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

By Herbert Gintis | Go to book overview

5
Extensive Form Rationalizability

The heart has its reasons of which reason
knows nothing.

Blaise Pascal

The extensive form of a game is informationally richer than the normal form since players gather information that allows them to update their subjective priors as the game progresses. For this reason, the study of rationalizability in extensive form games is more complex than the corresponding study in normal form games. There are two ways to use the added information to eliminate strategies that would not be chosen by a rational agent: backward induction and forward induction. The latter is relatively exotic (although more defensible) and will be addressed in chapter 9. Backward induction, by far the most popular technique, employs the iterated elimination of weakly dominated strategies, arriving at the subgame perfect Nash equilibria—the equilibria that remain Nash equilibria in all subgames. We shall call an extensive form game generic if it has a unique subgame perfect Nash equilibrium.

In this chapter we develop the tools of modal logic and present Robert Aumann's famous proof (Aumann 1995) that CKR implies backward induction. This theorem has been widely criticized, as well as widely misinterpreted. I will try to sort out the issues, which are among the most important in contemporary game theory. I conclude that Aumann is perfectly correct, and the real culprit is CKR itself.


5.1 Backward Induction and Dominated Strategies

Backward induction in extensive form games with perfect information (i.e., where each information set is a single node) operates as follows. Choose any terminal node τΤ and find the parent node of this terminal node, say node v. Suppose player i chooses at ν and suppose i's highest payoff at ν is attained at terminal node τ'T. Erase all the branches from ν so ν becomes a terminal node and attach the payoffs from τ' to the new terminal

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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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