Game theory 1

Game Theory

So Far…

• Understood environment where actions of agents were independent of each other.

• Game Theory: Analyzing environment of strategic interactions

• Examples: Your performance versus learning

What are Games?

“No Man is an Island Entire of Itself” -John Donne, 1624

Smartphone Market

• Spate of law suits Protecting patents filed by Apple• What determines the leadership of

Smartphone market?• Implications of the verdict on – Apple,

Samsung, Google• Android share: 64.1%, Apple – 18.8%

Lesson 1:

Strategic Interaction: Scenario where outcomes of actions dependent on that of others

MoneyBall

• Billy Beane Manager Oakland A Baseball team

•On a massive losing spree

•To win the World Series he must strategize

•Hires a Yale Economist to strategize the optimal bid for each player

Politics

• Who wins presidential elections?- Voters, Policy announcements, funds and campaign resources

• Result also depends upon the sequence in which elections are held in different primaries

Princess Bride

• Strategic interaction

• What does his choice Depend upon?

• Choices depend upon actions of others as well as beliefs of actions of others (under uncertainty)

Browser Wars• October 1994, Netscape released one of the

first commercial browsers- Navigator 1.0• It was priced at $39- 60% market share• August 1995, Microsoft introduces Internet

Explorer available only with Windows 95• Netscape has to decide its next move.• Reduce price, improve the speed, reduce the

size?

Smartphone Market

• Spate of law suits Protecting patents filed by Apple• What determines the leadership of

Smartphone market?• Implications of the verdict on – Apple,

Samsung, Google• Android share: 64.1%, Apple – 18.8%

Sequential Interaction

• Players make alternating moves

• Each player must look ahead. Anticipate rival’s future response

• Reason back and decide optimal action

Movie Release

• Dreamworks- The Antz : 2nd Oct 1998• Pixar- A Bug’s life: 25th Nov 1998

• Movie houses decide on what kind of movie to make

• When to release the movie• In this case simultaneous decision

Competing over Market Share

• Coke and Pepsi compete for market share• Every week they decide prices

• Every week Coke meets behind close doors

• Coke knows that Pepsi meets behind close doors.

Strategic Moves Game

• No sequence of alternating moves• Simultaneous moves

• Each player acts in ignorance of the other player’s actual choice

• Can not look ahead and reason backwards

Lesson 2:

Modeling Strategic Environment: Sequential or Simultaneous Interaction

IPL Auction

• Bid on a player reveals private information

• Winning a player depends upon your bid and that of the others

Information Rules

Lesson 3 : Actions and Outcome of the game depends

upon private information

Lesson 4: Actions reveal private information

Use the fact that others use the fact that your bids reveal private information about the course

Information Rules

What is Game Theory?• Finding the right strategy in Strategic

Environments

• Bad news: Knowing game theory does not guarantee winning

• Good news: Framework for thinking about strategic interaction

Games Business PlayExample Game

Price Wars Prisoner’s Dilemma

Pollution Abatement Free Riding

FCC Spectrum Auctions

Coke and Pepsi : Price War

• Coke has to decide whether to increase its price or to reduce it

• What should it do?• What will happen if it increased its price?

• What if it increased the price and Pepsi did not?

• What if Pepsi also increased its price?

Games Business PlayExample Game

Price Wars Prisoner’s Dilemma

Pollution Abatement Free Riding

FCC Spectrum Auctions

Games that we playExample Game

Doing the Dishes War of Attrition

Group Projects Free Riding

Dating Hidden Information

Free Riding

• Collective project• Ideal if you get a high grade • However would prefer not to exert effort

• Should you put effort or not?

Why Study Game Theory?

Because the press tells us to…

• “Game Theory, long an intellectual pastime, came into its own as a business tool.” Forbes, July 3, 1995, p. 62.

• “Game theory is hot.” The Wall Street Journal, 13 February 1995, p. A14

26

Why Study Game Theory?Because recruiters tell us to …

• “Game theory forces you to see a business situation

over many periods from two perspectives: yours and your competitor’s.”– Judy Lewent – CFO, Merck

27

Decision Theory vs. Game Theory

• Ten of you go to a restaurant

• If each of you pays for your own meal…– What to order is then a decision problem

• If you all agree to split the bill...– Now, this is a game

28

Restaurant Decision-Making

• Check splitting policy changes incentives.

May I recommend that with the Bleu Cheese for ten dollars more?

Sure!

It is only a

dollar more for

me!

ELEMENTS OF GAMES

Rules of the Game

• The strategic environment– Players– Strategies– Payoffs

• The assumptions– Rationality– Common knowledge

• The rules– Timing of moves– Informational conditions

The Strategic Environment

• Players: Who is interacting?

• Strategies: What are their options?

• Payoffs: What are their incentives?

Cigarette Advertising on TV

• All US tobacco companies advertised heavily on TV

• Surgeon General issues official warning• Cigarette smoking may be hazardous

• Cigarette companies fear lawsuits• Government may recover healthcare costs

• Companies strike agreement• Carry the warning label and cease TV

advertising in exchange for immunity from federal lawsuits.

1964

1970

Strategic Interaction• Players: Reynolds and Philip Morris• Strategies: Advertise or Not Advertise• Payoffs: Companies’ Profits

• Strategic Landscape:– Each firm earns $50 million from its customers– Advertising costs a firm $20 million– Advertising captures $30 million from competitor

The Strategic Environment

• Players• Everyone who has an effect on your PAYOFFS

• Strategies• Actions available to each player• Define a plan of action for every contingency

• Payoffs• Well being as a result of a set of actions of each player• Different payoffs for different set of actions.

ASSUMPTIONS

Rationality

• Most economic analysis assumes “rationality” of decision-makers, i.e. that you make decisions by

1. choosing an action that maximizes your welfare2. forming correct belief about the world

• In principle, requires enormous powers of imagination and computation.

Common Knowledge

• Players know the rules of the game

• Players know the payoffs, strategies, number of players and the sequence in which the game is played.

And Common Knowledge of Rationality??

• Each player is rational

• Each player knows that each player is rational

• Each player knows that each player knows that each player is rational

• Each player knows that each player knows that each player knows that each player is rational

• Each player knows that each player knows that each player knows that each player knows that each player is rational

• Etc. etc. etc.

EQUILIBRIUM

Equilibrium

• The likely outcome of a game when rational, strategic agents interact

– Each player is playing his or her best strategy given the strategy choices of all other players

– No player has incentive to change his or her action unilaterally

SEQUENCE OF PLAY

Simultaneous Move Games

• Scenario where agents take actions simultaneously not the same action

• Agents may take an action at different points in time

• Action is taken without the knowledge of the action taken by the other agent

• You can not think ahead and rational backwards

SIMULTANEOUS MOVE GAMES

REPRESENTING THE GAME

Cigarette Advertising on TV

• All US tobacco companies advertised heavily on TV

• Surgeon General issues official warning• Cigarette smoking may be hazardous

• Cigarette companies fear lawsuits• Government may recover healthcare costs

• Companies strike agreement• Carry the warning label and cease TV

advertising in exchange for immunity from federal lawsuits.

1964

1970

Strategic Interaction• Players: Reynolds and Philip Morris• Strategies: Advertise or Not Advertise• Payoffs: Companies’ Profits

• Strategic Landscape:– Each firm earns $50 million from its customers– Advertising costs a firm $20 million– Advertising captures $30 million from competitor

• How to represent this game?

Representing a Game

PLAYERS

STRATEGIESPAYOFFS

Philip Morris

No Ad Ad

Reynolds No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

What to Do?

If you are advising Reynolds, what strategy do you recommend?

Philip Morris

No Ad Ad

ReynoldsNo Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Solving the Game

• Best reply for Reynolds:• If Philip Morris advertises:• If Philip Morris does not advertise:

Philip Morris

No Ad Ad

ReynoldsNo Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

SOLVING THE GAME

How to solve the game? L

C

U 8, 3 0, 2

M 4, 2 1, 5

SOLVING THE GAME

• What strategy would you choose if you were player Row?

• Your decision would depend upon?• Your belief about Player column’s action• If you believe that Player column would play

L=> U • If you believe that Player column would play

M=> C

SOLVING THE GAME

• How do you get around the beliefs?

• You must put yourself in their shoes to see what they would do.

• They are also trying to guess what you are doing.

SOLVING THE GAME

• Your action depends upon your belief about other’s actions.

• Your belief about other’s actions in turn depends upon their belief about your actions

Solving the game

1. Dominated/Dominant Strategy

2. Best Response Functions/Nash Equilibrium

3. Iterated Elimination of Weakly Dominated Strategies (IEWDS)

1. Dominated/Dominant Strategies

Dominated Strategy

Strategy, say x is said to be dominated if there exists another strategy, say y which

gives higher payoff than x under all strategy profiles of other players

No Ad Ad

No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

No Ad Ad

No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

L C

R

U 8, 3 0, 4 4,4

M 4,2 1,5 5,3

D 3,7 0,1 2,0

L C

R

U 8, 3 0, 4 4,4

M 4,2 1,5 5,3

D 3,7 0,1 2,0

L C

R

U 8, 3 0, 4 4,4

M 4,2 1 ,5 5,3

D 3,7 0,1 2,0

Dominated/Dominant Strategies

Dominant Strategy

Strategy, say y is said to dominates x if it gives higher payoff than x under all strategy profiles of other players.

No Ad Ad

No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Dominance• A strategy is dominant if it

outperforms all other choices no matter what opposing players do

• Technical aside: strict dominance: Profit (Ad , any) > Profit (No , any)weak dominance: some inequalities are weak ()

• Games with dominant strategies are easy to play – No need for “what if …” thinking

Dominance

If you have a dominant strategy, use it.

Expect your opponent to use her dominant strategy if she has one.

Prisoner’s Dilemma

• Both players have a dominant strategy• The equilibrium results in lower payoffs for each player

No Ad Ad

No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Equilibrium

Optimal


• Two prisoner’s caught on the crime scene• Interrogated behind two separate cells

• If one confesses to crime, its proved they both were involved.

Prisoner’s DilemmaConfess Not Confess

Confess -3, -3 0, -8

Not Confess -8, 0 -1,-1

Example: SUV Price Wars

“General Motors Corp. and Ford Motor Co. slapped larger incentives on popular sport utility vehicles,

escalating a discounting war in the light-truck category …

Ford added a $500 rebate on SUVs, boosting cash discounts to $2,500.

The Dearborn, Mich., auto maker followed GM, which earlier in the week began offering $2,500 rebates on many of its SUVs.”

-- Wall Street Journal, January 31, 2003

SUV Price WarsDiscount No Discount

Discount 3,3 8, 0

No Discount 0, 8 5,5

• No Discount is a ………… strategy

• Discount is a ………. strategy

• No Discount is a dominated strategy

• Discount is a dominant strategy

• In this case Discount strictly dominates No Discount

Strictly Dominant Strategy:

Strategy, say x is said to strictly dominate y if it gives strictly higher payoff than x under all

strategy profiles of other players.


• Both players have a dominant strategy-Discount

• Neither player would play No discount

• However, each would be better off if they selected No Discount.

Strategic Tension

Prisoner’s Dilemma:• Clash between individual and group interests• The players each realize that they are jointly

better off if they play NO DISCOUNT- Group interest.

• However, each has the individual incentive to choose Discount- Individual interest.

Efficiency

• Strategy profile s is more efficient than s’ if all of the players prefer s to the outcome s’.

• Pareto Efficient: A strategy profile s is pareto efficient if there is no other strategy profile s’ such that every player gets more from s’ than s.


Discount 3,3 8, 0



Discount 3,3 8, 0


How to find dominated strategies

To check for the dominated strategies of a player represented by row

• Scan across the rows, look at all columns• If the payoff in one row is greater than the

payoff in another, and this is true for all columns, then the former dominates the latter.

L C

R

U 8, 3 0, 4 4,4

M 4,2 1,5 5,3

D 3,7 0,1 2,0

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Game theory 1