The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences

By Herbert Gintis | Go to book overview

2
Game Theory: Basic Concepts

High-rationality solution concepts in game theory can
emerge in a world populated by low-rationality agents.

Young (1998)

The philosophers kick up the dust and then complain that
they cannot see.

Bishop Berkeley


2.1 The Extensive Form

An extensive form game G consists of a number οι players, a game tree, and a set οf payoffs. A game tree consists of a number of nodes connected by branches. Each branch connects a head node to a distinct tail node. If b is a branch of the game tree, we denote the head node of b by bh, and the tail node of b by bt.

A path from node a to node a' in the game tree is a connected sequence of branches starting at a and ending at a'.1 If there is a path from node a to node a', we say a is an ancestor of a', and a' is a successor to a. We call the number of branches between a and a' the length of the path. If a path from a to a' has length 1, we call a the parent of a', and a' is a child of a.

We require that the game tree have a unique node r, called the root node, that has no parent, and a set Τ of nodes, called terminal nodes or leaf nodes, that have no children. We associate with each terminal node t e Τ (e means “is an element of”), and each player i, a payoff πi (t) ∈ R (R is the set of real numbers). We say the game is finite if it has a finite number of nodes. We assume all games are finite unless otherwise stated.

We also require that the graph of G have the following tree property. There must be exactly one path from the root node to any given terminal

1Technically, a path is a sequence b1, …, bk of branches such that


for i = 1,…, k − 1, and ; i.e., the path starts at a, the tail of each branch is the
head of the next branch, and the path ends at a'. The length of the path is k.

-30-

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The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences
Table of contents

Table of contents

  • Title Page iii
  • Contents vii
  • Preface xiii
  • 1: Decision Theory and Human Behavior 1
  • 2: Game Theory: Basic Concepts 30
  • 3: Game Theory and Human Behavior 45
  • 4: Rationalizability and Common Knowledge of Rationality 83
  • 5: Extensive Form Rationalizability 102
  • 6: The Mixing Problem 121
  • 7: Bayesian Rationality and Social Epistemology 132
  • 8: Common Knowledge and Nash Equilibrium 146
  • 9: Reflective Reason and Equilibrium Refinements 164
  • 10: The Analytics of Human Sociality 181
  • 11: The Evolution of Property Rights 201
  • 12: The Unification of the Behavioral Sciences 221
  • 13: Summary 248
  • 14: Table of Symbols 250
  • Symbols for Chapter 11 251
  • References 253
  • Index 283
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