[자료] chapter 2: Nash Equilibrium: Theory

자료: http://www.economics.utoronto.ca/osborne/igt/nash.pdf

지은이: Martin J. Osborne, Version: 2002/7/23

Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23.

Martin.Osborne@utoronto.ca. http://www.economics.utoronto.ca/osborne 

Copyright ⓒ 1995.2002 by Martin J. Osborne. All rights reserved. No part of this book may be reproduced by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from Oxford University Press, except that one copy of up to six chapters may be made by any individual for private study.

2.1 Strategic games 11

2.2 Example: the Prisoner's Dilemma 12

2.3 Example: Bach or Stravinsky? 16

2.4 Example: Matching Pennies 17

2.5 Example: the Stag Hunt 18

2.6 Nash equilibrium 19

2.7 Examples of Nash equilibrium 24

2.8 Best response functions 33

2.9 Dominated actions 43

2.10 Equilibrium in a single population: symmetric games and

symmetric equilibria 49

Prerequisite: Chapter 1.

* * *

In summary, the solution theory we study has two components. First, each player chooses her action according to the model of rational choice, given her belief about the other players' actions. Second, every player's belief about the other players' actions is correct. These two components are embodied in the following definition.

A Nash equilibrium is an action profile a* with the property that no player i can do better by choosing an action different from a_i* , given that every other player j adheres to a_j*.

In the idealized setting in which the players in any given play of the game are drawn randomly from a collection of populations, a Nash equilibrium corresponds to a steady state. If, whenever the game is played, the action profi le is the same Nash equilibrium a* , then no player has a reason to choose any action different from her component of a *; there is no pressure on the action profi le to change. Expressed differently, a Nash equilibrium embodies a stable “``social norm”``: if everyone else adheres to it, no individual wishes to deviate from it.

The second component of the theory of Nash equilibrium —that the players' beliefs about each other's actions are correct— implies, in particular, that two players' beliefs about a third player's action are the same. For this reason, the condition is sometimes said to be that the players' “``expectations are coordinated”.``