Game Theory: Static and Dynamic Games, Auctions, and Equilibria, Lecture notes of Game Theory

Download Game Theory: Static and Dynamic Games, Auctions, and Equilibria and more Lecture notes Game Theory in PDF only on Docsity!

Game Theory

Lecture 8

Reading: Perlo§ Chapter 13

August 2015

Introduction

Game theory is the study of strategic interaction. Game theory is applicable in so many Öelds other than economics. Evolutionary biology, international relations, whether or not to open the door for the old lady behind you.

Overview of Game Theory

A game is a situation in which your payo§ depends not only on what you do, but what the others do. Tick-tack-toe, whether or not to under price a rival Örm, or to cut carbon emissions are all games.

Overview of Game Theory

An action is a move you can make at a stage in a game. Cut emissions or donít cut emissions are actions. A strategy is a plan conditional on any possible contingency. Cut emissions if everybody else cuts emissions, donít otherwise is a strategy. An equilibrium is a set of strategies such that neither player wishes to deviate.

Overview of Game Theory

EXAMPLE

Consider the game rock-paper-scissors. Is this a static or a dynamic game? What are the actions and the strategies in this game?

Static Games

It is useful to summarize a game using normal-form representation. We show the players, their strategies and the payo§ as a combination of strategies in a payo§ matrix. It is just the summary of the game using a table.

Static Games

The row playerís payo§s are written Örst, and the column playerís payo§s are written second

Nick Silent Betray Tom Silent -1,-1 -12, 0 Betray 0, -12 -3, -

Static Games

Sometimes players will have a dominant strategy A dominant strategy is a strategy that produces a higher payo§ than any other possible strategy. No matter what your opponent might do, you play the dominant strategy.

Static Games

EXAMPLE

Do either players have a dominant strategy? Is there a dominant strategy equilibrium?

Column Left Right Row Up 2,4 10, Down 1,1 9,-

Static Games

In many games, the players donít have a dominant strategy. But players might have dominated strategies. A dominated strategy is a strategy the players will never play, so we can delete them. Denying the crime is a dominated strategy in the prisonerís dilemma.

Static Games

Consider the following game.

Player 2 Left Middle Right Player 1 Up 1,0 1,2 0, Down 0,3 0,1 2,

Static Games

Player 2 will never play Right. Player 1 can eliminate Right from the game.

Player 2 Left Middle Player 1 Up 1,0 1, Down 0,3 0,

Player 1 will never play Down. Player 2 can eliminate down. Keep going and we are left with only fup, middleg. This is called iterated deletion of dominated strategies.

Static Games

There are no dominant or dominated strategies in this game. We need to have something stronger.

Nick Right Left Stranger Right 5,5 -5, 0 Left -5,0 5, 5

Static Games

We can look at best responses. A best response is a best strategy given what you think the other player will do. A dominant strategy is one that is a best response to all possible strategies.