If the expected utility function
π([π1,π2,...,ππ])=π1π’1 +π2π’2 +β―+πππ’π represents preferences over lotteries βΏ, then expected utility function
π([π1,π2,...,ππ])=π1π£1 +π2π£2 +β―+πππ£π
also represents the same preferences βΏ if π£π = ππ’π + π for all π, where π > 0 and π are real
numbers.
Note: This shows that applying a strictly increasing linear transformation to the π’πβs does not change the preferences over lotteries being represented by the expected utility function. (Compare this to utility functions over outcomes, for which any strictly increasing transformation, not necessarily linear, doesnβt change the preferences over outcomes being represented.)