# How to have same utility function for two persons?

I have a question regarding utility functions:

Utility can be defined as follows:

$U=1+e^{\frac{x}{RT}}$

U:Utility

x: What we want to find the utility for (Certain equivalent)

RT: Risk tolerance

My question is what conditions must be upheld in order for two persons to create the same utility functions for a bet?

Example:

Person A bet \$5 that horse number 1 wins. Person B bet \$20 that horse number 2 do not win. Is it then possible that their utility functions is the same given that their probabilities for horse 1 winning is the same and that they have the same risk tolerance?

• Do you really want to know if they have the same utility functions? Because then they have to have, you know, the same utility functions. I suppose what you are really asking is "what conditions must hold for two people to have the same preferences over lotteries?". Please confirm, as the question doesn't make much sense at written. – BKay Nov 21 '16 at 16:00

Strictly speaking, infinitely different utility functions can give the same preferences over lotteries. If person A has utility function $f(x)$ then person B with utility function $g(x)=A+B\cdot f(x)$, $B>0$ will make identical decisions over all lotteries. There would be no way to know which individual was actually using utility function $f(x)$ and which was using $g(x)$ because they would always make the same decisions.