Survey sampling : a linear game

A linear game consists of two subsets of a vector space with a scalar product. The idea is that players 1 and 2 select, independently, elements of the first and second set, respectively. Then, player 2 has to pay to player 1 the value of the scalar product of the selected elements. We will discuss survey sampling within the framework of linear games with the statistician in the role of player 2. The vector space to be considered is the set of all symmetric matrices of order N x N with a scalar product identical with the usual mean squared error. The subset from which the statistician's selection is to be made is neither convex nor compact. Standard results of the theory of linear games have to be modified appropriately. The existence of minimax strategies will be established. At the same time we hope to improve our understanding of random selection and of the duality between the fixed population approach and model based approaches to the theory of survey sampling.