Why Linear Expressions in Discounting and in Empathy: A Symmetry-Based Explanation

People’s preferences depend not only on the decision-maker’s immediate gain, they are also affected by the decision-maker’s expectation of future gains. A person’s decisions are also affected by possible consequences for others. In decision theory, people’s preferences are described by special quantities called utilities. In utility terms, the above phenomena mean that the person’s overall utility of an action depends not only on the utility corresponding to the action’s immediate consequences for this person, it also depends on utilities corresponding to future consequences and on utilities corresponding to consequences for others. These dependencies reflect discounting of future consequences in comparison with the current ones and to empathy (or lack of) of the person toward others. In general, many formulas involving utility are nonlinear, even formulas describing the dependence of utility on money. However, surprisingly, for discounting and for empathy, linear formulas work very well. In this paper, we show that natural symmetry requirements can explain this linearity.