A Game Theoretical Approach to Design of Experiments

In an experiment, we change one or more process variables (or factors) in order to observe the effect their changes have on one or more response variables. The design of experiments (DOE) is an efficient procedure for planning experiments so that the data obtained can be analyzed to yield valid and objective conclusions. DOE begins with determining the objectives of an experiment and selecting the process factors for the study. An Experimental Design is the laying out of a detailed experimental plan in advance of doing the experiment. Well chosen experimental designs maximize the amount of "information" that can be obtained for a given amount of experimental effort. Design of experiments is thus a discipline that has very broad application across all the natural and social sciences and engineering. In this paper we present an experimental design problem as a Nash equilibrium problem in the context of Game Theory: the choice of two variables in n experiments is made by two players, each of them has to decide the location of his process variable far as possible from the opponent. This requirement allows to obtain a distribution of the points with high dispersion, in such a way to better explore all the available possible region, and can be interpreted as a particular non-cooperative game, a spatial competition, also known as Hotelling competition. The facility is identified with each single test point and the spatial domain corresponds to the admissible region. The optimal distribution is a Nash equilibrium solution of this game. In our model it turns out that the game has a peculiar structure, namely it is a potential game and the Nash equilibrium solutions will be the minimum points of a function that is called the potential function or potential in short. This aspect allow to prove existence results for the equilibrium solutions, and also to compute them by using a numerical procedure based on a genetic algorithm. A concrete application in aerospace engineering is discussed: the design of a test matrix in an envelope expansion flight test activity. We consider that the requirements of two different engineers categories can be formalized with two different objective functions. Structural Engineers, interested in assessment of aeroelastic properties, want to optimize distribution in Mach number and equivalent airspeed range (the latter being representative of dynamic pressure), while Systems Engineers, focused on environmental effects on the aircraft, want to optimize distribution in pressure altitude and equivalent airspeed range. We deal with an experiment with two design variables. Both categories of engineers share a common goal: exploring the entire envelope, thus distributing the test points as widely as possible in the flight envelope. We present the non-cooperative game facing this experimental design problem, together with existence and computational results.