Warranty problem: its statistical and game theoretic aspects

Manufacturers of consumer products, such as automobiles, usually offer warranties guaranteeing the product or its parts, for example, for five years or 50,000 miles, whichever comes first. There are at least two issues of interest to applied mathematicians and statisticians that arise from warranty considerations. The first is the specification of an optimum price-warranty combination, and the second is the forecast of a reserve fund to meet warranty claims against the product. The former involves a consideration of the item's reliability, its rate of usage, the consumer's attitude toward a specific warranty, and the competitor's actions, all of which lead towards a game theoretic formulation of the problem. The latter involves the analysis of a special kind of time series in two-dimensions, for which the use of dynamic linear models appears to be natural. This is an expository paper. Due to the timeliness of the topic, the focus here is on problem definition, its scope, and its formulation in a manner that facilitates analysis. Imbedded in the discussion are strategies for addressing some of the ensuing issues. The overall goal is to enlighten the reader about an area of research in the mathematical sciences that has an impact on industrial competitiveness and the manufacturing sciences.