Blackwell-Monotone Information Costs
Xiaoyu Cheng Yonggyun Kim

Adaptive Mixed-Scale Feature Fusion Network for Blind AI-Generated Image Quality Assessment
    Tianwei Zhou, Songbai Tan, Wei Zhou, Yu Luo, Yuan-Gen Wang, and Guanghui Yue
    April 29, 2024
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A Blackwell-monotone information cost function assigns higher costs to Blackwell more informative experiments. This paper provides simple necessary and sufficient conditions for Blackwell monotonicity over finite experiments. 
The key condition is a system of linear differential inequalities that are convenient to check given an arbitrary cost function. When the cost function is additively separable across signals, our characterization implies that Blackwell monotonicity is equivalent to sublinearity. This identifies a wide range of practical information cost functions. 
Finally, we apply our results to bargaining and persuasion problems with costly information.


JEL Classification:  C78, D81, D82, D83
	
Keywords:  Blackwell Information Order, Costly Information, Additively Separable Cost, Bargaining, Costly Persuasion
	




§ INTRODUCTION


A central agenda in contemporary economic theory is the integration of costly information across various fields. This raises the question of which information cost function should or could be used. One common principle, shared by almost all recent developments of information cost functions,[ These include entropy costs <cit.>, posterior separable costs <cit.>, and log-likelihood ratio (LLR) costs <cit.>, among many others.] is Blackwell monotonicity: a statistical experiment is more costly than another if it is more informative according to the classical information order by <cit.>. 
While Blackwell monotonicity is recognized as a desirable feature and has been extensively studied, it is usually analyzed in combination with other properties. 
Therefore, the implications and requirements of solely imposing Blackwell monotonicity on information costs remain underexplored. 

In this paper, we characterize simple necessary and sufficient conditions of Blackwell monotonicity for information cost functions defined over finite experiments.[A finite experiment stands for a map from finite states to probability distributions over finite signals. ] 
This characterization provides a tractable method for verifying Blackwell monotonicity given an arbitrary information cost function. Additionally, it enables us to construct a novel and broad class of Blackwell-monotone information cost functions that are additively separable across signals. Notably, this class encompasses well-known existing information costs such as all cost functions mentioned in footnote <ref>.

Our characterization is motivated by the garbling interpretation of the Blackwell order. <cit.> shows that for any Bayesian decision problem, the expected payoff under a statistical experiment f is no less than that under another experiment g if and only if g can be generated by adding some noise to f. That is, g is a garbling of f. If this condition holds, f is called to be Blackwell more informative than g. 

Consider the Euclidean subspace of statistical experiments with n states and m signals, and a Blackwell-monotone information cost function C defined over this space. An experiment f in this space can be represented as an n × m matrix, where each entry f_ij denotes the probability of generating signal s_j when the state is ω_i. 
Now, imagine the following garbling process: whenever signal s_j is generated, s_j is replaced by another signal s_k with probability ϵ, while the generation of other signals remains unchanged. Since this process worsens Blackwell informativeness, any Blackwell-monotone cost function should result in a lower cost after its application. By sending ϵ to zero, we can derive a local necessary condition for Blackwell monotonicity when the following limit exists: 

    D^+C(f; f^j(k)) ≡lim_ϵ↓ 0C(f + ϵ· f^j(k) ) - C(f) ϵ≤ 0 ,   ∀ j ≠ k, ∗

where f^j is the j-th column vector of f and f^j(k) = [0, ⋯, -f^j, ⋯ , f^j , ⋯ ,0] denotes an n × m matrix where j-th column is -f^j, k-th column is f^j, and all other entries are zero. Notice f^j(k) represents the direction of the aforementioned garbling. When C is differentiable at f, (<ref>) reduces to simple first-order inequalities: 

    ∑_i=1^n( ∂ C∂ f_ik - ∂ C∂ f_ij) f_ij≤ 0,   ∀ j ≠ k.


Our main results establish that this local condition is not only necessary but also sufficient for Blackwell monotonicity when combined with additional technical conditions.
When the signal is binary, under Lipschitz continuity, we show that an information cost function is Blackwell monotone if and only if it satisfies (<ref>) and permutation invariance, i.e., relabeling signals results in the same cost (Theorem <ref>). When there are more than two signals, we additionally impose quasiconvexity and show that the local condition and permutation invariance serve as necessary and sufficient conditions for Blackwell monotonicity (Theorem <ref>).[Quasiconvexity is imposed to address a technical issue that arises when extending to the case of more than two signals (see Proposition <ref> for details).]

The local condition (<ref>) becomes especially useful when the cost function is additively separable across signals, i.e., when there exists ψ: [0,1]^n→_+ such that C(f) = ∑_j=1^mψ(f^j). Using our characterization result, we show that an additively separable cost function is Blackwell monotone if and only if its component function ψ is sublinear, i.e., positively homogeneous and subadditive (Theorem <ref>). This result enables the construction of Blackwell-monotone information cost functions by simply choosing a sublinear function ψ. As a demonstration, we identify novel Blackwell-monotone information cost functions with simple functional forms, the norm costs and absolute-linear costs.

Equipped with these new tools, we revisit two economic applications involving costly information. In these applications, assumptions other than Blackwell monotonicity are often imposed on the information cost. Our characterizations allow us to study these problems in a more general framework without the necessity of these additional assumptions and thus help to strengthen the existing insights.

We first consider <cit.>'s ultimatum bargaining model where the buyer can acquire costly information about the unknown value of the seller's object before accepting an offer. To model costly information, they exogenously restrict the buyer's feasible set of experiments and define an information cost function over this restricted set. Using our characterization, we are able to extend their cost function to a Blackwell-monotone cost function over all experiments. This allows us to examine their results in a more general setting where the buyer's ability to acquire information is unrestricted. We show that while the exogenous restriction is crucial for their main result, its main insight remains true in the general setting when considering a different Blackwell-monotone information cost function (Proposition <ref>, <ref>). 

As another application, we consider the costly persuasion problem proposed by <cit.>. They extended their celebrated Bayesian persuasion model <cit.> by assuming that it is costly for the sender to commit to a persuasion policy, which is in the form of statistical experiments. 
To apply the concavification technique, they focus on cases where the information cost function is posterior separable, and the literature follows this tradition. 
We propose another way of solving the costly persuasion problem (without using concavification) that can be applicable to any Blackwell-monotone information costs. By using this, we provide a solution for the standard prosecutor-judge example with a specific non-posterior separable cost.



 §.§ Related Literature

The classical information order by <cit.> has recently regained prominence in light of the rapid expansion of information design and costly information acquisition literature.[For example, <cit.> study Blackwell dominance in large samples, and  <cit.> investigate Blackwell's theorem with infinite states. ] Therefore, it has brought more attention to understand how Blackwell's information order can be integrated into the cost of information. 
When defining information costs, there are mainly two approaches: posterior-based costs (defined over distributions of posteriors); and experiment-based costs (defined over statistical experiments). We refer to the introduction of <cit.> for a formal discussion. 

Ever since <cit.> introduced the entropy cost to the economic literature, posterior-based information costs have gained widespread recognition. Blackwell monotonicity of such costs (with a concave measure of uncertainty, e.g., entropy) is implied by one of Blackwell's sufficient condition.[See, for example, the discussion of Assumption 1 in <cit.>.] 
These costs have been applied to diverse problems including bargaining <cit.> and dynamic information acquisition <cit.>, among many others.
However, some recent papers found that their results derived from posterior-based costs can qualitatively change when experiment-based costs are employed instead <cit.>. 
Furthermore, as <cit.> and <cit.> point out, experiment-based costs may be more compatible with applications where priors are endogenously determined. 

These observations highlight the importance of determining which experiment-based costs ought to be utilized, with Blackwell monotonicity in consideration. As a pioneering work in this direction, <cit.> characterize the log-likelihood ratio (LLR) cost with Blackwell monotonicity and several other axioms. We contribute to the literature on experiment-based costs by deriving necessary and sufficient conditions for Blackwell monotonicity alone.[Additionally, the LLR cost is not defined over experiments that fully reveal some state as it would incur an infinite cost. Our characterization does not have this limitation, and it turns out that such boundary experiments play important roles in our applications.] 

Finally, there is a strand of literature in decision theory that focuses on the rationalization of revealed choice data with information costs. There, Blackwell monotonicity often appears as a central property in the preference representations. See, for example, <cit.>; and <cit.>. Our paper differs methodologically from these papers as our primitives are experiments instead of choice data. 



 §.§ Outline

Section <ref> introduces the formal framework. Section <ref> provides the main characterization results of Blackwell-monotone information costs. Section <ref> applies the characterization to study the class of additively separable costs. Section <ref> presents two applications. Section <ref> provides additional discussions and Section <ref> concludes. All omitted proofs can be found in the Appendix.



§ PRELIMINARIES

Let Ω = {ω_1, ⋯, ω_n} be a finite set of states. Fix a finite set of signals 𝒮 = {s_1, ⋯, s_m}, a statistical experiment f: Ω→Δ(𝒮) is represented by the n × m matrix 

    f = [ f_11    ⋯ f_1m;    ⋮    ⋱    ⋮; f_n1    ⋯ f_nm ],

where f_ij = f(s_j|ω_i) is the probability of generating signal s_j in state ω_i. Let f^j = [ f_1j, ⋯, f_nj]^⊺∈_+^n denote the j-th column vector of f for j = 1, ⋯ m. Using this notation, we can rewrite 

    f = [f^1, ⋯, f^m] ∈_+^n × m,

and sometimes write f_ij as f^j_i. Notice that ∑_j=1^m f^j = 1 where 1 = [1,⋯,1]^⊺. Let ℰ_m⊂^n × m denote the space of all experiments with m possible signals, equipped with the Euclidean topology. It is worth noting that, for all m' ≤ m, ℰ_m' can be embedded into ℰ_m. Let ℰ = ⋃_m<∞ℰ_m denote the class of all (finite) experiments.[The terminology of class follows from <cit.>.]



  
Blackwell Informativeness An experiment f is Blackwell more informative than another experiment g, denoted by f ≽_B g, if there exists a stochastic matrix M  (i.e., M_ij≥ 0 and ∑_j M_ij = 1 for all i) such that g = f M. This matrix M is also called a garbling matrix.

When both f and g are in ℰ_m, any potential garbling matrix is an m × m square stochastic matrix. Let ℳ_m denote the set of all such stochastic matrices. Notice ℳ_m is a convex subset of _+^m × m and its extreme points are given by the matrices with exactly one non-zero entry in each row (see e.g., <cit.>). Let ext(ℳ_m) = {E_1, ⋯, E_m^m} denote the set of all extreme points of ℳ_m. For any k ≤ m, let ext_k(ℳ_m) denote those extreme-point matrices with rank k. 

A permutation matrix P is a stochastic matrix with exactly one non-zero entry in each row and each column. Let 𝒫_m be the set of all m × m permutation matrices and observe that ext_m(ℳ_m) = 𝒫_m. Since when P ∈𝒫, P^-1 is also a permutation matrix, we have f ≽_B fP ≽_B fPP^-1 = f. Therefore, f and fP are equivalent in Blackwell informativeness, denoted by f ≃_B fP. Intuitively, permuting an experiment involves merely relabeling signals, so it should remain equally informative after permutation.
 


  
Information Costs An information cost function is given by C: ℰ_m→_+, i.e., defined over the space of experiments with m signals. Let 𝒞_m denote the set of all such functions. Under this formalization, each C ∈𝒞_m is a mapping over an Euclidean space which facilitates analysis. All applicable results can be carried over to information cost functions defined over ℰ by considering their restriction to every ℰ_m.

For each C ∈𝒞_m, let D^+C(f;h) denote its (one-sided) directional derivative at f ∈ℰ_m in the direction of h ∈^n × m, if exists. When C is differentiable at f, let ∇ C(f) ∈^n × m denote its gradient and we have

    D^+C(f;h) = ⟨∇ C(f), h⟩.

In addition, it is convenient to define ∇^j C(f) ∈^n as the j-th column vector of ∇ C(f), i.e., ∇^j C(f) = [∂ C(f)/∂ f^j_1, ⋯, ∂ C(f)/∂ f^j_n]^⊺. Thus, we can similarly write

    ∇ C(f) = [∇^1 C(f), ⋯, ∇^m C(f)].

Because for each f ∈ℰ_m, ∑_j=1^m f^j = 1, it is without loss of generality to let ∇^m C(f) = 0 if needed. 

Our results require a few standard functional assumptions: C ∈𝒞_m is Lipschitz if there exists a constant L > 0 such that for all f, g ∈ℰ_m, |C(f) - C(g)| ≤ L ‖ f - g‖.[Local Lipschitz continuity is sufficient for all our results. Since ℰ_m is compact, these two notions are equivalent for C over ℰ_m. However, this distinction becomes important when considering information costs defined over only the interior of ℰ_m, i.e., experiments where f_ij > 0 for all i,j, as in <cit.>. All our results still hold in this case under local Lipschitz continuity, thus including their LLR cost.] It is quasiconvex if for any f, g ∈ℰ_m and λ∈ [0,1], C(λ f + (1-λ)g) ≤max{C(f), C(g)}. 

Finally, say that it is permutation invariant if for any f ∈ℰ_m, C(f) = C(fP) for all P ∈𝒫_m.



§ BLACKWELL-MONOTONE INFORMATION COSTS

In this section, we provide our main characterizations of information cost functions that are consistent with the Blackwell information order. 


    An information cost function C ∈𝒞_m is Blackwell monotone if for all f,g ∈ℰ_m, C(f) ≥ C(g) whenever f ≽_B g. 


For any C ∈𝒞_m, let S_C(f) = {g ∈ℰ_m: C(f) ≥ C(g)} denote its sublevel set at f ∈ℰ_m. In addition, let S_B(f) = {g ∈ℰ_m: f ≽_B g} denote the sublevel set under the Blackwell information order. By definition, Blackwell monotonicity is equivalent to 

    S_C(f) ⊇ S_B(f)  for all  f ∈ℰ_m.

We characterize Blackwell monotonicity relying on a novel geometric characterization of the set S_B(f) and then link it to the set S_C(f). To better illustrate the key idea, we start by characterizing Blackwell monotonicity over ℰ_2, i.e., binary experiments.



 §.§ Blackwell Monotonicity over Binary Experiments

For any f ∈ℰ_2, since f^1 + f^2 = 1, it suffices to represent f by the vector f^1. For simplicity, denote a binary experiment as f = [f_1, ⋯, f_n]^⊺∈ [0,1]^n. Similarly, for any C ∈𝒞_2, we can also let C: [0,1]^n→_+. 


    Suppose C ∈𝒞_2 is Lipschitz. C is Blackwell monotone if and only if it is (i) permutation invariant; and (ii) for all f ∈ℰ_2,
    
    D^+C(f; -f) ≤ 0,  and  D^+C(f; 1 - f) ≤ 0,  whenever exists.

    When C is differentiable at f, (<ref>) simplifies to
    
    ⟨∇ C(f), f ⟩≥ 0 ≥⟨∇ C(f), 1 - f ⟩.





In the following, we provide a geometric characterization of the set S_B(f) for binary experiments and use it to illustrate the proof sketch of Theorem <ref>. After the sketch, we show that the geometric characterization leads to a further characterization of Blackwell monotonicity in the case of binary states that do not require any functional assumption. 



  §.§.§ Proof Sketch




  
Parallelogram Hull
For any f, g ∈ℰ_2 with f ≽_B g, there exists M ∈ℳ_2 such that [g, 1-g] = [f, 1-f] M. 
For a stochastic matrix M ∈ℳ_2, there exists [a,b]^⊺∈ [0,1]^2 such that

    M = [   a 1-a;   b 1-b ],

which implies 

    g = a f + b (1 - f).

In other words, f ≽_B g if and only if g is in the parallelogram hull of the vectors, f and 1-f, defined by 

    (f, 1-f) = { a f + b(1-f): a, b∈ [0,1] }.

This parallelogram hull is precisely the sublevel set S_B(f) when considered as a subset in ^n. In the case of binary states, i.e., n = 2, it is depicted in Figure <ref> by the parallelogram (ABCD).




The parallelogram highlights two extreme directions of decreasing informativeness: -f and 1-f, depicted by AB and AD, respectively. For any ϵ > 0, notice moving from f in the direction of -f resulting the following experiment 

    [f- ϵ f, 1 - f + ϵ f].

From the perspective of garbling, this experiment is derived from f by applying the following garbling: whenever s_1 is generated, with probability ϵ, generate s_2 instead, while s_2 continues to generate itself. The other direction, 1 - f, captures the opposite type of garbling: s_2 is sometimes replaced by s_1. 



  
Necessity 

Permutation invariance follows straightforwardly from Blackwell monotonicity. If D^+C(f;-f) exists, then 

    D^+C(f; -f) = lim_ϵ↓ 0C(f - ϵ f) - C(f)ϵ≤ 0,

where the inequality follows from C(f-ϵ  f) ≤ C(f) for all ϵ > 0 under Blackwell monotonicity, as illustrated above. Similarly, D^+C(f; 1 - f) ≤ 0 also holds. □


	When C is differentiable at f, D^+C(f;h) is linear in h and given by ⟨∇ C(f), h ⟩. In this case, notice (<ref>) is equivalent to
	
    ⟨∇ C(f), g - f ⟩≤ 0,   ∀ g ∈(f, 1-f) = S_B(f).

	Geometrically, this says that ∇ C(f) lies in the polar cone of S_B(f) at f, depicted in Figure <ref> by the blue cone. In other words, when C is differentiable at f, Blackwell monotonicity can be translated into a local constraint on the feasible directions of its gradient. 

	For a graphical illustration, in Figure <ref>, we draw a curve passing through the point A to illustrate a potential isocost curve, indicating the same information cost, of a smooth cost function. As the gradient of such a function is tangent to its isocost curve, the gradient of this cost function (the purple arrow) lies outside the polar cone of S_B(f) and thus violates Blackwell monotonicity. This is confirmed by noticing that the cost increases in the direction of AD near A.




  
Sufficiency

Because (<ref>) is a local property, sufficiency requires additional regularity conditions on the cost function. Lipschitz continuity ensures that when restricting the cost function to any line segment, it is differentiable almost everywhere and the Fundamental Theorem of Calculus (FTC) applies. 
Consider any f ≽_B g, i.e., g lies inside the parallelogram ABCD. If g is above the line BD, as illustrated in Figure <ref>, we can find a two-segment path from f to g, which moves only in the extreme directions required by (<ref>): moving f in the direction of -f to reach f', and then moving f' in the direction of 1-f' to reach g. Then, as derivatives of the function when restricting to line segments coincide with the corresponding directional derivatives, (<ref>) implies it is non-positive almost everywhere along this path. Therefore, applying FTC implies C(f) ≥ C(g).
If g lies below the line BD, the permutation of g, namely g P, lies above the line BD and has the same cost as g, follows from permutation invariance. Then, the same argument applies to gP implying C(f) ≥ C(g P) = C(g). 
Lemma <ref> in the appendix formally shows this argument. 



  §.§.§ A Further Characterization with Binary States 

In the binary-binary case (n=m=2), restrict attention to the following set of experiments (the same cost function can be applied to the other experiments by permutation invariance):  

    ℰ̂_2 ≡{ (f_1, f_2)  : 0 ≤ f_1 ≤ f_2 ≤ 1  }.

For any f, g ∈ℰ̂_2, the parallelogram in Figure <ref> suggests that f ≽_B g if and only if the slope of AB for f is steeper than that of g, and the slope of AD for f is shallower than that of g. In other words, f ≽_B g if and only if

    α≡f_2f_1≥g_2g_1≡α'    and  β≡1-f_11-f_2≥1-g_11-g_2≡β'.[Let x/0 = +∞ for all x>0 and 0/0 = 1.]

Note that α is the likelihood ratio for generating the signal s_1 and 1/β is the likelihood ratio for signal s_2. Thus, (<ref>) implies that if both α and β increase, Blackwell informativeness increases. Also note that α and β can take any value in [1, +∞] and[If α = +∞, then f_1 = 0 and f_2 = β-1/β. If α = β = 1, let f_1 = f_2 = 0.]

    f_1 = β-1 αβ - 1  and   f_2 = α (β-1) αβ - 1.

Define C̃ : [1,∞]^2 →_+ as follows: 

    C̃(α, β) ≡ C ( β - 1αβ - 1, α (β - 1)αβ -1 ).

Thus, we have the following proposition.

	When n=m=2, for any C: ℰ̂_2 →_+, C is Blackwell monotone if and only if C̃ as defined in (<ref>) is increasing in α and β. 


These likelihood ratios can also be used to provide a different interpretation of Blackwell monotonicity. Notice when C is differentiable and ∂ C/∂ f_2≠ 0, (<ref>) can be rewritten as 

    α = f_2f_1_cthe slope of AB≥-∂ C/∂ f_1 ∂ C/∂ f_2_cthe slope of
    the isocost curve≥1-f_21-f_1_cthe slope of AD  = 1β	 .[With some algebra, we can show that f_2 ≥ f_1 and (<ref>) imply ∂ C/∂ f_2≥ 0 ≥∂ C/∂ f_1. ]

The slope of the isocost curve can be considered as the marginal rate of information transformation (MRIT). Thus, inequality (<ref>) provides another interpretation for Blackwell monotonicity: the MRIT should fall between the two likelihood ratios provided by the experiment.



 §.§ Blackwell Monotonicity over Finite Experiments

In this section, we characterize Blackwell monotonicity over ℰ_m with an arbitrary number m of signals. For necessity, much of the intuition from binary experiments carries over to the general case. Permutation invariance is clearly required for Blackwell monotonicity. An analogous condition to (<ref>) can be derived again by considering the extreme directions of decreasing informativeness in the space ℰ_m (Lemma <ref>). Specifically, such extreme directions are characterized by the following vectors: 
 
Given f ∈ℰ_m, for j ≠ k, define f^j(k) ∈^n × m as the matrix with f^j in the k-th column, -f^j in the j-th column and zeros elsewhere, i.e., 

    f^j(k) ≡[    0    ⋯ -f^j    ⋯    0    ⋯  f^j    ⋯    0 ].


Observe that for any ϵ∈ [0,1], f + ϵ f^j(k) ∈ℰ_m and is obtained again by the garbling where with a probability ϵ, s_j is replaced by s_k, i.e., merging the signal s_j into s_k. Thus, Blackwell monotonicity would require the cost function to be decreasing along these extreme directions. 

When establishing sufficiency over binary experiments, the key step was to construct a path along which Blackwell informativeness decreases. With more than two signals, however, such a path within the space ℰ_m does not always exist, as shown by the following proposition. 

 
	Suppose that n=m=3 and let
	
    I_3= [ 1 0 0; 0 1 0; 0 0 1 ]≽_B 
    		g = [ 4/5 1/5   0;   0 4/5 1/5; 1/5   0 4/5 ]∈ℰ_3.

	If f ∈ℰ_3 is Blackwell more informative than g, then f is a permutation of I_3 or g. 


Proposition <ref> suggests that there is no path in ℰ_3 connecting I_3 and g along which Blackwell informativeness decreases. 
Because had there exist such a path, there must exist some experiment other than permutations of I_3 or g that is more informative than g but less informative than I_3, which contradicts the proposition.

In this paper, we identify two approaches to overcome this issue and establish sufficiency. One approach is to embed ℰ_m into a higher-dimensional space, ℰ_2m, where a decreasing path connecting experiments in ℰ_m can be shown to always exist. In other words, we can use the necessary condition in ℰ_2m to characterize sufficiency in ℰ_m. This characterization, however, might be difficult to apply when it is not obvious how to extend a given cost function defined on ℰ_m to ℰ_2m. For this reason, we delay the presentation of this idea to Section <ref>.

The more convenient and direct approach, still within the original space ℰ_m, is to impose quasiconvexity, a standard condition in cost minimization problems ensuring the local minimum is also global.[However, quasiconvexity is strictly non-necessary for Blackwell monotonicity. See Section <ref>.] To see how quasiconvexity is able to address the problem raised in Proposition <ref>, observe that there is a permutation matrix P such that 

    g = 45 I_3 + 15 I_3P.

As permutation invariance implies C(I_3) = C(I_3 P), we have C(g) ≤ C(I_3) when C is quasiconvex. Our next result shows that, with quasiconvexity, the same type of necessary and sufficient condition as in Theorem <ref> can be established for all experiments.


    Suppose C ∈𝒞_m is Lipschitz and quasiconvex. C is Blackwell monotone if and only if it is (i) permutation invariant, and (ii) for all f ∈ℰ_m,  
    
    D^+C(f; f^j(k)) ≤ 0,   ∀ j ≠ k,  whenever exists.

    When C is differentiable at f, (<ref>) simplifies to 
    
    ⟨∇^k C(f) - ∇^j C(f), f^j⟩ = ∑_i=1^n( ∂ C∂ f_ik - ∂ C∂ f_ij) f_ij≤ 0,   ∀ j ≠ k.



First of all, notice that when m = 2, condition (ii) reduces to those in Theorem <ref> by letting f^1 = f, f^2 = 1 - f and C(f) ≡ C(f, 1 - f).

In the proof of Theorem <ref>, the key step is to show that any extreme point of S_B(f) is either a permutation of f or can be reached from f by a sequence of segments in the directions of f^j(k)'s required by (ii) (Lemma <ref>). Intuitively, these directions of merging signals are precisely those directions from an extreme point E ∈ext_k(ℳ_m) to some extreme point in E ∈ext_k-1(ℳ_m). Once this is established, similar to the proof of sufficiency in Theorem <ref>, FTC along these segments along with (i) and (ii) together imply that all extreme points of S_B(f) are in S_C(f). Then, quasiconvexity ensures that the entire set S_B(f) is in S_C(f), and thus C is Blackwell monotone.


	From Theorem <ref>, checking Blackwell monotonicity over non-binary experiments requires one more step. Namely, one needs to first verify whether C is quasiconvex. It is worth noting that when C is twice differentiable, quasiconvexity can be checked by verifying the determinants of every order of its bordered Hessian matrices are non-positive, a similar procedure as checking convexity. See <cit.> and also Proposition 3.4.4 in <cit.> for a reference. 




§ ADDITIVELY SEPARABLE COSTS

Results in Section <ref> provide valuable and tractable means to verify whether an arbitrary information cost function adheres to Blackwell monotonicity, essentially by checking the local condition. This becomes even simpler when the cost function is additively separable across signals, as the differential inequalities specify directions where only two signals are changing. In this section, we characterize Blackwell monotonicity for such additively separable information costs and identify a wide range of such costs including both new and well-known existing forms.


	An information cost function C: ℰ→_+ is additively separable if there exists a Lipschitz function ψ: [0,1]^n→_+ with ψ(0) = 0 such that, for all m and f ∈ℰ_m,
	
    C(f) = ∑_j=1^mψ(f^j).



Notice the definition of additive separability has already built-in permutation invariance (same ψ for all signals) and ψ(0) = 0, naturally required by Blackwell monotonicity.[For ψ(0) = 0, adding columns of zeros does not change the informativeness of an experiment and thus should not change its cost.] 
As a result, an additively separable cost function can be easily defined over all experiments ℰ as its restriction to ℰ_m is obvious. In this case, Blackwell monotonicity over ℰ is equivalent to Blackwell monotonicity over all ℰ_m. 

For additively separable costs, Blackwell monotonicity can be characterized by a simple condition on the component function ψ.


    Suppose C: ℰ→_+ is additively separable. C is Blackwell monotone if and only if ψ is sublinear, i.e., 
    
    	
  * ψ is positive homogeneous, i.e., ψ (γ· h) = γ·ψ (h) for all h ∈ [0,1]^n and γ > 0 with γ· h ∈ [0,1]^n;
    	
    	
  * ψ is subadditive, i.e., ψ (h+ l) ≤ψ(h) + ψ (l) for all h, l ∈ [0,1]^n with h+l ∈ [0,1]^n. 
    


We next present the proof of sublinearity implying Blackwell monotonicity, which highlights the application of sufficient conditions in Theorem <ref>.


    If ψ is sublinear, then for any f, g ∈ℰ_m and λ∈ (0,1), 
    
    C(λ f + (1-λ) g) =   ∑_j=1^mψ (λ f^j + (1-λ)g^j ) 
    ≤   ∑_j=1^m (λ·ψ ( f^j  ) + (1-λ) ·ψ (g^j) )= λ C(f) + (1-λ) C(g).

    Therefore, C is convex, and thus quasiconvex. It then remains to verify (<ref>). For any f ∈ℰ_m, any j ≠ k, and ϵ > 0, we have 
	
    C(f + ϵ f^j(k)) - C(f)     = ψ(f^k + ϵ f^j) - ψ(f^k) + ψ( (1-ϵ) f^j ) - ψ (f^j) 
       ≤ϵψ(f^j) - ϵψ(f^j) = 0,

	where the first equality follows from additive separability of C and the inequality follows from sublinearity of ψ. As this holds for all ϵ > 0, it follws that D^+ C(f; f^j(k)) ≤ 0 whenever exists, thus establishing Blackwell monotonicity by Theorem <ref>.

Additive separability plays a key role in the proof, as it implies the difference C(f + ϵ f^j(k)) - C(f) depends only on the j-th and k-th column vectors in both experiments. In other words, a crucial property of additively separable costs is that, when holding the other signals fixed, the decrease in information cost by garbling between two signals is independent of all other signals. While this property might seem strong, it is actually shared by many well-known information costs, as we show in the next section.



 §.§ Subclasses of Additively Separable Costs

Theorem <ref> enables the construction of Blackwell-monotone information costs by simply finding a sublinear component function ψ. Sublinear functions include all norms, seminorms, and linear functions. Additionally, any pointwise maximum of sublinear functions is also sublinear. As an illustration, we highlight the following subclasses of such costs. 



  
Norm Costs Norms are natural choices of sublinear functions. For any norm ‖·‖ on ^n, the following cost function is Blackwell monotone:

    C(f) = ∑_j=1^m‖ f^j‖ - ‖1‖,

where subtracting ‖1‖ is to ensure that the cost is grounded, i.e., the cost of an uninformative experiment is zero. Among norm costs, the supnorm can be used to construct probably the simplest example of a Blackwell-monotone cost function: 

    C(f) = ∑_j=1^mmax_i f_ij - 1,

For binary experiments represented by a single vector f ∈^n_+, the supnorm cost function is simply given by 

    C(f) = max_i f_i - min_i f_i.





  
Absolute-Linear Costs The absolute value of a linear function is a seminorm, and thus sublinear. Therefore, given any a ∈^n, the following cost function is Blackwell monotone:

    C(f) = ∑_j=1^m|⟨ a, f^j⟩| = ∑_j=1^m |∑_i=1^n a_i f_ij|.

Notice that when a ∈^n is arbitrary, this cost function can potentially be a constant over all experiments. For example, when a ∈_+^n, one has C(f) = ∑_j=1^m|⟨ a, f^j⟩| = ∑_j=1^m⟨ a, f^j⟩ = ⟨ a, 1⟩. 

To avoid this issue, say that a Blackwell-monotone cost function is strictly grounded if it assigns zero costs only to uninformative experiments. Our next result identifies necessary and sufficient conditions to ensure strict groundedness of absolute-linear costs.


    An absolute-linear cost function C(f) = ∑_j=1^m|⟨ a, f^j⟩| is strictly grounded if and only if ∑_i=1^n a_i = 0 and a_i≠ 0 for all i.


Absolute-linear costs prove instrumental in applications as they provide a tractable and general class of Blackwell-monotone costs to start with. Especially, in the case of binary states and signals, as an experiment can be represented by f = [f_1, f_2]^⊺, any strictly grounded absolute-linear cost function is given by 

    C(f) = λ| f_2 - f_1|,  for some λ > 0.

Furthermore, any monotone (not necessarily linear) transformation of this cost function is still Blackwell monotone, providing freedom in choosing functional forms in applications. For example, the following quadratic cost function will be used in our applications: 

    C(f) = λ (f_2 - f_1)^2,  for some λ > 0.






  
Linear ϕ-divergence Costs Let ϕ_ii': [0,∞] →∪{+∞} be a convex function with ϕ_ii'(1) = 0 and β_ii'≥ 0. The linear ϕ-divergence cost function is defined as

    C(f) = ∑_j=1^m∑_i,i'β_ii' f^j_i'ϕ_ii'( f^j_i/f^j_i') = ∑_i,i'β_ii'∑_j=1^m f^j_i'ϕ_ii'( f^j_i/f^j_i'),

where ∑_j=1^m f^j_i'ϕ_ii'( f^j_i/f^j_i') is the ϕ-divergence (with generator ϕ_ii') between the probability distributions over signals in state ω_i and ω_i'. The LLR cost axiomatized in <cit.> is a special case with ϕ_ii'(x) = x log x. Blackwell monotonicity of such costs is well-known following from the data-processing inequality for ϕ-divergence (see Theorem 7.4 in <cit.>). Here, as we show it can be rewritten as an additively separable cost, Blackwell monotonicity can also be established by sublinearity of ∑_i,i'β_ii' f^j_i'ϕ_ii'( f^j_i/f^j_i'), which follows from Jensen's inequality given convexity of ϕ_ii' (See Theorem 2.7.1 in <cit.>).





  
Entroy Costs and Posterior-Separable Costs As already mentioned, another popular strand of defining costs of information is based on distribution over posteriors. This type of information cost is prior-dependent and thus cannot be directly defined as a function over only experiments. Nonetheless, once the prior is fixed, it would induce a cost function over experiments. 

Among such cost functions, the entropy cost studied in <cit.> and <cit.> can be shown to be additively separable: Let μ∈Δ(Ω) be a given prior, 

    C_μ(f)         = λ·∑_j=1^m∑_i=1^n μ_i f^j_i ·log ( μ_if^j_i/τ (f^j)  ) - λ(∑_i=1^nμ_ilogμ_i).

where τ (f^j) is the probability of receiving signal j, i.e.,  τ (f^j) ≡∑_i=1^nμ_i · f^j_i.
Notice for each j, the term ∑_i=1^n μ_i f^j_i ·log( μ_if^j_i/τ (f^j)) depends only on the column f^j, thus the entropy cost is additively separable.

Moreover, a generalization of the entropy cost, the posterior-separable cost, is defined by letting H (the measure of uncertainty) be any concave and upper semi-continuous function (see, e.g., <cit.> and <cit.>). Then notice that 

    C_μ(f)     = H(μ) - 𝔼[H(μ(·|s_j))] = H(μ ) + ∑_j=1^mψ ( f^j )

where ψ:[0,1]^n →ℝ is defined by

    ψ(f^j) ≡ - τ (f^j) · H[ ( μ_if^j_i/τ (f^j) )_i].

Thus, posterior-separable costs are also additively separable.

To verify whether C_μ (f) is Blackwell-monotone, it is sufficient to check the sublinearity of ψ. For positive homogeneity, from τ (γ· f^j) = γ·τ(f^j), we have  ψ (γ· f^j) = γ·ψ (f^j) whenever γ· f^j ∈ [0,1]^n. Additionally, τ(f^j+ f^k) = τ(f^j) + τ(f^k) and the concavity of H imply that

    ψ(f^j) + ψ(f^k) =    - τ(f^j)  · H[( μ_if^j_i/τ(f^j))_i] 
    	- τ(f^k)  · H[ ( μ_if^k_i/τ(f^k))_i]  
    ≥     -  (τ(f^j) + τ(f^k) ) · H[ ( μ_if^j_i + μ_i f^k_i /τ(f^j + f^k) )_i] =  ψ (f^j+ f^k),

thus, ψ is subadditive. 


To summarize, we identify the class of additively separable costs that encompasses a wide range of well-known information costs. Theorem <ref> provides a simple characterization of Blackwell monotonicity for such costs. This not only provides a simple method to construct Blackwell-monotone costs, but also, as highlighted in linear ϕ-divergence costs and posterior-separable costs, provides an alternative approach to verify Blackwell monotonicity of costs in this form. 



§ APPLICATIONS

In this section, we study two applications with costly information. Importantly, we highlight how our characterization of Blackwell monotonicity and additively separable costs can provide a general framework and tractable tools to analyze these problems. 



 §.§ Bargaining with Information Acquisition

<cit.> study an ultimatum bargaining model where the buyer can acquire costly information about the unknown value of an object before accepting the seller's offer.[Their model differs from <cit.> in that the buyer observes the seller's offer and acquires information about his valuation, whereas in <cit.>, the buyer chooses an attention strategy, which is a map from the valuation and the offer to the acceptance probability.]
To model costly information, they exogenously restrict the buyer's feasible set of experiments and define an information cost function over the restricted set. Using our characterization, we are able to extend their cost function to a Blackwell-monotone cost function over all experiments, allowing us to examine their results in a more general setting. In this section, we show that while the exogenous restriction is crucial for their main result, their Theorem 1, the same conclusion can still be shown in the general setting under a different Blackwell-monotone information cost function. 

Formally, a seller possesses an object (which holds zero value to herself) that has two possible values (for the buyer), denoted as v ∈ V = {H, L} with H > L > 0, according to a common prior π = Pr(H) ∈ (0,1). The seller observes the value of the object and offers a price p ∈ to the buyer. The buyer observes only the price p and can acquire information about v with experiments. Under Blackwell monotonicity, it is without loss of generality to restrict attention to binary experiments.[Suppose that the buyer utilizes an experiment with more than two signals. Then, consider a garbling of the signals, where signals inducing the buyer to accept the offer are assigned to h, and signals inducing the buyer to decline the offer are assigned to l. After applying this garbling, the expected material payoff remains the same, but it is less costly since it is less Blackwell informative.  ] 
Let f = [f_L, f_H]^⊺∈ℰ_2 where f_L and f_H denote the probability of generating signal h in states L and H, respectively. Let the information cost be denoted by C_λ(f), parametrized by λ > 0.  

<cit.> exogenously restrict the buyer's feasible set of experiments to H-focused experiments, i.e., f_L = 0.[They also separately consider cases with L-focused experiments with f_H = 1, and a mix of both that does not span the space of all experiments.] With this restriction, they could define information cost functions simply as an increasing function of f_H. One example of their cost function is given by

    C_λ((f_H, 0)) = λ/2 f_H^2.

Based on discussions in the previous section, this cost function can be extended to a Blackwell-monotone cost function over all experiments as follows: 

    C_λ(f) = λ/2 (f_H - f_L)^2.

This enables us to examine their results without restricting the buyer's ability to acquire information.

Let σ: V →Δ(_+) denote the seller's strategy. After the buyer observes the seller's offer p, the buyer forms a belief μ∈ [0,1] about the value of the object, chooses an experiment f ∈ℰ_2 and takes an action contingent on the signal. The buyer's optimal strategy f^* given (p,μ) can be solved by the following program:

    max_[f_L, f_H]^⊺∈ℰ_2μ f_H (H-p) + (1-μ)f_L(L - p) - C_λ(f),

under which the buyer accepts the offer if receives signal h and rejects otherwise. A strategy profile (σ^*, f^*(p,μ)) and a belief system (μ_p)_p ∈_+ constitute a Perfect Bayesian equilibrium if:[Same as in their model, we focus on Pareto-undominated equilibria.]

	
  * f^*(p,μ_p) solves (<ref>) given all (p, μ_p) ; σ^* is optimal given f^*. 
	
  * μ_p is obtained via Bayes' rule on path. 


Theorem 1 in <cit.> claims that when only H-focused experiments are available to the buyer, as information becomes arbitrarily cheap, i.e., λ→ 0, all equilibria are pooling equilibrium under which both types of seller offer the same price close to L and the buyer accepts the offer without acquiring any information. Once the buyer's feasible experiment is not restricted, however, this result no longer holds under (at least) the quadratic cost function as other equilibria would emerge:


	When C_λ(f) = λ/2 (f_H - f_L)^2, for all λ < π(1-π)H, there always exists non-pooling equilibria under which the buyer acquires information. 


The main intuition of Proposition <ref> is similar to Proposition 1 in <cit.> where the buyer is restricted to acquire information using L-focused experiments. When λ < π(1-π)H, we show that there always exists a semi-separating equilibrium where the buyer acquires information using an experiment with 1 = f_H^* > f_L^* > 0. This confirms that their intuition holds even when the restriction of L-focused experiments is lifted. More importantly, this also implies that their Theorem 1 hinges crucially on the exogenous restriction of using only H-focused experiments.

Despite this fact, our next proposition reestablishes their Theorem 1 without any restriction on experiments by considering cost functions that are not covered in their model, the absolute-linear costs. 


    When C(f) = λ| f_H - f_L|, for any λ > 0, there exists ϵ > 0 such that every equilibrium is a pooling equilibrium where
    
        
  * σ(L) = σ(H) = δ_p^* with p^*∈ [L, L + ϵ).
        
  * On the equilibrium path, [f_L^*, f_H^*] = [1,1], i.e., the buyer acquires no information and buys at price p^* with certainty.
    
    Moreover, ϵ→ 0 as λ→ 0, and thus eventually, the buyer extracts the full surplus. 	


For an intuition of Proposition <ref>, under absolute-linear costs, we show that the buyer's optimal information acquisition is either no information or full information, a special feature driven by linearity (Lemma <ref>). This fact removes the possibility of semi-separating equilibrium as in Proposition <ref> and thus ensures all equilibria are pooling. 



 §.§ Costly Persuasion 


Consider the standard prosecutor-judge example with costly information provision in <cit.>. 
The judge (Receiver) chooses between two actions: either aquits or convicts.
There are two states of the world: the defendant is innocent (ω = i) or guilty (ω = g).  The payoff of the prosecutor (Sender) is state-independent with u_S(c) = 1 and u_S (a) =0, whereas the judge's payoff is to match the state and the action: u_R(a,i) = u_R(c,g) =1 and u_R(c,i) = u_R(a,g) =0. 

The prosecutor commits to a persuasion policy at some Blackwell-monotone information cost C. Since the judge's action is binary, by using the same argument in footnote <ref>, it is without loss to consider binary experiments (f_1, f_2) ∈ℰ_2 where f_2 = (c|g) and f_1 =  (c|i).
When the prior belief  is μ =  (g), the prosecutor's problem is

    max_[f_1, f_2]^⊺∈ℰ_2μ·  f_2 + (1-μ) · f_1 - C(f_1, f_2)

subject to the posterior belief upon receiving c is greater than or equal to 1/2:

    μ f_2μ f_2 +(1-μ)f_1≥12  ⇔  μ f_2 ≥ (1-μ) f_1 .

When μ≥ 1/2, setting f_1 = f_2=1 yields the highest material payoff and the least information cost, and satisfies (<ref>), i.e., always sending the signal c is optimal. 

Now assume that μ < 1/2. 
As an intermediate step to solve the problem, we consider an auxiliary cost minimization problem:

    min_[f_1, f_2]^⊺∈ℰ_2 C(f)     s.t.   μ· f_2 + (1-μ) · f_1 = w    and   (<ref>).

In other words, the auxiliary problem is to solve for the least costly information needed for the prosecutor to achieve a given material payoff level w. Note that from (<ref>), the range of w is [0, 2μ].

 
	When C is Blackwell monotone and 1 > 2μ≥ w ≥ 0, (<ref>) binds for the solution of (<ref>). 




This lemma is illustrated in Figure <ref>. From the results of Section <ref>, as f_1 increases, Blackwell informativeness decreases along the line of μ f_2 + (1-μ)f_1 = w. Therefore, to minimize the Blackwell-monotone cost, (<ref>) needs to bind: 

    f_1 = w/2(1-μ)   and     f_2 = w/2μ.

Intuitively, when (<ref>) is non-binding, it implies that the posterior belief after receiving c is greater than 1/2. Thus, the prosecutor can save on information cost by making the experiment less persuasive, while still ensuring the judge convicts. 

Next, by plugging (<ref>) in, the prosecutor's problem becomes

    max_0 ≤ w ≤ 2μ   w - C ( w/2(1-μ), w/2μ ).

Therefore, given the cost function, the prosecutor's problem becomes an one-dimensional maximization problem. 
As an example, the following proposition characterizes the optimal persuasion policy for the quadratic cost function.

 
	Suppose that  C(f_1, f_2) = (f_2 - f_1)^2. There exists 0 < μ̂<1/2 such that the prosecutor's optimal persuasion policy is as follows: 
	
    f_1 = 
    			1,						   if μ≥ 1/2,
    μ/1-μ, 	   if μ∈ (μ̂, 1/2 ),
    μ^2(1-μ)/(1 - 2μ)^2   if μ≤μ̂,
    		  and  
    		f_2 = 
    			1,						   if μ≥ 1/2,
    
    			1, 	   if μ∈ (μ̂, 1/2 ),
    μ(1-μ)^2/(1 - 2μ)^2   if μ≤μ̂.



This result is illustrated in Figure <ref> and <ref>. 
When μ≥μ̂, the optimal persuasion policy is the same as the one without the cost: the posterior belief is either 1/2 or 0. In this case, the prosecutor always convicts guilty defendants and, with some positive probability, convicts innocent defendants. 
When μ < μ̂, this policy is no longer optimal as it becomes too expensive. Instead, the prosecutor sacrifices the probability of convicting the guilty defendant to lower the costs. Observe that the posterior belief upon receiving a depends on μ, while the posterior belief upon receiving c is constant (1/2).  
This result differs qualitatively from the one with posterior separable costs, where the posterior beliefs are independent of prior belief whenever the information is provided. 




§ ADDITIONAL RESULTS AND DISCUSSIONS




 §.§ Non-necessity of Quasiconvexity


As already mentioned, quasiconvexity is stirctly non-necessary. The following example illustrates a cost function over binary experiments that is Blackwell monotone but not quasiconvex.


	Suppose n = m = 2. Denote any experiment f ∈ℰ_2 by f = [f_1, f_2]^⊺. As before, we restrict attention to the set ℰ̂_2 = { (f_1, f_2)  : 0 ≤ f_1 ≤ f_2 ≤ 1 }. Consider C: ℰ̂_2 →_+ defined by 
	
    C(f) = min{f_2/f_1, 1-f_1/1-f_2}.

	By using (<ref>), we can easily see that f ≽_B g implies C(f) ≥ C(g), i.e., C is Blackwell monotone. 
	
	Consider f = [0, 1/2]^⊺ and g = [1/2, 1]^⊺ with costs C(f) = C(g) = 2. For the average convex combination of them, given by h = [1/4, 3/4]^⊺, the cost is C(h) = 3 > C(f) = C(g).
	Hence, this cost function is not quasiconvex. 




 §.§ Binary Experiments with Quasiconvexity

Quasiconvexity was not needed in establishing Blackwell monotonicity over binary experiments ℰ_2. However, when impose quasiconvexity in this case, we can establish a sufficient condition for Blackwell monotonicity without imposing Lipschitz continuity and the local condition in Theorem <ref>: 

Recall that any binary experiment can be represented by f = [f_1, ⋯, f_n]^⊺∈ [0,1]^n, and 0 and 1 are completely uninformative experiments. Say that C is non-null if for any f ∈ [0,1]^n, C(f) ≥ C(1) =C(0). 

 
	If C ∈ℰ_2 is quasiconvex, permutation invariant, and non-null, then C is Blackwell monotone. 



	As an intermediate result of Theorem <ref>, we have S_B(f) =  (f, 1-f), that is, f ≽_B g implies that g = a f + b(1 -f) for some (a, b) ∈ [0,1]^2. Then, if a ≥ b, g= (1-a) ·0 +  (a-b) · f + b ·1; and if a < b, g = (1-b) ·0 + (b-a) · (1 - f) + a ·1. From quasiconvexity and non-nullness, we have C(f) ≥ C(g) or C(1-f) ≥ C(g). Then, by permutation invariance, C(f) = C(1 -f), thus, C(f) ≥ C(g). 




 §.§ Sufficiency via Higher Dimensions
 
Proposition <ref> demonstrates an example in ℰ_3 where f = I_3 ≽_B g but there does not exist a path in ℰ_3 connecting f and g along which Blackwell informativeness decreases.
Nevertheless, when both f and g are considered as experiments in ℰ_6, a decreasing path can be found. To see this, we first embed f and g into ℰ_6 by adding three columns of zeros. Then consider the following experiment 

    f = [ 4/5   0   0 1/5   0   0;   0 4/5   0   0 1/5   0;   0   0 4/5   0   0 1/5 ],

It is not hard to see that f ≃_Bf by finding stochastic matrices connecting them. Thus, f≽_B g. Then for any λ∈ [0,1], let 

    f_λ = (1-λ) f + λ g = [       4/5       λ/5         0 (1 - λ)/5         0         0;         0       4/5       λ/5         0 (1 - λ)/5         0;       λ/5         0       4/5         0         0 (1 - λ)/5 ].

It can also be shown that f≽_B f_λ≽_B g. Thus, a decreasing path connecting f and g now can be found in ℰ_6. 

The previous observation actually holds for all f ≽_B g in ℰ_m. That is, there always exists a decreasing path connecting f and g in the space ℰ_2m (Lemma <ref>). However, such a decreasing path does not necessarily move in the extreme directions, f^j(k)'s. Therefore, to establish a similar argument as in Theorem <ref> in this case, we need to rely on the linearity of directional derivaitives, and thus requires a stronger functional assumption on C:


	Suppose C ∈𝒞_2m is continuously differentiable. Conditions (i) and (ii) in Theorem <ref> over ℰ_2m are sufficient for C to be Blackwell monotone over ℰ_m. 





§ CONCLUDING REMARKS

Information is costly and more information should cost more. Building upon this premise, this paper characterizes necessary and sufficient conditions for information cost functions to be monotone when informativeness is compared using Blackwell's information order. 
This characterization allows us to study the implications of Blackwell monotonicity in various economic applications. 
For some applications exhibiting monotonicity between signals and actions, another well-known information order proposed by <cit.>, which refines the Blackwell order, becomes more relevant. We believe the methodology developed in this paper can be also extended to characterize Lehmann-monotone costs. However, we leave this for future research.



econ







equationsection
lemmasection


§ PROOFS FOR SECTION <REF>




 §.§ Proof of Theorem <ref>




  §.§.§ A Lemma



	For any f, g ∈ℰ_2 such that f ≽_B g, there always exists 1 ≥ a ≥ b ≥ 0 such that either 
	
    g = a f + b (1 - f)    or   1 - g = a f + b (1 - f).

	Without loss, let g satisfy the first equation of (<ref>) and  f' = a - b/1-b f.[When b=1, define f' = 1.]  Then, for all λ∈ [0,1], the followings hold:
	
    f ≽_B (1-λ) f + λ f' ≽_B f',  and 
       f' ≽_B (1-λ) f' + λ g ≽_B g.

	Moreover, f' - f is in the direction of -f, and g - f' is in the direction of 1-f'.



	Recall that f ≽_B g implies that there exist (a,b) ∈ [0,1]^2 such that g = a f +b (1-f). 
	If a ≥ b, the first equation of (<ref>) holds. If a < b, we have a' = 1-a > 1-b = b' and 1 - g = a' f + b' (1-f ). 
	
	When b =1, a is also equal to 1 and g = f + (1-f) = 1 = f'. Then, (<ref>) trivially holds. Notice that (1-λ) f + λ1 = 1 · f + λ (1 -f) ∈(f, 1 - f), thus, f ≽_B  (1-λ) f + λ1. Next, we have 1 = 1 · ( (1-λ) f + λ1 ) + 1 ·{1 - ( (1-λ) f + λ1 ) }, which implies  (1-λ) f + λ1≽_B 1. Therefore, (<ref>) holds. 
	
	When b < 1, we have a-b/1-b∈ [0,1] and f ≽_B f'. For any λ∈ [0,1], f ≽_Bλ f + (1-λ) f' simply follows from convexity of (f, 1 - f). On the other hand, notice that 
	
    f' = a-b/1-b/1-λ + λa-b/1-b ((1-λ) f + λ f').

	Since 
	
    a-b/1-b/1-λ + λa-b/1-b∈ [0,1],

	we have f' ∈(((1-λ) f + λ f'), 1 - ((1-λ) f + λ f')), and thus (1-λ) f + λ f' ≽_B f'. 
	
	From g = a f + b (1 - f), we have 
	
    g = a-b/1-b f + b (1 - a-b/1-b f ) = f' + b (1 - f'),

	thus f' ≽_B g and g - f' = b (1 - f'). By a similar argument, f' ≽_B (1-λ) f' + λ g ≽_B g. The last statement also follows from the above argument.




  §.§.§ Proof



	Necessity is proved in the main text. 
	
	For sufficiency, take any f ≽_B g. First, permutate and relabel g if needed to have g satisfy the first equation of (<ref>). Permutation invariance ensures that the cost stays the same. 	
	Define ϕ_1 (λ) ≡ C( (1-λ) f + λ f' ) and ϕ_2 (λ) ≡ C((1-λ) f' + λ g ). They are both Lipschitz continuous function in λ as Lipschitz continuity is preserved under composition. 
	Therefore, ϕ_1 is differentiable almost everywhere and satisfies, when differentiable, 
	
    ϕ_1'(λ)     = D^+C ( (1-λ) f + λ f' ; - f + f' ).

	On the other hand, observe that 
	
    -f + f'       = - 1-a/1-b1-λ + λa-b /1-b ( (1-λ) f + λ f' ).

	Therefore, ϕ_1'(λ) has the same sign as D^+C ( (1-λ)f + λ f'; - ((1-λ)f + λ f') ) and it is non-positive from (<ref>). Lipschitz continuity in λ enables FTC, which gives: 
	
    C(f') = ϕ_1(1) = ϕ_1(0) + ∫_0^1ϕ_1'(λ) dλ≤ϕ_1(0) = C(f).

	
	Similarly, observe that 
	
    ϕ_2'(λ)     = D^+C ( (1-λ) f' + λ g ; - f' + g ),  
    
    		-f' + g       = b (1- f') = b1-λ b ( 1 - ( (1-λ) f' + λ g )  ).

	Then,  ϕ_2'(λ) is non-positive since it has the same sign with D^+C( (1-λ)f'+λ g; 1 - ( (1-λ)f'+λ g ) ). By applying the FTC, we also have C(g) = ϕ_2(1) ≤ϕ_2(0) = C(f'). Therefore, we have C(f) ≤ C(g), thus C is Blackwell monotone.






 §.§ Proof of Proposition <ref>



	Suppose that f ≽_B g, i.e., there exists a 3 × 3 stochastic matrix B = (b_ij) such that fB = g. 
	
	Observe that at least one of f_11,  f_12 and f_13 is positive—if not, every entry of the first row of fB is equal to zero. Without loss of generality, let f_11 be positive (we can obtain it by permuting f).
	Note that f_11 b_13+ f_12 b_23+f_13b_33 =0. 
	Since every entry of f and B are nonnegative, b_13=0. 
	
	Next, observe that 4/5 = f_21 b_13+ f_22 b_23+f_23 b_33. From b_13=0, at least one of f_22 and f_23 is positive. Without loss, let f_22 be positive. Then, from g_21=0, we have b_21=0. Then, it gives us f_21 b_11+ f_23 b_31=0. We consider two cases: b_11=0 or f_21=0. 
	
	

  
1. b_11=0:
	From b_11=b_13=0, we have b_12= 1. From g_32=0 and b_12>0, we have f_31 =0. 
	
	Additionally, we have f_13 b_31 = 4/5, f_23 b_31 =0 and f_33 b_31 = 1/5. Therefore, b_31, f_13, f_33≠ 0 and f_23=0. From g_13=0 and f_13≠ 0, we have b_33=0. Likewise, from g_32=0 and f _33 > 0, b_32=0. Then, it gives us b_31 = 1.  
	
	From b_11=0,  b_21=0 and b_31=1, we have f_13 = 4/5 and f_33 = 1/5. From f_31=0, f_32 = 4/5. From g_32=0 and f_32>0, we have b_22 =0. It gives us b_23=1. Therefore, B is a permutation matrix and f is a permutation of g. 
	
	

  
2. b_11>0 and f _21 =0:
	
	Observe that f_23 b_31 =0, f_22b_22+f_23b_32 = 4/5 and f_22b_23+f_23b_33 = 1/5. We consider two subcases: f_23=0 or b_31=0. 
	
	

  
(a) f_23 =0:	
	
	From f_21=f_23=0, we have f_22=1. Additionally, we have b_22=4/5 and b_23 = 1/5. From 0 = g_13 = f_12b_23 + f_13b_33 and 0= g_32= f_31 b_12 +f_32 b_22+f_33 b_32, we have f_12 = f_32 =0. Observe that 0 = g_13 = f_13 b_33 and 4/5 = g_33 = f_33 b_33. Then, we have b_33 >0 and f_13=0. From f_12 = f_13=0, we have f_11= 1. This also gives b_11 = 4/5 and b_12 = 1/5. 
	Again from 0= g_32 and b_12=1/5, we have f_31 =0. Therefore, from f_31=f_32=0, we have f_33=1, i.e., f is I_3. 
	
	

  
(b) b_31=0:	
	
	From b_21=b_31=0, we have f_11· b_11 = 4/5 and f_31 · b_11 = 1/5. Therefore, b_11, f_11, f_31 >0. Next, 0 = g_32 = f_31 b_12 + f_32 b_22 + f_33b_32 gives b_12 =0. From b_12=b_13 =0, we have b_11=1. 
	
	Suppose that both b_22 and b_32 are positive. Then, from 0 = g_32 = f_32b_22+ f_33b_32, we have f_32=f_33=0. It contradicts 4/5=g_33 = f_31 b_13 + f_32 b_23 + f_33 b_33 since b_13= f_32=f_33=0. Therefore, at least one of b_22 and b_32 is equal to zero. Likewise, if both b_23 and b_33 are positive, we have f_12=f_13=0 from g_13=0, but it contradicts g_12 = 1/5>0. Thus, at least one of b_23 and b_33 is equal to zero. Also, note that B needs to be a full rank matrix (as g has a full rank). To have that, there are two possibilities: (i) b_22= b_33=1 and b_23=b_32=0; or (ii) b_23=b_32 =1 and b_22= b_33=0. Then, B is either I_3 or a permutation of I_3. Therefore, f is g or a permutation of g.  	




 §.§ Proof of Theorem <ref>




  §.§.§ Useful Lemmas



	For any f, g∈ℰ_m, f ≽_B g if and only if 
	
    g - f ∈{∑_j=1^mλ_j h_j: λ_j∈ [0,1], h_j∈co({f^j(k): k ≠ j}), ∀ j },

	where co(·) denote the convex hull. 
	
	Moreover, if C ∈𝒞_m is Blackwell monotone, then for all f ∈ℰ_m and all j ≠ k,
	
    D^+C(f; f^j(k)) ≤ 0,  whenever exists.




	For the first part, for any f, g∈ℰ_m. If f ≽_B g, then there exists a stochastic matrix M ∈ℳ_m such that g = fM. Then we have 
	
    g - f     = f(M - I) 
        = [ f^1   ⋯ f^m ][   m_11-1        ⋯     m_1m;        ⋮        ⋱        ⋮;     m_m1        ⋯ m_mm - 1 ]
        = [ f^1   ⋯ f^m ][   - ∑_k = 2^m m_1k                  ⋯               m_1m;                  ⋮                  ⋱                  ⋮;               m_m1                  ⋯ - ∑_k = 1^m-1 m_mk ]
        = ∑_j=1^m∑_k ≠ j m_jk f^j(k)

	with m_jk≥ 0 and ∑_k=1^m m_jk = 1 for all j. It thus can be further written as 
	
    g-f     = ∑_j=1^m(∑_k ≠ j m_jk)(∑_k ≠ jm_jk/∑_k ≠ j m_jk f^j(k) ).

	Notice that ∑_k ≠ j m_jk∈ [0,1] and the term in the second parentheses is a convex combination of f^j(k) for k ≠ j. Thus, we have
	
    g-f = ∑_j=1^mλ_j h_j,

	for λ_j∈ [0,1] and h_j∈co({f^j(k): k ≠ j}). 
	
	Conversely, it suffices to reverse the above steps to construct a stochastic matrix M such that g = fM. In particular, we can take m_jk = λ_j·μ_jk where ∑_k ≠ jμ_jkf^j_k = h_j and let m_jj = 1 - ∑_k ≠ j m_jk. Then, such M is a stochastic matrix and g = fM.
	
	
	For the second part, fix any f ∈ℰ_m and any j ≠ k, let
	
    f_λ≡ f + λ f^j(k), λ∈ [0,1].

	Since f_λ - f satisfies (<ref>) for all λ, Blackwell monotonicity implies 
	
    C(f) ≥ C(f_λ),   ∀λ∈ [0,1].

	If D^+C(f; f^j(k)) exists, then 
	
    D^+C(f; f^j(k)) = lim_λ↓ 0C(f_λ) - C(f)/λ≤ 0.




	Let B_jk be an m × m matrix such that b_jj = -1, b_jk = 1, and all other entries are equal to zero. Then for any f ∈ℰ_m,
	
    fB_jk = f^j(k).


The proof of Lemma <ref> is by algebra and thus omitted.


	Suppose C ∈𝒞_m is Lipschitz and satisfies (ii) in Theorem <ref>. Then for any 1 ≤ k ≤ m and E ∈ext_k-1(ℳ_m), there exists E' ∈ext_k (ℳ_m) such that for all λ∈ [0,1],
	
    fE' ≽_B (1-λ) fE' + λ fE ≽_B fE.

	And it further implies C(fE') ≥ C(fE).



	For any 1 ≤ k ≤ m and E ∈ext_k-1(ℳ_m), we show that there exists E' ∈ext_k (ℳ_m) such that (<ref>) holds.
	
	Since E is not a full rank matrix, there exists a column e^i such that at least two entries are equal to 1. Let e_ji = e_j'i = 1. Additionally, there are n-k+1 columns such that all the entries are equal to zero. Let one of such columns be e^i'. Let E' be a matrix such that e'_j'i'=1, e'_j'i=0 and all other entries are same as E. Note that E' has exactly n-k empty columns, i.e., E' ∈ext_k (ℳ_m). 
	
	Let B denote B_i'i as defined in Lemma <ref>. Note that when I_m is the identity matrix of size m,  I_m + λ B is a stochastic matrix for any λ∈ [0,1]. Observe that B^2 = - B and (I_m + λ B) · (I_m + B) = I_m+ B. Additionally, E' (I_m+B) = E and E'(I_m + λ B) = (1-λ) E' + λ E. Therefore, we have 
	
    (1-λ)fE' + λ f E = f E'(I_m + λ B), 
       fE = f E' (I_m + B) = fE'(I_m + λ B) ·  (I_m+B).

	Since I_m + λ B and I_m + B are stochastic matrices, (<ref>) holds. 
	
	Recall by Lemma <ref>, 
	
    fB = f^i'(i).

	(ii) then implies that for all λ∈ [0,1], 
	
    D^+(C((1-λ)fE' + λ fE), fE - ((1-λ)fE' + λ fE)) 
    
    		=    D^+(C((1-λ)fE' + λ fE), ((1-λ)fE' + λ fE)B)   ≤ 0.

	
	Finally, we show for such E and E', 
	
    C(fE') ≥ C(fE).

	For λ∈ [0,1], define the function ϕ(λ) = C((1-λ)fE' + λ fE). Notice that ϕ(0) = C(fE') and ϕ(1) = C(fE). ϕ is Lipschitz continuous over [0,1] as a composition of Lipschitz continuous functions. Thus ϕ is differentiable almost everywhere on [0,1] and satisfy 
	
    ϕ'(λ) = D^+C((1-λ)fE' + λ fE; fE - fE').

	Moreover, Lipschitz continuity implies absolute continuity and thus the Fundamental Theorem of Calculus (FTC) holds for ϕ on [0,1], which further implies that 
	
    C(fE) - C(fE')     = ϕ(1) - ϕ(0) 
    		= ∫_0^1ϕ'(λ) dλ
        =∫_0^1 D^+((1-λ)fE' + λ fE; fE - fE') dλ
        = ∫_0^11/1-λ D^+(C((1-λ)fE' + λ fE), fE - ((1-λ)fE' + λ fE)) dλ
       ≤ 0,

	where the second last equality uses positive homogeneity of D^+C(f;·) and the last inequality follows from that (<ref>) holds for all λ∈ [0,1].




  §.§.§ Proof



	Necessity follows directly from the definition and Lemma <ref>. 
	
	For sufficiency, Lemma <ref> implies that C(f) ≥ C(fE) for all E ∈ext(ℳ_m). Take any f,g ∈ℰ_m with f ≽_B g so that g = fM. By quasiconvexity of C, we have
	
    C(g) ≤max{C(fE): E ∈ext(ℳ_m) }≤ C(f),

	thus, C is Blackwell monotone.




§ PROOFS FOR SECTION <REF>




 §.§ Proof of Necessity in Theorem <ref>



	Suppose C is additively separable. First, given any f̂∈ [0,1]^n. For any k ∈, consider the following experiments, 
	
    f = [    f̂     0     ⋯     0 1- f̂ ]∈ℰ_k + 1,

	and 
	
    g = [  1/kf̂      ⋯  1/kf̂ 1 - f̂ ]∈ℰ_k+1.

	Observe that
	
    f  [ 1/k   ⋯ 1/k   0;   ⋮   ⋱   ⋮   0; 1/k   ⋯ 1/k   0;   0   ⋯   0   1;     ] = g   and    g   [ 1 0 ⋯ 0 0; ⋮ ⋮ ⋱ ⋮ 0; 1 0 ⋯ 0 0; 0 0 ⋯ 0 1;   ]= f,

	that is, f ≽_B g ≽_B f. Thus, Blackwell monotonicity implies that C(f) = C(g). Then additive separability implies 
	
    ψ(1/kf̂) = 1/kψ(f̂).

	
	Next, for any ℓ∈ such that ℓf̂∈ [0,1]^n. Consider the following experiments,
	
    f = [    ℓf̂ 1- ℓf̂ ]∈ℰ_2,

	and
	
    g = [     f̂      ⋯     f̂ 1- ℓf̂ ]∈ℰ_ℓ+1.

	By the same argument, Blackwell monotonicity implies that C(f) = C(g), and thus
	
    ψ(ℓf̂) = ℓψ(f̂).

	Together it implies that, for all f̂∈ [0,1]^n, for all z ∈ℚ such that z f̂∈ [0,1]^n,
	
    ψ(f̂) = z ψ(f̂).

	By density of ℚ in  and continuity of ψ(·), we have positive homogeneity of ψ over [0,1]^n. 
	
	Next, we show subadditivity, i.e., for any f̂, ĝ∈ [0,1]^n such that f̂ + ĝ∈ [0,1]^n, then 
	
    ψ(f̂ + ĝ) ≤ψ(f̂) + ψ(ĝ).

	Consider the following experiments, 
	
    f = [         f̂          ĝ 1 - f̂ - ĝ ]∈ℰ_3,

	and 
	
    g = [     f̂ + ĝ 1 - f̂ - ĝ ]∈ℰ_2.

	As g is obtained by merging the first two signals in f, we have f ≽_B g. Thus, Blackwell monotonicity implies that C(f) ≥ C(g), and thus sublinearity of ψ holds. 





 §.§ Proof of Proposition <ref>



	Let ℰ_m⊂ℰ_m denote the set of uninformative experiments in ℰ_m. Notice that f ∈ℰ_m if and only if 
	
    f^j∈{λ1: λ∈ [0,1]},   ∀ j.

	
	For sufficiency, given ∑_i=1^n a_i = 0 and a_i≠ 0 for all i, then for any f̂∈ [0,1]^n,
	
    ⟨ a, f̂⟩ = 0 ⇔f̂∈{λ1: λ∈ [0,1]}.

	This implies that C(f) is strictly grounded. 
	
	For necessity, C(f) = 0 for all f ∈ℰ_m implies that ∑_i=1^n a_i = 0. Next, towards a contradiction, suppose that ∑_i=1^n a_i = 0 yet a_1 = 0. Then the following experiment,
	
    f = [     λ 1 - λ;     1     0;     ⋮     ⋮;     1     0 ],

	is not in ℰ_2 for λ≠ 0 and C(f) = 0. Thus, a contradiction. 




§ PROOFS FOR SECTION <REF>




 §.§ Bargaining with Information Acquisition

Before proving Proposition <ref> and <ref>, we first show the following lemma which characterizes the buyer's optimal information acquisition strategy (we break indifference towards accepting the offer). 


	For all p ∈ [L, H] and μ∈ (0,1),
	
		
  * If C_λ(f) = λ(f_H - f_L)^2/2, then the buyer's optimal strategy satisfies either f_H^* = 1 or f_L^* = 0, or both. 
		
  * If C_λ(f) = λ| f_H - f_L|, then the buyer's optimal strategy is either full information acquisition, i.e., [f_L^*, f_H^*]= [0, 1] or no information acquisition, i.e., [f_L^*, f_H^*] = [0, 0] or [1,1].
	


Notice that when p ∉ [L,H] or μ = 0 or 1, the buyer's optimal strategy is always no information acquisition. 


	Recall the buyer's problem is to solve, for all p and μ, 
	
    max_[f_L, f_H]^⊺∈ℰ_2μ f_H (H-p) + (1-μ)f_L(L - p) - C_λ(f),

	under which the buyer accepts the offer after observing the signal h and rejects otherwise. 
	
	First, observe that for all μ∈ (0,1), if p = L, then the buyer's optimal strategy is no information acquisition and always accepts, i.e., f^*_L = f^*_H = 1; If p = H, then the buyer's optimal strategy is no information acquisition and always rejects, i.e., f^*_L = f^*_H = 0. 
	
	Next, we consider the case where p ∈ (L, H). Consider the auxiliary cost minimization problem that solves the minimum cost needed to achieve a given material payoff level w:
	
    min_[f_L, f_H]^⊺∈ℰ_2 C_λ(f)   s.t.  μ f_H (H-p) + (1-μ)f_L(L - p) = w.

	The feasible levels of w is given by [w, w] where w = max{0, μ(H-p)+(1-μ)(L-p)} and w = μ(H-p). That is, w is the optimal payoff level under no information, and w is the optimal payoff level under full information.
	
	Given w ∈ [w, w], the set of experiments that achieve w is given by the line segment
	
    f_H = - (1-μ)(L-p)/μ(H-p) f_L + w/μ(H-p).

	Because p ∈ (L, H) and μ∈ (0,1), the slope of this segment is positive and the intercept w/μ(H-p) is non-negative. 
	
	Notice that for both cost functions, their isocost curves are the same and in the form of f_H - f_L = c. Therefore, the buyer's optimal experiment must be on the boundary with either f^*_H = 1 or f^*_L = 0. Specifically, 
	
    [f_L^*, f_H^*] = 
    		
    			[μ(H-p) - w /(1-μ)(p-L), 1]    if μ(H-p) + (1-μ)(L - p)≥ 0,
    
    			[0, w/μ(H - p)]    if μ(H-p) + (1-μ)(L - p) < 0.

	This proves (i). 
	For (ii), let C_λ(f) = λ| f_H - f_L|. By (<ref>), it is without loss to restrict attention to experiments with f_H≥ f_L and thus the cost function can be simplified to C_λ(f) = λ (f_H - f_L). 
	
	Let [f^*_L(w), f^*_H(w)]^⊺ denote the optimal experiment that achieves payoff w solved from (<ref>). The buyer's problem is then to solve 
	
    max_w ∈ [w, w] w - λ (f_H^*(w) - f_L^*(w)).

	Notice that, both f^*_L(w) and f^*_H(w) are linear in w. We conclude that the optimal w is either w, under which [f_L^*,f_H^*] = [1,1] or [0,0]; or w, under which [f_L^*,f_H^*] = [0,1] .



	Let 
	
    C_λ(f) = λ/2(f_H - f_L)^2.

	Then notice that the buyer's optimal information acquisition strategy solves 
	
    max_w ∈ [w, w] w - λ/2(f_H^*(w) - f_L^*(w))^2.

	Suppose μ(H-p) + (1-μ)(L - p)≥ 0 and substituting (<ref>), we have 
	
    max_w ∈ [w, w] w - λ/2 (1 - μ(H-p) - w /(1-μ)(p-L) )^2.

	From this we can solve that the optimal w^* is given by 
	
    w^* = w   if  (1-μ)(p - L)≥λ ,
    w + (1-μ)^2(L - p)^2/λ   if  (1-μ)(p - L) < λ.

	Notice in the second case, the buyer acquires information in the optimal strategy. Specifically, the optimal experiment is given by f_H^* = 1 and 
	
    f_L^* = 1 - (1-μ)(p - L)/λ.

	The low-type seller is indifferent between offering p and L if and only if
	
    p ( 1 - (1-μ)(p - L)/λ) = L,

	which is equivalent to 
	
    p(1-μ) = λ.

	
	In summary, we claim that there is a non-pooling equilibrium where the buyer acquires information when there exists p and μ such that 
	
    p(1-μ) = λ, μ H ≥ p,  and μ > π.

	In this case, let σ(H) = δ_p, σ(L)(p) = π (1-μ)/μ (1-π) and σ(L)(L) = 1 - σ(L)(p). These are well-defined as μ > π. Then on the equilibrium path, if the buyer observes price p, the buyer's belief is exactly μ, and the other two conditions imply that 
	
    μ (H-p) ≥ p (1-μ) = λ≥ (1-μ)(p-L).

	The previous discussions thus imply that in this case, the buyer's optimal information acquisition exactly makes the low-type seller indifferent between offering p and L. 
	
	Finally, we show that it is always possible to find p and μ such that 
	
    p(1-μ) = λ, μ H ≥ p,  and μ > π,

	when λ < π(1-π)H. Letting p = λ/(1-μ), notice the second condition implies 
	
    λ≤μ(1-μ)H.

	As λ < π(1-π)H, by continuity, one can always find μ > π such that the above conditions hold.




	By Lemma <ref>, it is without loss to focus on experiments with f_H≥ f_L. In this case, we have 
	
    C_λ(f) = λ (f_H - f_L).

	
	We first show that such pooling equilibria are possible. Suppose both types of sellers offer price p^*≥ L. Then by Lemma <ref>, the buyer's optimal strategy is [f_L^*, f_H^*] = [1,1] if and only if
	
    π(H-p^*) + (1-π)(L-p^*) ≥ 0,

	and 
	
    π(H-p^*) + (1-π)(L-p^*) ≥π(H-p^*) - λ,

	where the first condition follows from (<ref>) and the second follows from achieving w = π(H-p^*) + (1-π)(L-p^*) ≥ 0 is more optimal than achieving w. From these two conditions, one can derive that p^*≤ L + ϵ where
	
    ϵ = min{π (H-L), λ/1-π}.

	This can be supported as a PBE by letting the buyer's off-path belief satisfy μ_p = 0 for all p ≠ p^* and thus the buyer accepts the offer only when p ≤ L. This gives the seller a worse payoff than offering p^*. 
	
	Next, we argue that there cannot be any separating equilibria. Suppose there is a separating equilibrium where the two types of sellers offer different prices H ≥ p_H > p_L≥ L. In this case, the buyer will not acquire any information and always accepts the offers. Then the low-type seller can profitably deviate by offering p_H instead of p_L, a contradiction. 
	
	Finally, we argue that there cannot be any equilibria where any type of seller mixes between two different prices. If the buyer always accepts both offers, then the seller will not be indifferent between these two prices, a contradiction. Thus, the only possibility is that the buyer does not accept with probability 1 under one of the offers, i.e., the buyer acquires information. By Lemma <ref>, the buyer must acquire full information and accept only the high-type seller's offer. As a result, neither type of seller would be indifferent between the two offers they possibly randomize, a contradiction.
	
	For the last statement of the proposition, notice (<ref>) implies that ϵ→ 0 as λ→ 0. 




 §.§ Costly Persuasion


 
	
	Consider α and β defined in Section <ref> as functions of f_1 along the line μ f_2 + (1-μ) f_1 = w:  
	
    α (f_1) ≡f_2f_1 = w - (1-μ) f_1 μ· f_1,    β (f_1) ≡1-f_11-f_2 = μ (1-f_1)μ - w + (1-μ) f_1.

	By taking derivatives and using w ≤  2μ < 1, we have 
	
    α'(f_1) = - wμ f_1^2 <0 ,  β'(f_1) = - (1-w)μ( μ -w+(1-μ)f_1 )^2 <0.

	Then, from (<ref>), Blackwell informativeness decreases along the line as f_1 increases. Then, from Blackwell monotonicity, C is minimized when f_1 is maximized. To achieve this maximization, (<ref>) needs to bind. 




	For μ≥ 1/2, we show that f_1 = f_2=1 is optimal in the main text.
	
	Now assume that μ < 1/2. By plugging the cost function in, (<ref>) is equivalent to 
	
    max_0 ≤ w ≤ 2μ w - w^24μ· h(μ)    where    h(μ) ≡μ (1-μ)^2  ( 1 - 2μ )^2  .

	Observe that for all 0 < μ  <1/2
	
    h'(μ) =2μ + (1-2μ) (1+μ^2) (1- 2μ)^3  >0 .

	Additionally, h(0) =0 and lim_μ→ 1/2 h(μ) = ∞. Therefore, there exists μ̂ such that h(μ̂) = 1. Then, the solution of the minimization problem (<ref>) subject to 0 ≤ w ≤ 2μ is 
	
    w^* = 
    			2μ, 				   if μ∈ (μ̂ , 1/2),	
    
    			2μ· h(μ), 	   if μ≤μ̂ .

	By plugging this into (<ref>), we have (<ref>). 





§ PROOFS FOR SECTION <REF>



 §.§ Proof of Theorem <ref>



	For any f,g ∈ℰ_m and λ∈ [0,1], define experiment
	
    λ f ⊕ (1-λ)g ∈ℰ_2m
 
	to be that with probability λ, it generates signals in {s_1, ⋯, s_m} according to f and with probability 1-λ, it generates signals in {s_m+1, ⋯, s_2m} according to g. Notice we can write such an experiment as 
	
    [    λ f (1-λ)g ]∈ℰ_2m.

	
	We next present a lemma showing that for every f,g ∈ℰ_m, if f ≽_B g, then there always exists a decreasing path from f to g in ℰ_2m.
	
	
		For any f, g ∈ℰ_m, if f≽_B g, then for all λ∈ [0,1],
		
    f ≽_Bλ f ⊕ (1-λ)g ≽_B g.

	
	
	
		We first show the lemma holds when f = g. Notice that for all λ∈ [0,1], 
		
    [ f 0 ][     λ I (1-λ) I;       0       I ] = 
    			[    λ f (1-λ)f ],

		and 
		
    [    λ f (1-λ)f ][ I 0; I 0 ] = 
    			[ f 0 ].

		
		Next, consider any f ≽_B g. Let g = fM for some stochastic matrix M ∈ℳ_m.  Consider the following stochasitc matrix in ℳ_2m: 
		
    [ λ I   0;   0   M ].

		Then we have 
		
    [    λ f (1-λ)f ][ I 0; 0 M ] = 
    			[     λ f (1-λ) g ].

		Thus, we have establishes that f ≽_Bλ f ⊕ (1-λ)g for all λ∈ [0,1]. 
		
		Consider another stochastic matrix in ℳ_2m:
		
    [ M 0; 0 I ].

		Then for all λ∈ [0,1], notice that 
		
    [    λ f (1-λ)g ][ M 0; 0 I ] = 
    			[      λ g (1-λ) g, ]

		where the last matrix is Blackwell equivalent to g. Thus, we have also established that λ f ⊕ (1-λ)g ≽_B g for all λ∈ [0,1].
	
	
	Suppose C ∈𝒞_2m is continuously differentiable and satisfies (i) and (ii) in Theorem <ref> over ℰ_2m. By differentaibility of C, we have for any f ∈ℰ_2m, 
	
    D^+C(f; f^j(k)) = ⟨∇ C(f), f^j(k) ⟩≤ 0, ∀ j ≠ k.

	Then by Lemma <ref>, for any g ∈ℰ_2m with f ≽_B g, we have 
	
    ⟨∇ C(f), g - f ⟩≤ 0,

	where the inequality follows as it is a positive linear combination of ⟨∇ C(f), f^j(k) ⟩. 
	
	Finally, for any f,g ∈ℰ_m with f ≽_B g, applying FTC along the decreasing path from f to g in ℰ_2m identified in Lemma <ref> implies C(f) ≥ C(g), and thus establishes the conclusion. 



