# Statistical game

A two-person zero-sum game in which player $ I $
is interpreted as nature, player $ II $
as a statistician, player $ I $'
s strategy is a random process, player $ II $'
s strategy is a decision rule (decision function), and the pay-off function of player $ I $
is the loss function ( "risk" ) of the statistician. Player $ I $'
s mixed strategy will then be a probability measure on the set of random processes, player $ II $'
s mixed strategy will be a randomized decision rule, and player $ I $'
s pay-off function is defined as the mathematical expectation of the statistician's loss function. Statistical games arose out of problems of mathematical statistics (for example, the problems of estimating parameters, testing hypotheses, etc.), in which the minimax test is used (see Minimax principle). A systematic description of problems of mathematical statistics (including problems of sequential analysis) as statistical games was first introduced by A. Wald, who proved the validity of the minimax theorem for a broad class of statistical games (see [1]). This theorem provides a method for solving problems of mathematical statistics, since in a number of cases it is easier and more convenient to find the maximin than the minimax, which, in turn, makes it easier to find an optimal randomized rule.

#### References

[1] | A. Wald, "Statistical decision functions" , Wiley (1950) |

[2] | D. Blackwell, M.A. Girshick, "Theory of games and statistical decisions" , Dover, reprint (1979) |

#### Comments

Statistical games are considered when one deals with worst case designs. A modern version of statistical games lies at the heart of $ H ^ \infty $ control (a subject within automatic control, cf. also $ H ^ \infty $ control theory), see [a1].

#### References

[a1] | J. Doyle, K. Glover, P. Khargonekar, B. Francis, "State-space solutions to standard and control problems" IEEE Trans. Autom. Control , AC34 : 8 (1989) pp. 831–847 |

[a2] | J.L. Savage, "The foundation of statistics" , Wiley (1954) |

[a3] | J.W. Milnor, "Games against nature" R.M. Thrall (ed.) C.H. Coombs (ed.) R.L. Davis (ed.) , Decision processes , Wiley (1954) pp. 49–59 |

**How to Cite This Entry:**

Statistical game.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Statistical_game&oldid=48813