I am an engineer who is working in networking and I have a question about the stuff that is called "Utility Function".
I have been researching over some scientific papers in my field and see that people ussually consider the transmission rate of each user as a kind of utility function.
After that, they formulate some sort of utility maximization problem in the form.
$\begin{array}{*{20}{c}} {{\rm{Max}}}&{{R_1} + {R_2} + {R_3}} \end{array}$
Where $R_1,R_2,R_3$ are transmission rates of each user. In my opinion, utility is seemingly something that the user really really prefer to have more.
So my questions are as follows:
1/ Are there any mathematical requirements (for example: convexity, shape, monotonicity, integrability, continuity, differentiability, etc) of the function $f(x)$ so that it can be considered as an utility function ?
2/ In my Virtual Reality engineering case, the users do not want to increase their transmission rate but they want to reduce their latency. As a consequence of that, if I have these latencies as $L_1$, $L_2$ and $L_3$, can I consider these so called "inverse latency" functions $\frac{1}{{{L_1}}}$, $\frac{1}{{{L_2}}}$, $\frac{1}{{{L_3}}}$ as three utility functions ?
I am worrying so much about what my happens at zero because it might blow up.
3/ If the three inverse latency function in question two can be considered as utility functions, can this problem be consider as a utility maximization problem
$\begin{array}{*{20}{c}} {{\rm{Max}}}&{\frac{1}{{{L_1}}} + \frac{1}{{{L_2}}} + \frac{1}{{{L_3}}}} \end{array}$
Thank you very much for your enthusiasm !
A utility function assigns a numerical value to each element in some choice/outcome set facing an individual, ranking these elements in accordance with the preferences of that person.
In order to be able to represent a preference relation by a utility function, the preference relation must be "rational". This has a specific meaning. It means that the preference relation is
a) Complete, in that the person can express a preference over any two elements of the choice/outcome set (including the "indifference" case)
b) Transitive, in that if I prefer A to B and B to C, then I necessarily prefer A to C.
This kind of preferences can be represented by a utility function $U$, and $U$ must be such that if whenever A is weakly preferred to $B$, we will have $U(A) \geq U(B)$.
In case $A$ and $B$ are themselves numerical values, a strictly increasing function will do.
In the OP's case, since "latency" is "undesirable" one could use their reciprocal as a utility function. But many other strictly monotonic functions could do, for example $U(L) = e^{-L}$ which ranges in $[0,1]$.
Summing utilities of different persons,
a) Implicitly makes use of the "cardinal utility" concept (where a utility of 20 means double a utility if 10), and not of "ordinal utility" (where we just rank the different outcomes but we do not make quantitative comparisons between them).
b) Creates implicitly a layer of preferences over the importance (or not) of each individual. If we simply sum, the low utility of one can be fully compensated by increasing the utility of the other, since we only care about the sum of all. In a social context, under simple summing, if there is wealth 100, then the distribution of this wealth, its allocation among individuals does not matter.