Why Expected Utility Is Preserved Only by Linear Transformations

Summary

The document distinguishes utility representations under certainty from expected utility under uncertainty. Under certainty, any increasing transformation of a utility function preserves the ranking of preferences. With lotteries, preferences are represented by probability-weighted averages of utility, so changing utility values nonlinearly can change the ranking of uncertain outcomes.

The example describes a lottery with two possible outcomes and computes its expected utility by weighting each outcome’s utility by its probability. This illustrates why positive affine transformations preserve the same expected-utility representation, whereas arbitrary increasing transformations generally do not. The discussion is conceptual and points readers toward a textbook chapter for a fuller treatment; it does not develop the formal axioms behind von Neumann–Morgenstern utility.

Key ideas

• Under certainty, any increasing transformation preserves the preference ranking represented by a utility function.
• Expected utility evaluates lotteries using probability-weighted utility values.
• Nonlinear transformations can change the ranking of lotteries.
• Positive affine transformations preserve an expected-utility representation.

Tags

# Expected Utility


# Expected Utility












We know that under certainty, any increasing monotone transformation of a utility function is also a utility function representing the same preferences. Under uncertainty, we must restrict this statement to linear transformations if we are to keep the same preference representation.

Now the problem is that I don't know how to give to this concept a mathematical and a economic explanation. I know that Von Neumann - Morgenstern utility function is used in these cases, but what this means? Can anybody help me, maybe give me an exhausting and understandable reference? Thanks in advance!

## Answer by phdstudent (score 2, accepted)

https://quant.stackexchange.com/a/19337

The best explanation I came across so far is the one in Gravelle and Rees (2003) chapter 17. I could exactly write here what they state, but that would be copying.

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/19341

Under uncertainty we have to deal with "lotteries" where for example with 75% chance you get A and with 25% chance you get B and you have to compute expected utility 0.75*U(A)+0.25*U(B). It is clear that transformations of the utility function are going to create problems (i.e. different outcomes) unless they are linear.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.