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I have the following utility function: $$ u(x_1, x_2, x_3) = med(x_1, x_2, x_3) $$ Given that $UMG_{i}$ ≥ 0, the utility function represents a strictly monotonic preference. Does this assertion make sense?"

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I guess by $med$ you mean the median value. For example: $med(2,7,19) = 7$.

The preferences are not strictly monotonic because increasing only one of the goods, doesn’t necessarily increase the utility value. (Only increasing the maximum or, only increasing the minimum value by a bit (not enough to surpass the former median value), keeps the same utility value).

The sufficient condition you meant is $UMG_i > 0$.

Also, here we have $UMG_i \geq 0$ wherever it is defined, but there are points where $UMG_i$ isn’t defined as the function is not differentiable where at least two of the $x_i$’s are equal.

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