# 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.

The only parameterization given in this utility function is the risk tolerance. When two agents happen to have the same risk tolerance they would also have the same utility function.

Regarding your Example: They could have the same risk tolerance depending on their wealth. If 5 Dollar depict 1 % of the wealth of person A and if 20 Dollar aswell amounts for 1 % for the wealth of person B they could have the same risk aversion. If however their wealth does not coincide the risk aversion of the two agents ought do be different from each other.

• Okay, thank you very much for your reply. Does this mean that the probability does not matter in the case when they have the same utility function or not? What about when we find the expected utility of the game for both person A and person B, and then putting this value into the utility formula above to calculate the certainty equivalent for the bet. Would not the probability change the utility functions (because it changes the expected utility and therefore also the certain equivalent)? – David May 23 '16 at 8:42
• What I meant was: Do they have the same utility functions if person A and person B have different probability that horse number 1 wins given that they have the same risk tolerance? – David May 23 '16 at 12:40