What is a standard term for utility functions of the type:

$$ u(x_1,\dots,x_m) = \max(\frac{x_1}{w_1},\dots,\frac{x_m}{w_m}) $$

where $x_i$ is the amount of commodity type $i$, and $w_i$ is a constant weight?

This is similar to Leontief utilities, only with max instead of min.

What term describes the type of goods with such a utility function? Initially I thought they were called "perfect substitutes", but now I see that this term is used for linear utility functions.

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    $\begingroup$ Perhaps anti-Rawlsian? :) since the Rawlsian social welfare function is of the form $\min(u_1,\dots,u_m)$. $\endgroup$ – Herr K. Nov 29 '15 at 19:29
  • $\begingroup$ Cant you transform this into a Leontieff? i.e. shouldn't this representation be equivalent to a min over the negative x's? $\endgroup$ – ChinG Nov 29 '15 at 21:17
  • $\begingroup$ @ChinG probably yes, but what is the meaning of negative quantities? I don't think such function has the properties of Leontief utility. For example, with Leontief utilities, the products are complementaries, and with the maximum utility function. they are not. $\endgroup$ – Erel Segal-Halevi Dec 1 '15 at 10:53

The Rawlsian welfare function, which takes the form of the min of all agent's utility, is often called the maximin function, because it maximizes minimum utility. In the spirit of that nomenclature, your function is a maximax utility function. And, apparently, this function does see some use. It is sometimes called the optimistic decision criteria, because a person evaluating projects based on their best outcome uses a similar evaluation function.

The maximax looks at the best that could happen under each action and then chooses the action with the largest value. They assume that they will get the most possible and then they take the action with the best best case scenario. The maximum of the maximums or the "best of the best". This is the lotto player; they see large payoffs and ignore the probabilities.

Jones (2002), Decision Theory Notes

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  • $\begingroup$ +1 although I disagree with the lotto player statement, at least in the context of this utility function. The $w_i$ could represent a weight according to probabilities. $\endgroup$ – Maarten Punt Apr 5 '19 at 18:44

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