GAME THEORY

NATURE AND SCOPE OF GAME THEORY

The essential nature of game theory is that it involves strategic behavior, which

means interdependent decision-making

ELEMENTS OF A GAME PLAYERS:

These are the relevant decision-making identities, whose utilities are interdependent.

STRATEGIES:

These are complete plans of action for playing the game.

PAYOFFS:

These represent changes in welfare or utility at the end of the game, and are determined by the choices of strategy of each player.

PRISONER’S DILEMMA (PD).

AT THIS STAGE WHAT THE TYPE OF SITUATION DESCRIBED ABOVE HAS TO DO WITH BUSINESS STRATEGY

To illustrate this, let us consider the situation

Of Coke and Pepsi.

At any given time period each firm has to decide

whether to maintain their existing price or to offer a

discount to the retailers who buy from them.

PRISONER’S DILEMMA FOR COKE AND PEPSI

PEPSI

Maintain Price Discount

Maintain Price

COKE

Discount

50 50

70 -10

-10 70

10 10

STRUCTURE OF PAYOFFS IN PRISONER’S DILEMMA

Strategy pair (self/other) Name of payoff

Defect/co-operate Temptation (70)Co-operate/co-operate Reward (50)Defect/defect Punishment (10)Co-operate/defect Sucker’s payoff (10)

TYPES OF GAME

Co-operative and non-cooperative games Two-player and multi-player games Zero-sum and non-zero-sum games Perfect and imperfect information Static and dynamic games Discrete and continuous strategies ‘One-off’ and repetitive games

STATIC GAMES

The nature of this type of game raises the following

questions:

How does a firm determine strategy in this type of situation?

What do we mean by an equilibrium strategy?

Is there anything that firms can do to change the equilibrium to a more favorable one, meaning to ensure co-operation?

EQUILIBRIUM

We can now consider three types of equilibrium and

appropriate strategies in situations involving

different payoffs.

Dominant strategy equilibrium, Iterated dominant strategy equilibrium, Nash equilibrium

DOMINANT STRATEGY EQUILIBRIUM

Strictly dominant strategy in a situation, will

always give at least as high a payoff as any other

strategy, whatever player other does

DOMINANT STRATEGY EQUILIBRIUM

PEPSI

Maintain Price Discount

Maintain Price

COKE

Discount

50 50

70 -10

-10 70

10 10

PARETO DOMINATED SITUATION

This means that there is some other outcome where

at least one of the players is better off while no other

player is worse off.

ITERATED DOMINANT STRATEGY EQUILIBRIUM

What would happen if one firm did not have a

dominant strategy?

CONT….. PEPSI

Maintain Price Discount

Maintain Price

COKE

Discount

80 50

70 -10

-10 70

10 10

NASH EQUILIBRIUM

The situation becomes more complicated when

neither player has a dominant strategy.

CONT…….

75 10

PEPSI

Maintain Price Discount

Maintain Price

COKE

Discount

60 50

15 10

10 15

CONTI……

If Coke maintains price, Pepsi will discount; and, given this best response, Coke’s best reply is to maintain price.

If Coke discounts, Pepsi will maintain price; and, given this best response, Coke’s best reply is to discount.

CONTI…….The same equilibrium could also be expressed from

Pepsi’s point of view:

If Pepsi discounts, Coke will maintain price; and, given this best response, Pepsi’s best reply is to discount.

If Pepsi maintains price, Coke will discount; and, given this best response, Pepsi’s best reply is to maintain price

OLIGOPOLY MODELS

Cournot model Bertrand model Contestable markets model

THE COURNOT MODEL There are few firms in the market and many

buyers.

The firms produce homogeneous products; therefore each firm has to charge the same market price (the model can be extended to cover differentiated products).

Competition is in the form of output, meaning that each firm determines its level of output based on its estimate of the level of output of the other firm.

Each firm believes that its own output strategy does not affect the strategy of its rival(s).

Barriers to entry exist.

Each firm aims to maximize profit, and assumes that the other firms do the same.

THE BERTRAND MODEL

There are few firms in the market and many buyers.

The firms produce homogeneous or differentiated products; therefore each firm has to charge the same market price in the case of homogeneous products, but there is some scope for charging different prices for differentiated products.

CONTI… Competition is in the form of price, meaning that

each firm determines its level of price based on its estimate of the level of price of the other firm. Each firm believes that its own pricing strategy does not affect the strategy of its rival(s).

Barriers to entry exist. Each firm has sufficient capacity to supply the

whole market. Each firm aims to maximize profit, and assumes

that the other firms do the same.

CONTESTABLE MARKETS

There are an unlimited number of potential firms that can produce a homogeneous product, with identical technology.

Consumers respond quickly to price changes. Incumbent firms cannot respond quickly to entry

by reducing price. Entry into the market does not involve any sunk

costs. Firms are price-setting Bertrand competitors.

DYNAMIC GAMES

Many business scenarios tend to involve

sequential moves rather than simultaneous

moves.

Example: Decision to invest in new plant

EQUILIBRIUM

Dynamic games are best examined by drawing a game tree.

An extensive-form game not only specifies the

players, possible strategies, and payoffs, as in the

normal-form game, but also specifies when players can move, and what information they have at the time of each move.

CONTI…….

In order to analyze this game tree we must derive the sub game perfect Nash equilibrium (SPNE).

This is the situation where each player selects an

optimal action at each stage of the game that it might reach, believing the other player(s) will act in the same way.

CONT…….

70 40

FIRM B

Expand No change

Expand

FIRM A

No change

50 20

85 25

95 30

LIMITATIONS OF GAME THEORY

As game theory applications have become more widespread throughout economics and the other social and natural sciences, certain criticisms have arisen regarding the validity of its conclusions