Multi-value Function: What & Why

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SUMMARY

Multi-valued functions, also known as correspondences in set-valued analysis, can be transformed into single-valued functions by mapping sets to the power set of another set. This transformation is particularly useful in microeconomics, where complex problems can be reframed as inquiries about these correspondences. In complex analysis, multi-valued functions often appear as inverses, with the "principal value" being a conventionally chosen single value. Understanding these concepts is essential for applying multi-valued functions effectively in various mathematical contexts.

PREREQUISITES
  • Understanding of set theory and power sets
  • Familiarity with multi-valued functions and their properties
  • Basic knowledge of microeconomics and its mathematical applications
  • Introduction to complex analysis and principal values
NEXT STEPS
  • Research the applications of correspondences in microeconomics
  • Explore the mathematical properties of multi-valued functions
  • Study the concept of principal values in complex analysis
  • Examine set-valued analysis techniques and their implications
USEFUL FOR

Mathematicians, economists, and students of advanced mathematics interested in the applications and implications of multi-valued functions in various fields.

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Why multi-valued function is a function?
 
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You can always make a multi-valued function $f$ between sets $X$ and $Y$ single-valued by considering the associated function (in the narrow, traditional sense) that maps $X$ to the power set of $Y$.

In the context of set-valued analysis (which has many applications in e.g. microeconomics), the multi-valued functions are often called "correspondences". Some problems from the application domain can then be translated elegantly into questions about those correspondences. If you are interested, I can provide more references.

In other contexts, such as complex analysis, multi-valued functions often arise as inverses, and then one typically make a choice by convention and calls it the "principal value".
 

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