The Savage principle and accounting for outcome in single-criterion nonlinear problem under uncertainty

Abstract

In the middle of the last century the American mathematician and statistician professor of Michigan University Leonard Savage (1917-1971) and the well-known economist, professor of Zurich University (Switzerland) Jurg Niehans (1919-2007) independently from each other suggested the approach to decision-making in one-criterion problem under uncertainty (OPU), called the principle of minimax regret. This principle along with Wald principle of guaranteed result (maximin) is playing the most important role in guaranteed under uncertainty decision-making in OPU. The main role in the principle of minimax regret is carrying out the regret function, which determines the Niehans-Savage risk in OPU. Such risk has received the broad extension in practical problems during last years. In the present article we suggest one of possible approaches to finding decision in OPU from the position of a decision-maker, which simultaneously tries to increase the payoff (outcome) and to reduce the risk (i.e., “to kill two birds with one stone in one throw”). As an application, an explicit form of such a solution was immediately found for a linear-quadratic variant of the OPU of a fairly general form.