Increasing Risk and Increasing Informativeness: Equivalence Theorems

When considering problems of sequential decision making under uncertainty, two of the most interesting questions are: How does the value of the optimal decision variable change with an increase in risk? How does the value of the optimal decision variable change with a more informative signal? In this paper we show that, if the payoff function is linear in the random variable, then one model can simultaneously answer both questions. This result holds for the reaction functions and equilibria of non-cooperative games, as well as for single decision makers, with virtually no restrictions on the payoff functions. This is useful because otherwise it is very difficult to get at general results on the impact of learning. Furthermore, we clarify why the impacts of risk and a more informative signal are different when the payoff function is non-linear in the random variable. It is because the directional impacts of informativeness are independent of risk attitude; the impacts of risk are not.