2

We find sufficient conditions which satisfy the next three claims.

Claim 1. Under two-dimensional uncertainty, there exists an equilibrium in which every type uses different messages for different signals, as shown in Figure 3.

Claim 2. Under two-dimensional uncertainty, there is no equilibrium in which different types use different messages for s = 1. (Language is vague).

Claim 3. Under one-dimensional uncertainty, there only exists a babbling equilibrium given type 1 while there is a separation equilibrium given type 2, as shown in Figure 4.

Claim 1 holds if b<1+L4 as follows. Under two-dimensional uncertainty, in the supposed equilibrium, each type induces a = 0 given s = 0 and a=1+L2 given s = 1. Thus, this equilibrium exists if both types prefer a = 0 to a = 1 given s = 0.

Claim 2 holds if b>p(L−1)2 as follows. Under two-dimensional uncertainty, if different types use different messages for s = 1, type 1 induces a = 0 given s = 0 and a=1+p+(1−p)L2 given s = 1 while type 2 induces a = 0 given s = 0 and a=(1+p)L+1−p2 given s = 1. However, if b>p(L−1)2, type 1’s most preferred action is not a=1+p+(1−p)L2 given s = 1 among all inducible actions.

Claim 3 holds if b∈(12,L2] as follows. Under one-dimensional uncertainty, each type of S knows the state (because s = θ) and induces a = θ by sending m = s in a separating equilibrium if it exists. We can show that a separating equilibrium exists for a type if this type wants to induce low action a = 0 for θ = 0. The condition is b≤12 for type 1 and b≤L2 for type 2.

Therefore, if 1 < L < 2, b∈(12,1+L4) suffices to satisfy the three claims.

Then, fix b∈(12,1+L4). We look at the most informative equilibrium. Under two-dimensional state-uncertainty, both types send m = Null given s = 0 and m = Large given s = 1, as shown in Figure 3. R’s ex-ante expected payoff is:

12(−(1−L)28)⏟F=F1+12(−(1−L)28)⏟F=F2=−(1−L)28,

where:

{12(−(0−0)2)⏟θ=0+12(−(1−1+L2)2)⏟θ=1=−(1−L)28if F=F1,12(−(0−0)2)⏟θ=0+12(−(L−1+L2)2)⏟θ=L=−(1−L)28if F=F2.

Under one-dimensional uncertainty, we consider a babbling equilibrium given type 1 and a separation equilibrium given type 2, as shown in Figure 4. R’s ex-ante expected payoff is:

12(−14)⏟F=F1+12(0)⏟F=F2=−18,

where:

{12(−(0−12)2)⏟θ=0+12(−(1−12)2)⏟θ=1=−14if F=F1,12(−(0−0)2)⏟θ=0+12(−(L−L)2)⏟θ=L=0if F=F2.

Hence, R is better-off under two-dimensional state-uncertainty.

Fix b<12. Under one-dimensional state-uncertainty, there is a separation equilibrium given each type. R’s ex-ante expected payoff is:

12(0)⏟F=F1+12(0)⏟F=F2=0.

Hence, R is worse-off under two-dimensional state-uncertainty.

Fix b>1+L4. Under two-dimensional state-uncertainty, there only exists a babbling equilibrium in which only a=1+L4 is induced by both types. R’s ex-ante expected payoff is:

12(−L2−2L+516)⏟F=F1+12(−5L2−2L+116)⏟F=F2=−3L2−2L+38(<−L2−2L+316).

Under one-dimensional state-uncertainty, there only exists a babbling equilibrium in which only a=12 (a=L2) is induced given type 1 (type 2). R’s ex-ante expected payoff is:

12(−14)⏟F=F1+12(−(1−L)28)⏟F=F2=−L2−2L+316.

Hence, R is worse-off under two-dimensional state-uncertainty.

A.3.2 Proof for Remark 3

Let a₀ = 0, a1=1+p+(1−p)L2 and a2=(1+p)L+1−p2 (i. e. a0<a1<a2). If the type is revealed in the first stage, type 1 can induce a₀ and a₁ while type 2 can induce a₀ and a₂, respectively, in the second stage. But type 1 knows he will prefer a₂ to a₁ given s = 1. Hence, type 1 pretends to be type 2.

A.4 Proofs for Section 5.2 (Different Prior Beliefs)

A.4.1 Proof for Remark 4

The proof and the outcome are similar to those for a model in Section 4. We only show differences in equilibrium outcome from the basic model.

Type 2’s partition y(N2) satisfies:

Δyi={Δyi−1+4(qR−qS)(1−p)2−qR(1−p)yifor i∈{2,...,N2}∖{N1,N1+1},Δyi−1+4(qR−qS)(1−p)2−qR(1−p)yi+2Gb,p(N1)(1+(1−p)(1−qR)+p)for i∈{N1,N1+1}.

The relationship between partitions of both types is:

xi={1+(1−p)(1−qS)+p1+(1−p)(1−qS)yifor i∈{N1,N1+1},1(≤1+(1−p)(1−qS)+p1+(1−p)(1−qS)yN1)for i=N1.

R selects a = a_i given m = m_i such that:

ai={(1+(1−p)(1−qR)+p)(yi−1+yi)2for i∈{1,...,N2}∖{N1},(1+(1−p)(1−qR)+p)(yi−1+yi)2−Gb,p(N1)for i=N1.

Due to the difference in prior beliefs |qR−qS|, any type of S cannot credibly separate messages for signals. Then, as in the basic model, the communication should have a partitional form. The players use a smaller number of messages as |qR−qS| increases. Moreover, in a partition equilibrium with N₁–N₂ intervals, both types of S commonly use messages m1,m2,...,mN1. Type 1’s ideal action given signal s = x_i is equivalent to type 2’s ideal action given s = y_i, which is a=(1+(1−p)(1−qS)+p)yi + b, for i∈{1,...,N1−1}.

A.4.2 Proof for Remark 5

In a partition equilibrium with N intervals, the partition satisfies:

Δxi=Δxi−1+42−qRb+4(qR−qS)2−qRxifori∈{2,...,N}.

S send m = m_igiven s∈(xi−1,xi) for i∈{1,...,N} so that mi≠mi′ for any i′≠i. Then, R selects a = a_i given m = m_isuch that ai=(2−qR)(xi−1+xi)4 for i∈{1,...,N}.

Therefore, we observe non-monotonic relationships unless qR=qS (common prior belief). For example, by fixing qR=14 and qS=34, a larger |b| can result in a smaller or larger NDP(b,qR,qS):

NDP(b,qR,qS)={4forb∈(33758416,223544),3forb∈(1748,225464)∖(33758416,223544),2forb∈(1116,1516)∖(1748,225464),1otherwise.

For another example, by fixing b=38 and qR=12, a larger |qR−qS| can result in a smaller or larger NDP(b,qR,qS):

NDP(b,qR,qS)={4for qS∈(29−32116,4+(476−121545])1/3+(476+121545])1/316),3for qS∈(78,1)∖(29−32116,4+(476−121545])1/3+(476+121545])1/316),2for b∈(12,78),1for b∈(0,12).

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Published Online: 2019-10-22

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Articles in the same Issue

https://doi.org/10.1515/bejte-2018-0093

Keywords for this article

information; language; state-uncertainty; vagueness