Download - GameTheory.ppt

7/27/2019 GameTheory.ppt

http://slidepdf.com/reader/full/gametheoryppt 1/54

GAME THEORY



Game Theory

• Basic Concepts

- The payoff matrix

- The Nash Equilibrium

- Dominant Strategies

- Dominated Strategies- Maximin Strategies

- Mixed Strategies



Game Theory and Oligopoly

• Non co-operative games :Prisoner’s

Dilemma

• Cooperative Games: Enforcing a Cartel

• Repeated games : dealing with cheaters

• Sequential games :The advantage of being

first



Game Theory and Economics

Game theory is the study of how people

behave in strategic situations.Strategic decisions are those in which

each person, in deciding what actions to

take, must consider how others mightrespond to that action.




If the market is composed by a small number of firms, each firm must act strategically.

Each firm affects the market price changing the

quantity produced.

Suppose 2 firms are producing 100 units.

If one of the firms decides to increase the

production by 10 units.

The market supply will increase from 200 to 210

and the price has to drop to reach an equilibrium.




Therefore, it also affects the profits of other

firms.

Each firm knows that its profit depends notonly on how much it produced but also on

how much the other firms produce.



What is a Game?

A game is a situation where the

participants’ payoffs depend not only on

their decisions, but also on their rivals’decisions.

This is called Strategic Interactions:

My optimal decisions will depend on whatothers do in the game.



A Game

Four elements to describe a game:

players;

rules: when each player moves, what are the

possible moves, what is known to each player

before moving;

outcomes of the moves;

payoffs of each possible outcome: how muchmoney each player receive for any specific

outcome.



A situation of competitive rivalry must involve two or more players whose choice of actions affect eachother.

Players and their actions

•A “player” can be a firm, an interest group or coalition,

a military leader, government official.

•Games generally consider only one kind of action —

e.g., number of daily departures, fares, in-flight services,schedules, advertising, choice of hubs, ordering planes,

expanding terminals, use of computerized reservations

systems, mergers and acquisitions, and human resource

decisions.



Outcomes and Payoffs

The firm’s action, together with the actions ofits rivals, determine its payoff

•In the standard “business” game, the payoff can

be in the form of profit, market share, ratings

points,

•In war games, the payoff might be measured in

enemy killed or territory seized.

•In political games, payoffs may be measured in

votes or campaign contributions.



Underlying “rules”

The rules of the game define the range ofpossible outcomes and payoffs

•For example, collusion to fix prices or a merger

among direct rivals in a concentrated market

structure may be against the rules.

•Another set of rules specifies whether players move

sequentially or simultaneously, who moves first, and

what does each player know about the other players’

preference and prior to actions?



Game theory in business

In applying game theory to the behaviour of firms we can suggest that

firms face a number of strategic choices which govern their ability to

achieve a desired pay-off, including:

Decisions on price and output, such as whether to:

• Raise• Lower

• Hold

Decisions on products, such as whether to:

• Keep existing products

• Develop new ones



.

Decisions on promoting products, such as whether to:

• Spend more on advertising

• Spend less

• Keep spending constant

Firms could derive a range of possible pay-offs from theirstrategy choices, including:

• More profits for shareholders

• Greater market share

• Improved chances of survival• Getting rid of a rival



Matching pennies

Each player selects one side of a coin;

if the coins match player 1 wins and gets 1dollar from player 2;

if the coins don’t match player 2 wins and

gets 1 dollar from player 1.



Matrix Representation of

Matching Pennies

Player 2

Head Tail

Player 1

Head 1,-1 -1,1

Tail -1,1 1,-1



Boeing-Airbus game

Boeing and Airbus have to decide whether toinvest in the development of a Super Jumbo for long distance travel;

if they both develop successfully the new plane,

their profits will drop by 50 millions a year;if only one develop the Super Jumbo, it will

make 80 millions a year in additional profits,

whereas the profits of the other firm will drop by30 millions a year;

if no firm develops the plane, nothing changes.




Boeing-Airbus game

Airbus

Develop Do notdevelop

Boeing

Develop -50,-50 80,-30

Do not

develop-30,80 0,0



Solutions of the Games

To predict what will be the

solution/outcome of the game we needsome tools:

dominated and dominant strategies;

Nash equilibrium.



Prisoners Dilemma

Two individuals have been arrested for possession of guns. The police suspects thatthey have committed 10 bank robberies;

if nobody confesses the police, they will be jailed for 2 years.

if only one confesses, he’ll go free and his

partner will be jailed for 40 years.if they both confess, they get 16 years




Prisoners Dilemma

B

Confess Do notConfess

C

Confess 16,16 0,40

Do not

Confess40,0 2,2



We want to predict the outcome of the game

B

ConfessDo not

Confess

C

Confess 16,16 0,40

Do not

Confess40,0 2,2

Suppose that C decides to confess. What is the best decision for B?




B

ConfessDo not

Confess

C

Confess 16,16 0,40

Do not

Confess40,0 2,2

Suppose that C decides to remain silent. What is the best decision

for B?




B

ConfessDo not

Confess

C

Confess 16,16 0,40

Do not

Confess40,0 2,2

Suppose that B decides to confess. What is the best decision for C?




B

ConfessDo not

Confess

C

Confess 16,16 0,40

Do not

Confess40,0 2,2

Suppose that B decides to remain silent. What is the best decision

for C?



Outcome of the Game

B

ConfessDo not

Confess

C

Confess 16,16 0,40

Do not

Confess40,0 2,2



Dominated and Dominant Strategy

Dominant Strategy:

a strategy that gives best possible payoffs no

matter what the opponent does;Dominated Strategy:

a strategy is dominated if there exists another strategy that is dominant.

So far we have only assumed that each player is rational to determine the outcomeof the game.



Nash Equilibrium

A list of strategies, one for each player, is a Nashequilibrium if each player’s strategy maximizes his

(or her) payoff given the strategies selected by the

other players and if this condition holds for all

players simultaneously.

Any outcome that is best response for both

players is a Nash equilibrium



Nash Equilibrium

The decisions of the players are a Nash

Equilibrium if no individual prefers a

different choice.In other words, each player is choosing the

best strategy, given the strategies chosen by

the other players.



Dominant strategy

Confess is the dominant strategy in this case,

since it gives the shortest sentence

irrespective of whether the other prisoner selects the “confess” or “do not confess”

strategy



Price wars in a duopoly

The preceding is what we call a “non-

cooperative” game. Cooperation among

duopolists is a strategy that maximizes joint profits. So why do price wars break out?



The payoff matrix for a running

shoe duopoly

HighPrice LowPrice

High

Price

$10 million,

$10 million

$5 million,

$12 million

LowPrice

$12 million,$5 million

$7 million,$7 million

REEBOK

NIKE

Notice that “low price” is the

dominant strategy



Pizza Planet and Luigi’s are rivals in the

market for home-delivered pizza. Each rivalseeks to gain an advantage through advertising(product differentiation).

Advertising is presumed NOT to affect marketdemand--only market share.

Market share depends on the intensity of advertising relative to one’s rival.

Advertising rivalry



Let

P = $15

Q = 100 pizzas (market quantity-demanded)

AC (w/o advertising expense) = $5.

Hence:

/Pizza = (TR - TC)/Q = ($1500 - $500)/100 = $10



If neither seller advertises, each will

sell 50 pizzas and earn a profit of $500. However, advertising could

potentially increase sales to 75

pizzas.



The payoff matrix for

a pizza duopoly

Low

Advertising

High

Advertising

Low

Advertising

$400,

$400

$150,

$550

High

Advertising

$550,

$150

$300,

$300

LUIGI’S

PIZZA PLANET

Notice that “high advertising” is the

dominant strategy



Nash Equilibrium and the Shoe,

Pizza Duopoly Games•A Nash equilibrium is given by the “low-

low” strategy in the running shoe duopoly

game •A Nash equilibrium is given by the “high-

high strategy” in the pizza duopoly game.



Dominated Strategy

• Example

• Example of an iterated deletion of

dominated strategy equilibrium.doc

http://localhost/var/www/apps/conversion/tmp/scratch_10/Example%20of%20an%20iterated%20deletion%20of%20dominated%20strategy%20equilibrium.doc






Maximin Strategy

• This strategy is based on the fact that

sometimes it is more prudent to maximize

the minimum gains achievable in a gamingsituation. Let's take the Meat and Potatoes

example to appreciate this strategy.



Dominant Strategy -? ;

Nash Equilibrium - ?

Nash Equilibriumis based on the fact

that both M and P

behave rationally.

However, can Palways count on M

being rational?

Can P take the risk of selling potatoes if he is conservative and afraid

that M may not behave rationally ? No.

So, what does P do? P will sell meat, for he is assured of making at least

$80,000. In this scenario, P is pursuing a Maximin Strategy of

maximizing the minimum gains that can be earned.



Mixed Strategies

• A mixed strategy is an assignment of a

probability to each pure strategy. This

allows for a player to randomly select a pure strategy.



• ..\Reference\mixed strategies.pdf

http://localhost/var/www/apps/conversion/tmp/Reference/mixed%20strategies.pdf








Game theory and Oligopoly

• Non Cooperative Games

• Cooperative Games

• Repeated Games

• Sequential games



Non Cooperative games• The Prisoner’s Dilemma model

• This model explains a number of interesting

phenomena in business

Example:

30, 30 10,40

40,10 20,20

Firm 2

Low- level Advg High level Advg

Low level advg

High level advgFirm 1



Cooperative Games: Enforcing a Cartel

The experience of the pan-masala industry during the 1980s is an

interesting example of how a non cooperative game was turned, bygovernment policy, into a cooperative game. For many years, the

major pan-masala manufacturers had been spending large sums on TV

advertising to promote their products. Much of this advertising was

mutually offsetting, so the firms probably were in the Prisoner’s

Dilemma. They would have been better off if they had reduced their advertising expenditures, but no one firm could afford to do so unless

there was some assurance that their lead would be followed by the

other firms.

In mid-1980s, the government banned advertising of pan-masala on

Doordarshan. Initially, the companies fought the ban, but it soon became apparent that any lost sales were more than made up by the

savings in advertising. With this ban, the government did for pan-

masala manufacturers what they were unable to do for themselves-

enforce a reduction in advertising expenditure.



Repeated Games- Dealing with

cheaters• What is the optimal strategy for firms playing non

cooperaive repeated games?

AN approach described as “tit-for-tat”

TIT FOR TAT is a very simple strategy, which

cooperates on the first round, and thereafter

simply copies what its opponent did on the

previous round — thereby rewarding cooperationwith more cooperation and punishing defection

with defection.



• EXAMPLE of tit for tat pricing.doc

http://localhost/var/www/apps/conversion/tmp/scratch_10/EXAMPLE%20of%20tit%20for%20tat%20pricing.doc

http://localhost/var/www/apps/conversion/tmp/scratch_10/EXAMPLE%20of%20tit%20for%20tat%20pricing.doc



Sequential Games

• The advantage of being first

No new

product

Introduce new

product

No New product 2,2 (5), 10

Introduce new

product 10, (5) (7), (7)Firm 1

Firm2

Assumption: Maximin Strategy

Download - GameTheory.ppt

Top Related