Chapter 21 Game theory and statistics Game theory in particular the theory of two person zero sum games has played a multiple role in statistics. Its principal role has been to provide a unifying framework for the various branches of statistical inference. Statistics regarded from the game theoretic point of view became known as decision theory. While unifying statistical inference decision theory has also proved useful as a tool for weeding out procedures and approaches that have taken hold in statistics without good reason. On a less fundamental level game theory has contributed to statistical inference the minimax criterion. While the role of this criterion in two person zero sum games is central its application in statistics is problematic. Its justification in game theory is based on the direct opposition of interests between the players as expressed by the zero sum assumption. Together with the minimax criterion randomized or mixed strategies also appear in decision theory. The degree of importance of randomization in statistics differs according to which player is randomizing. Mixed strategies for Nature are a priori distributions. In the Bayes approach these are assumed to represent the Statisticians states of knowledge prior to seeing the data rather than Natures way of playing the game. Therefore they are often assumed to be known to the Statistician before he or she makes his or her move unlike the situation in the typical game theoretic set up. Mixed strategies for the Statistician on the other hand are strictly speaking superfluous from the Bayesian point of view while according to the minimax criterion it may be advantageous for the Statistician to randomize and it is certainly reasonable to grant him or her this option.