Utility

[HalRVarian]

Consumer has a utility function U(x; y; z) =
median{x; y; z}.

 Find the demand function for x.

[Granpa Simpson]

The utility derived from consumption of x,y,z can be differentiated as:
U(x,y,z)=x if y<x<z or z<x<y................................(1).
U(x,y,z)=y if z<y<x or x<y<z................................(2).
U(x,y,z)=z if x<z<y or y<z<x................................(3).
For (1) utility is dependent solely on x so the demand function will be x=M/Px, y=0, z=0.
Similarly for (2) it will be y=M/py, x=0, z=0.
for (3) it will be z=M/Pz, y=0, x=0.
Dont know weder the approach is correct or not..plz do check and let me know..!!!!

[Dreyfus]

I m not 100% sure for this,
x = Px/M  for y<x<z or z<x<y
Where Px is price of good x and M is income

[Gayyam]

Let the prices be px, py, pz.

So, given that the consumer wants to maximise his utility of med{x,y,z}, he'll care about only maximising the median value of x,y,z.
Since the median value is to maximised, by definition, all the numbers above the median should be at least as large as the median.
Now, in a set of three numbers, the middle value will be the median and one other value will be greater than or equal to the median. So, if you want to maximise the median, then the best strategy would be to increase the median value only so much so that it remains the median. i.e. not greater than the largest value.
Thus, without loss of generality, if px≤py≤pz then the best strategy would be to consume the least expensive two goods, to maximise the median. Thus, in this case, the maximum would occur at x=y.