Evaluating Decision-Weighted Integrals Requires Function Definitions

Summary

The document asks how to evaluate an integral involving a utility function, a cumulative probability, and a decision-weighting function. The response explains that the expression cannot generally be evaluated without explicit definitions of these functions or relationships among them. This is a basic but important point for probability-weighted utility models: symbolic notation alone does not determine a numerical value or an integrable form.

It then gives the differential chain rule for a weighting function composed with a probability function, including the sign change when the argument is one minus that probability. These identities convert the differential into derivatives with respect to the underlying variable, which may make ordinary integration techniques applicable once the functions are specified. The answer also notes ambiguity in the original notation. It does not evaluate a particular integral or discuss boundary conditions, convergence, or assumptions on differentiability, so those would need to be supplied for a concrete calculation.

Key ideas

• An integral involving unspecified utility, weighting, and probability functions cannot be evaluated uniquely.
• The differential of a composed weighting function can be expanded using the chain rule.
• Differentiating a weighting function of one minus a probability introduces a negative sign.
• The resulting integral still depends on explicit functions and suitable regularity assumptions.

Tags

# How to evaluate the following integral?


# How to evaluate the following integral?












I stumbled across an expression and I wonder how to evaluate this: $-\int_ {0} ^ {+\infty} {v(x)} dw^{+} (1-p(x))$ where $v(x)$ is some utility function and $w(p(x)) $ is a decision weighting function, dependent on some cumulative probability $p(x)$? Perhaps it's obvious but I am too blind to see it. Thanks a lot for any ideas! EDIT:Expression has been corrected.

## Answer by Kurt G. (score 1, accepted)

https://quant.stackexchange.com/a/68252

Without explicit knowledge of the functions $v,w,p$ or some relationships between them you cannot evaluate this integral. With explicit knowledge (or relationships) you can apply the rules $$ dw(p(x))=w'(p(x))\,p'(x)\,dx\,,\quad\quad dw(1-p(x))=-w'(1-p(x))\,p'(x)\,dx\, $$ to see if you get something that can be integrated. I gave both formulas because your formulas involving $w$ and $p$ are ambiguous.

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