رؤى - Computational Complexity - # Blackwell Monotonicity of Information Cost Functions

Characterizing Blackwell-Monotone Information Cost Functions

Q: How can the characterization of Blackwell-monotone costs be extended to the case of infinite states and/or signals

To extend the characterization of Blackwell-monotone costs to the case of infinite states and/or signals, we can leverage the concept of embedding the space of experiments into a higher-dimensional space. By embedding the space of experiments with a finite number of states and signals into a higher-dimensional space, such as E2m, where m is the number of signals, we can ensure the existence of a decreasing path connecting experiments in the original space. This approach allows us to utilize the necessary conditions derived for finite experiments to establish sufficiency for infinite states and/or signals. By extending the cost function defined on the original space to the higher-dimensional space, we can apply the characterization results for Blackwell-monotone costs in a more general setting.

Q: What are the implications of relaxing the quasiconvexity assumption in the general case with more than two signals

Relaxing the quasiconvexity assumption in the general case with more than two signals can have significant implications for the characterization of Blackwell-monotone costs. Quasiconvexity is a key condition that ensures the local minimum is also a global minimum, which is essential for establishing sufficiency for Blackwell monotonicity. When the quasiconvexity assumption is relaxed, the characterization of Blackwell-monotone costs may become more challenging. Without the guarantee of global optimality, the identification of all extreme points of the sublevel set under the Blackwell information order may not be straightforward. This relaxation could lead to the existence of multiple local minima, potentially resulting in a more complex analysis of Blackwell-monotone costs with more than two signals.

Q: Are there connections between the sublinearity condition for additively separable Blackwell-monotone costs and the properties of the underlying decision problems where these costs are applied

The sublinearity condition for additively separable Blackwell-monotone costs has significant implications for the underlying decision problems where these costs are applied. Sublinearity of the component function ψ in additively separable costs ensures that the cost function is positively homogeneous and subadditive, which are essential properties for Blackwell monotonicity. In decision problems involving costly information acquisition, the sublinearity of the cost function implies that the cost decreases at a decreasing rate as the informativeness of the experiment increases. This property aligns with the intuitive notion that acquiring additional information becomes less costly as the information becomes more informative. Therefore, the sublinearity condition not only characterizes Blackwell-monotone costs but also reflects the nature of information costs in decision-making scenarios.

المفاهيم الأساسية

Necessary and sufficient conditions for information cost functions to be Blackwell monotone, including a characterization of additively separable Blackwell-monotone costs.

الملخص

The paper provides a characterization of Blackwell-monotone information cost functions, which assign higher costs to more informative statistical experiments according to Blackwell's classical information order.

Key highlights:

For binary experiments, Blackwell monotonicity is equivalent to the cost function being permutation invariant and satisfying certain directional derivative conditions.
For experiments with more than two signals, Blackwell monotonicity is characterized by the cost function being permutation invariant and satisfying a system of linear differential inequalities, along with a quasiconvexity assumption.
For additively separable information cost functions, Blackwell monotonicity is equivalent to the component function being sublinear (positively homogeneous and subadditive). This identifies a broad class of Blackwell-monotone costs, including norm costs, absolute-linear costs, and existing costs like entropy and posterior-separable costs.
The characterizations provide a tractable method to verify Blackwell monotonicity of arbitrary information cost functions and enable the construction of novel Blackwell-monotone costs.
The results are applied to study bargaining and persuasion problems with costly information, showing how the general framework can strengthen existing insights by relaxing additional assumptions.

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الرؤى الأساسية المستخلصة من

Blackwell-Monotone Information Costs

by Xiaoyu Cheng... في arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.15158.pdf

استفسارات أعمق

How can the characterization of Blackwell-monotone costs be extended to the case of infinite states and/or signals

To extend the characterization of Blackwell-monotone costs to the case of infinite states and/or signals, we can leverage the concept of embedding the space of experiments into a higher-dimensional space. By embedding the space of experiments with a finite number of states and signals into a higher-dimensional space, such as E2m, where m is the number of signals, we can ensure the existence of a decreasing path connecting experiments in the original space. This approach allows us to utilize the necessary conditions derived for finite experiments to establish sufficiency for infinite states and/or signals. By extending the cost function defined on the original space to the higher-dimensional space, we can apply the characterization results for Blackwell-monotone costs in a more general setting.

What are the implications of relaxing the quasiconvexity assumption in the general case with more than two signals

Relaxing the quasiconvexity assumption in the general case with more than two signals can have significant implications for the characterization of Blackwell-monotone costs. Quasiconvexity is a key condition that ensures the local minimum is also a global minimum, which is essential for establishing sufficiency for Blackwell monotonicity. When the quasiconvexity assumption is relaxed, the characterization of Blackwell-monotone costs may become more challenging. Without the guarantee of global optimality, the identification of all extreme points of the sublevel set under the Blackwell information order may not be straightforward. This relaxation could lead to the existence of multiple local minima, potentially resulting in a more complex analysis of Blackwell-monotone costs with more than two signals.

Are there connections between the sublinearity condition for additively separable Blackwell-monotone costs and the properties of the underlying decision problems where these costs are applied

The sublinearity condition for additively separable Blackwell-monotone costs has significant implications for the underlying decision problems where these costs are applied. Sublinearity of the component function ψ in additively separable costs ensures that the cost function is positively homogeneous and subadditive, which are essential properties for Blackwell monotonicity. In decision problems involving costly information acquisition, the sublinearity of the cost function implies that the cost decreases at a decreasing rate as the informativeness of the experiment increases. This property aligns with the intuitive notion that acquiring additional information becomes less costly as the information becomes more informative. Therefore, the sublinearity condition not only characterizes Blackwell-monotone costs but also reflects the nature of information costs in decision-making scenarios.