<p>The Rawlsian welfare function, which takes the form of the min of all agent's utility, is often called the <a href="https://plato.stanford.edu/entries/original-position/" rel="nofollow noreferrer">maximin function</a>, 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. </p>

<blockquote>
  <p>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.</p>
</blockquote>

<p><a href="https://people.richland.edu/james/summer02/m160/decision.html" rel="nofollow noreferrer">Jones (2002), Decision Theory Notes</a> </p>

CC BY-SA 4.0
2019-04-05T18:44:17.070Z
+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.
Maarten Punt
CC BY-SA 4.0
2019-04-05T15:48:01.043Z
2019-04-05T15:48:01.043Z
BKay
<p>What is a standard term for utility functions of the type:</p>

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

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

<p>This is similar to <a href="https://en.wikipedia.org/wiki/Leontief_utilities" rel="nofollow">Leontief utilities</a>, only with max instead of min.</p>

<p>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.</p>

CC BY-SA 3.0
2015-11-29T19:29:01.060Z
Perhaps *anti-Rawlsian*? `:)` since the [Rawlsian social welfare function](https://en.wikipedia.org/wiki/Social_welfare_function#Cardinal_social_welfare_functions) is of the form $\min(u_1,\dots,u_m)$.
Herr K.
CC BY-SA 3.0
2015-11-29T21:17:23.837Z
Cant you transform this into a Leontieff? i.e. shouldn't this representation be equivalent to a min over the  negative x's?
ChinG
CC BY-SA 3.0
2015-12-01T10:53:43.757Z
@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.
Erel Segal-Halevi
<utility><definition>
A term for utility functions based on the max operator
Erel Segal-Halevi
# A term for utility functions based on the max operator

(Asked by: username_0 on 2015-11-29)

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](https://en.wikipedia.org/wiki/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.

**Tags:** utility, definition

> **Comments:**
> **username_1**: Perhaps *anti-Rawlsian*? `:)` since the [Rawlsian social welfare function](https://en.wikipedia.org/wiki/Social_welfare_function#Cardinal_social_welfare_functions) is of the form $\min(u_1,\dots,u_m)$.  
> **username_2**: Cant you transform this into a Leontieff? i.e. shouldn't this representation be equivalent to a min over the  negative x's?  
> **username_0**: @username_2 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.  

---

## 1 answers

### Answer by username_3 on 2019-04-05. Score: 4 | Accepted answer

The Rawlsian welfare function, which takes the form of the min of all agent's utility, is often called the [maximin function](https://plato.stanford.edu/entries/original-position/), 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](https://people.richland.edu/james/summer02/m160/decision.html)

> **Comments:**
> **username_4**: +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.  

---

