Theory of Games and Statistical Decisions

David A. Blackwell, M. A. Girshick

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486638317
• eBook ISBN: 9780486150895
• Topic: Mathematics
• Subtopic: Game Theory
• Index: Mathematics

Evaluating statistical procedures through decision and game theory, as first proposed by Neyman and Pearson and extended by Wald, is the goal of this problem-oriented text in mathematical statistics. First-year graduate students in statistics and other students with a background in statistical theory and advanced calculus will find a rigorous, thorough presentation of statistical decision theory treated as a special case of game theory.
The work of Borel, von Neumann, and Morgenstern in game theory, of prime importance to decision theory, is covered in its relevant aspects: reduction of games to normal forms, the minimax theorem, and the utility theorem. With this introduction, Blackwell and Professor Girshick look at: Values and Optimal Strategies in Games; General Structure of Statistical Games; Utility and Principles of Choice; Classes of Optimal Strategies; Fixed Sample-Size Games with Finite Ω and with Finite A; Sufficient Statistics and the Invariance Principle; Sequential Games; Bayes and Minimax Sequential Procedures; Estimation; and Comparison of Experiments.
A few topics not directly applicable to statistics, such as perfect information theory, are also discussed. Prerequisites for full understanding of the procedures in this book include knowledge of elementary analysis, and some familiarity with matrices, determinants, and linear dependence. For purposes of formal development, only discrete distributions are used, though continuous distributions are employed as illustrations.
The number and variety of problems presented will be welcomed by all students, computer experts, and others using statistics and game theory. This comprehensive and sophisticated introduction remains one of the strongest and most useful approaches to a field which today touches areas as diverse as gambling and particle physics.

CHAPTER 1

Games in Normal Form

1.1. Introduction

A game is characterized by a set of rules having a certain formal structure, governing the behavior of certain individuals or groups, the players. Chess and bridge are examples of games in the sense considered here, and will be used for illustrative purposes.

Broadly speaking, the rules provide that the game shall consist of a finite sequence of moves in a specified order, and the nature of each move is prescribed. Moves are of two kinds, personal moves and chance moves. A personal move is a choice by one of the players of one of a specified, possibly infinite, set of alternatives; for instance, each move in chess is a personal move; the first move is a choice by White of 1 of 20 specified alternatives. The actual decision made in a particular play of a game at a given personal move we shall call the choice at that move. A chance move also results in the choice of one of a specified set of alternatives; here the alternative is selected not by one of the players, but by a chance mechanism, with the probabilities with which the mechanism selects the various alternatives specified by the rules of the game. For instance, the first move in bridge consists of dealing the first card to a specified player. This is a chance move with 52 alternatives; the rules require that each alternative shall have probability

of being selected. The actual selection made in a particular play of a game at a given chance move we shall call the outcome at that move.

In terms of moves, the rules of a game have the following structure. For the first move, the rules specify whether it is to be a personal move or a chance move. If it is a personal move, the rules list the available alternatives and specify which player is to make the choice; if it is a chance move, the rules list the available alternatives and specify the probabilities with which they are to be selected. For moves after the first, say the kth move with k > 1, the rules specify, as a function of the choices and outcomes at the first k – 1 moves (a) whether the kth move is to be a personal move or a chance move, (b) if a chance move, the alternatives and their probabilities of selection, and (c) if a personal move, the alternatives, the player who is to make the choice, and the information concerning the choices and outcomes at the first k – 1 moves that is given to him before he makes his choice. Finally, the rules specify, as a function of the choices and outcomes at the successive moves, when the game shall terminate and the score, not necessarily numerical, that is to be assigned to each player.

Various points should be noted in connection with the above description. (1) For k > 1, the characteristics of the kth move depend on the results of previous moves. In bridge, for instance, when the first three bids are “pass,” the move after the fourth bid will be a chance move, i.e., a new deal, if the fourth bid is “pass,” and a personal move, i.e., a bid by the next player, if the fourth bid is anything other than “pass.” (2) The information given to a player when he is to make a certain move does not necessarily include the information that was given to him a...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Contents
• 1 Games in Normal Form
• 2 Values and Optimal Strategies in Games
• 3 General Structure of Statistical Games
• 4 Utility and Principles of Choice
• 5 Classes of Optimal Strategies
• 6 Fixed Sample-Size Games with Finite Ω
• 7 Fixed Sample-Size Games with Finite A
• 8 Sufficient Statistics and the Invariance Principle in Statistical Games
• 9 Sequential Games
• 10 Bayes and Minimax Sequential Procedures When Both Ω and A Are Finite
• 11 Estimation
• 12 Comparison of Experiments
• References
• Index