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If someone has the attitude that they want to maximize their worst possible outcome (so they are maximally risk-averse), what does the utility function for that look like? Can this attitude be expressed as a utility function?

Maybe as a function defined by a limit?

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Gilboa and Schmeidler (1989) is the canonical reference for such a function. The paper provides an axiomatic foundation for such a function being used to represent a maximin preference.

The following (taken from this lecture note) is a succinct version:

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It should be noted that maximin expected utility usually appears in the context of ambiguity aversion, not risk aversion, although the two types of aversion are indeed related.

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  • $\begingroup$ I am being nit-picky here. But I claim that Gilbo and Schmeidler's function is not the maxmin function OP asked about. OP asked about an agent who is maximally risk-averse, while Gilboa and Schmeidler axiomatize an agent who is maximally (knightian) uncertainty-averse. Risk and Knightian uncertainty are different things. $\endgroup$ – brunosalcedo Aug 12 '19 at 15:05
  • $\begingroup$ @brunosalcedo: Thanks for the comment. As the last sentence in my answer makes clear, GS's function is typically used in the context of ambiguity aversion, not risk aversion. In my interpretation of OP's question, I saw the possibility of OP being confused about the notions of risk aversion and ambiguity aversion, as such confusion is quite common. That was what led me to infer that GS's function is what the OP asked for. OP's acceptance of my answer is an indication that I was not far off. $\endgroup$ – Herr K. Aug 12 '19 at 17:28
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You can simply apply what you know about perfect complements. If we have $C_i$ for $i=1,2,...,n$, where $C_i$ is consumption in the $i$th state of the world, each state of the world occuring with some probability, the utility function $U=\min\{C_1,C_2,...,C_n\}$ will ensure that the consumer will choose to have equal consumption in every state of the world.

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