game theory

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Game Theory

Chapter 10

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Applications of Game Theory National Defense – Terrorism and Cold War Movie Release Dates and Program

Scheduling Auctions http://en.wikipedia.org/wiki/Spectrum_auction

http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction

Sports – Cards, Cycling, and race car driving Politics – positions taken and $$/time spent

on campaigning Nanny Monitoring Group of Birds Feeding Mating Habits

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Game Theory and TerrorismGame theory helps insurers to judge the risks of terror Financial Times Jenny Wiggins September 8, 2004

Shortly after September 11 2001, a small group of companies that specialise in assessing risk for the insurance industry launched US terrorism risk models.

These combine technology and data to predict likely terrorist targets and methods of attack, and possible losses to life and property.

They are aimed at the insurance and reinsurance industry, which already uses similar models to assess potential losses from natural catastrophes such as hurricanes and earthquakes.

"Most major commercial insurers and reinsurers are using terrorism modelling today," says Robert Hartwig, chief economist at the Insurance Information Institute.

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Game Theory and Terrorism (cont.)Andrew Coburn, director of terrorism research at RMS,

says the company can pinpoint possible targets because it believes terrorists make rational decisions.

"Their methods and targeting are very systematic," he says.

RMS uses game theory - analytical tools designed to observe interactions among people - in its models. It argues that, as security increases around prime targets, rational terrorists will seek out softer targets.

Industry participants, however, say the predictive abilities of the models are limited, given the difficulty of foreshadowing human behaviour.

The development of the models has attracted the interest of the US government…

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Game Theory and Randomization

Random Checks Newsweek October 22, 2007

Security officials at Los Angeles International Airport now have a new weapon in their fight against terrorism: randomness. Anxious to thwart future terror attacks in the early stages while plotters are casing the airport, security patrols have begun using a computer program called ARMOR (Assistant for Randomized Monitoring of Routes) to make the placement of security checkpoints completely unpredictable.

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Game Theory and Randomization (cont.) Randomness isn't easy. Even when they want to be

unpredictable, people follow patterns. That's why the folks at LAX turned to the computer scientists at USC.

The idea began as an academic question in game theory: how do you find a way for one "agent" (or robot or company) to react to an adversary who has perfect information about the agent's decisions? Using artificial intelligence and game theory, researchers wrote a set of algorithms to randomize the actions of the first agent. Academic colleagues couldn't appreciate how the technology could be useful. "It was very disappointing," says Milind Tambe, the USC engineering professor who led the ARMOR team.

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Applications of Game Theory National Defense – Terrorism and Cold War Movie Release Dates and Program

Scheduling Auctions http://en.wikipedia.org/wiki/Spectrum_auction

http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction

Sports – Cards, Cycling, and race car driving Politics – positions taken and $$/time spent

on campaigning Nanny Monitoring Group of Birds Feeding Mating Habits

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Grey’s Anatomy vs. The DonaldNBC delays 'Apprentice' premiereBy Nellie Andreeva Dec 20, 2007

NBC is taking the premiere of "Celebrity Apprentice" out of the cross-hairs of the last original episode of ABC's "Grey's Anatomy"... or so it seems.NBC on Wednesday said that it will push the launch of "Apprentice" from Jan. 3 to Jan. 10, expanding "Deal or No Deal" to two hours on Thursday, Jan. 3.The move follows ABC's midseason schedule announcement Friday that included the last original episode of "Grey's" airing Jan. 3,…

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Grey’s Anatomy vs. The Donald'Grey' move has NBC red Peacock shifts 'Apprentice' backBy Nellie Andreeva Dec 21, 2007

The Thursday night scheduling tango between NBC and ABC continued Thursday morning when ABC officially announced that it will move the last original episode of "Grey's Anatomy" from Jan. 3 to Jan. 10.That led to a reversal in NBC's Wednesday decision to push the premiere of "Celebrity Apprentice" from Jan. 3 toJan. 10 to avoid the first-run "Grey's."NBC said Thursday afternoon that "Apprentice," hosted by Donald Trump, will now launch Jan. 3 as originally planned.

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Game Theory and Movie Release DatesThe Imperfect Science of Release DatesNew York Times November 9, 2003

On Dec. 25, which this year happens to be a Thursday, five new movies will be released in theaters -- six, if you count a new Disney IMAX film called ''Young Black Stallion.'' As with the Fourth of July and Thanksgiving, there is a special cachet to opening a film on Christmas Day…. The casual moviegoer rarely ponders why a particular bubbly romantic comedy, serial-killer thriller, literary costume drama or animated talking-farm-animals movie opens on the day it does. Movies come; movies go; movies wind up on video. To those responsible for putting those films on the screen, however, nothing about the timing of their releases is arbitrary.

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Game Theory and Movie Release Dates (cont.)

Last December featured one of the most dramatic games of chicken in recent memory, when two films starring Leonardo DiCaprio were both slated to open on Christmas weekend. Ultimately, Miramax blinked first, moving the release of Martin Scorsese's ''Gangs of New York'' five days earlier and ceding the holiday to the other DiCaprio film, DreamWorks' ''Catch Me if You Can.'' ''We didn't think about moving,'' says Terry Press, the head of marketing for DreamWorks. ''We had been there first, and 'Catch Me if You Can' was perfect for that date.'' This year, DreamWorks chose to schedule a somber psychological drama, ''House of Sand and Fog,'' for the day after Christmas, deferring a bit to Miramax. ''I don't want our reviews to run on the same day as 'Cold Mountain,''' Press says.

Ever wonder why a movie theater shows a preview of an upcoming movie that is to be released in 2 years?

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Applications of Game Theory National Defense – Terrorism and Cold War Movie Release Dates and Program

Scheduling Auctions http://en.wikipedia.org/wiki/Spectrum_auction

http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction

Sports – Cards, Cycling, and race car driving Politics – positions taken and $$/time spent

on campaigning Nanny Monitoring Group of Birds Feeding Mating Habits

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FAA AuctionsBlame rests with the FAA USA TODAY December 18, 2007

The Federal Aviation Administration (FAA) is the gang that couldn't shoot straight. After years of ignoring airspace that is too crowded and near-collisions that are too common, the agency is now plotting a response that would make a bad problem worse. The problem is of the agency's own making. Air congestion has increased, but the issue could have been handled better by federal officials. Across the country, air traffic control towers are dangerously understaffed because FAA bean-counters have not prioritized the hiring of more personnel. As a result, the New York area airports have 20% fewer controllers on duty than they should.

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FAA Auctions (cont.) Blame rests with the FAA USA TODAY December 18, 2007

Now, the Transportation Department is set to unveil a proposal to cut flights and sell hourly slots to the highest bidder. But auctioning flights would raise fares, limit consumer choice and strike a blow to the economy. It wouldn't shorten the wait at the gates or increase capacity. It would force airlines to pay a premium to fly that will surely be passed on to travelers. And it would reduce options for those flying to small and midsize cities.Flight rationing, like congestion pricing, is not a viable solution. It is experimental game theory. America's busiest airports should not be the guinea pigs for an ideological solution that has never been tested at any airport, let alone the nation's busiest.

http://www.aviationairportdevelopmentlaw.com/2009/10/articles/faa-1/it-is-official-the-faa-rescinds-slot-auction-rule/

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Applications of Game Theory National Defense – Terrorism and Cold War Movie Release Dates and Program

Scheduling Auctions http://en.wikipedia.org/wiki/Spectrum_auction

http://en.wikipedia.org/wiki/United_States_2008_wireless_spectrum_auction

Sports – Cards, Cycling, and race car driving Politics – positions taken and $$/time spent

on campaigning Nanny Monitoring Group of Birds Feeding Mating Habits

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Game Theory Terminology Simultaneous Move Game – Game in

which each player makes decisions without knowledge of the other players’ decisions (ex. Cournot or Bertrand Oligopoly).

Sequential Move Game – Game in which one player makes a move after observing the other player’s move (ex. Stackelberg Oligopoly).

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Game Theory Terminology

Strategy – In game theory, a decision rule that describes the actions a player will take at each decision point.

Normal Form Game – A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies.

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Example 1: Prisoner’s Dilemma(Normal Form of Simultaneous Move Game)

Martha’s options

Don’t Confess Confess

Peter’s Options

Don’t Confess M: 2 years P: 2 years

M: 1 yearP: 10 years

Confess M: 10 yearsP: 1 year

M: 6 years P: 6 years

What is Peter’s best option if Martha doesn’t confess?

What is Peter’s best option if Martha confess?

Confess (1<2)Confess (6<10)

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Example 1: Prisoner’s Dilemma

Martha’s options

Don’t Confess Confess

Peter’s Options

Don’t Confess M: 2 years P: 2 years

M: 1 yearP: 10 years

Confess M: 10 yearsP: 1 year

M: 6 years P: 6 years

What is Martha’s best option if Peter doesn’t confess?

What is Martha’s best option if Peter Confesses?

Confess (1<2)Confess (6<10)

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Example 1: Prisoner’s Dilemma

Martha’s options

Don’t Confess Confess

Peter’s Options

Don’t Confess 2 years , 2 years 10 years , 1 year

Confess 1 year , 10 years 6 years , 6 years

Dominant Strategy – A strategy that results in the highest payoff to a player regardless of the opponent’s action.

First Payoff in each “Box” is Row Player’s Payoff .

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Example 2: Price Setting Game

Firm B’s options

Low Price High Price

Firm A’s Options

Low Price 0 , 0 50 , -10

High Price -10 , 50 10 , 10

Is there a dominant strategy for Firm B? Is there a dominant strategy for Firm A?

Low Price Low Price

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Nash Equilibrium

A condition describing a set of strategies in which no player can improve her payoff by unilaterally changing her own strategy, given the other player’s strategy. (Every player is doing the best they possibly can given the other player’s strategy.)

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Example 1: Nash?

Martha’s options

Don’t Confess Confess

Peter’s Options

Don’t Confess 2 years , 2 years 10 years , 1 year

Confess 1 year , 10 years 6 years , 6 years

Nash Equilibrium: (Confess, Confess)

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Example 2: Nash?

Firm B’s options

Low Price High Price

Firm A’s Options

Low Price 0 , 0 50 , -10

High Price -10 , 50 10 , 10

Nash Equilibrium: (Low Price, Low Price)

Chump, Chump, Chump

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http://videosift.com/video/Game-Theory-in-British-Game-Show-is-Tense?loadcomm=1

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Traffic and Nash EquilibriumQueuing conundrums; Traffic jams The Economist, September 13, 2008

Strange as it might seem, closing roads can cut delaysDRIVERS are becoming better informed, thanks to more accurate and timely advice on traffic conditions. Some services now use sophisticated computer-modelling which is fed with real-time data from road sensors, satellite-navigation systems and the analysis of how quickly anonymous mobile phones pass from one phone mast to another. Providing motorists with such information is supposed to help them pick faster routes. But the latest research shows that in some cases it may slow everybody down.Hyejin Youn and Hawoong Jeong, of the Korea Advanced Institute of Science and Technology, and Michael Gastner, of the Santa Fe Institute, analysed the effects of drivers taking different routes on journeys in Boston, New York and London. Their study, to be published in a forthcoming edition of Physical Review Letters, found that when individual drivers each try to choose the quickest route it can cause delays for others and even increase hold-ups in the entire road network.

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Traffic and Nash Equilibrium (cont.)

The physicists give a simplified example of how this can happen: trying to reach a destination either by using a short but narrow bridge or a longer but wide motorway. In their hypothetical case, the combined travel time of all the drivers is minimised if half use the bridge and half the motorway. But that is not what happens. Some drivers will switch to the bridge to shorten their commute, but as the traffic builds up there the motorway starts to look like a better bet, so some switch back. Eventually the traffic flow on the two routes settles into what game theory calls a Nash equilibrium, named after John Nash, the mathematician who described it. This is the point where no individual driver could arrive any faster by switching routes.

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Traffic and Nash Equilibrium (cont.)The researchers looked at how this equilibrium could arise if travelling across Boston from Harvard Square to Boston Common. They analysed 246 different links in the road network that could be used for the journey and calculated traffic flows at different volumes to produce what they call a "price of anarchy" (POA). This is the ratio of the total cost of the Nash equilibrium to the total cost of an optimal traffic flow directed by an omniscient traffic controller. In Boston they found that at high traffic levels drivers face a POA which results in journey times 30% longer than if motorists were co-ordinated into an optimal traffic flow. Much the same thing was found in London (a POA of up to 24% for journeys between Borough and Farringdon Underground stations) and New York (a POA of up to 28% from Washington Market Park to Queens Midtown Tunnel). Modifying the road network could reduce delays. And contrary to popular belief, a simple way to do that might be to close certain roads. This is known as Braess’s paradox, after another mathematician, Dietrich Braess, who found that adding extra capacity to a network can sometimes reduce its overall efficiency.

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Game Theory and PoliticsGame Theory for Swingers: What states should the candidates visit before Election Day? Oct. 25, 2004

Some campaign decisions are easy, even near the finish of a deadlocked race. Bush won't be making campaign stops in Maryland, and Kerry won't be running ads in Montana. The hot venues are Florida, Ohio, and Pennsylvania, which have in common rich caches of electoral votes and a coquettish reluctance to settle on one of their increasingly fervent suitors. Unsurprisingly, these states have been the three most frequent stops for both candidates. Conventional wisdom says Kerry can't win without Pennsylvania, which suggests he should concentrate all his energy there. But doing that would leave Florida and Ohio undefended and make it easier for Bush to win both. Maybe Kerry should foray into Ohio too, which might lead Bush to try to pick off Pennsylvania, which might divert his campaign's energy from Florida just enough for Kerry to snatch it away. ... You see the difficulty: As in any tactical problem, the best thing for Kerry to do depends on what Bush does, and the best thing for Bush to do depends on what Kerry does. At times like this, the division of mathematics that comes to our aid is game theory.

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Game Theory and Politics (cont.)To simplify our problem, let's suppose it's the weekend

before Election Day and each candidate can only schedule one more visit. We'll concede Pennsylvania to Kerry; then for Bush to win the election, he must win both Florida and Ohio. Let's say that Bush has a 30 percent chance of winning Ohio and a 70 percent chance at Florida. Furthermore, we'll assume that Bush can increase his chances by 10 percent in either state by making a last-minute visit there, and that Kerry can do the same. If Bush and Kerry both visit the same state, then Bush's chances remain 30 percent in Ohio and 70 percent in Florida, and his chance of winning the election is 0.3 x 0.7, or 21 percent. If Bush visits Ohio and Kerry goes to Florida, Bush has a 40 percent chance in Ohio and a 60 percent chance in Florida, giving him a 0.4 x 0.6, or 24 percent chance of an overall win. Finally, if Bush visits Florida and Kerry visits Ohio, Bush's chances are 20 percent and 80 percent, and his chance of winning drops to 16 percent.

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Example 3: Bush and Kerry

Kerry’s options

Ohio Florida

Bush’s Options

Ohio 21% , 79% 24% , 76%

Florida 16% , 84% 21% , 79%

Nash Equilibrium: (Ohio, Ohio)

.3*.7 .4*.6

.2*.8 .3*.7

Bush’s dominant strategy is to visit Ohio.

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EXAMPLE 4: Entry into a fast food market:

Burger King’s options

Enter Skaneateles

Don’t Enter Skaneateles

McDonalds’ Options

Enter Skaneateles BK= -40

M= -30

BK= 0

M= 50

Don’t Enter Skaneateles

BK= 40

M= 0

BK= 0

M= 0

Is there a dominant strategy for BK? NO Is there a dominant strategy for McD? NO

Is there a Nash Equilibrium(ia)?

Yes, there are 2 – (Enter, Don’t Enter) and (Don’t Enter, Enter). Implies, no need for a dominant strategy to have NE.

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Example 5: Cournot Example from Last Class

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90 100Q1

Q2

r1(Q2)

r2(Q1)26.67

26.67

Nash Equilibrium is Q1=26.67 and Q2=26.6

Do Firms have a dominant Strategy? No, output that maximizes

profits depends on output of other firm.

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EXAMPLE 6: Monitoring Workers

Worker’s options

Work Shirk

Manager’s Options

Monitor W: 1M: -1

W: -1M: 1

Don’t Monitor W: -1M: 1

W: 1M: -1

Is there a dominant strategy for the worker? Is there a dominant strategy for the manager?

NO NO

Is there a Nash Equilibrium(ia)? Not a pure strategy Nash Equilibrium– player chooses to take one action with probability 1

Randomize the actions yields a Nash = mixed strategy

John Nash proved an equilibrium always exists

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Mixed (randomized) Strategy

Definition:

A strategy whereby a player randomizes over two or more available actions in order to keep rivals from being able to predict his or her actions.

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Calculating Mixed Strategy EXAMPLE 6: Monitoring Workers Manager randomizes (i.e. monitors with

probability PM) in such a way to make the worker indifferent between working and shirking.

Worker randomizes (i.e. works with probability Pw) in such a way as to make the manager indifferent between monitoring and not monitoring.

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Example 6: Mixed Strategy

Worker’s options

Work Shirk

Manager’s Options

Monitor W: 1M: -1

W: -1M: 1

Don’t Monitor W: -1M: 1

W: 1M: -1

PM

1-PM

PW1-PW

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Manager selects PM to make Worker indifferent between working and shirking (i.e., same expected payoff)

Worker’s expected payoff from working

PM*(1)+(1- PM)*(-1) = -1+2*PM

Worker’s expected payoff from shirking

PM*(-1)+(1- PM)*(1) = 1-2*PM

Worker’s expected payoff the same from working and shirking if PM=.5. This expected payoff is 0 (-1+2*.5=0 and 1-2*.5=0). Therefore, worker’s best response is to either work or shirk or randomize between working and shirking.

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Worker selects PW to make Manager indifferent between monitoring and not monitoring.

Manager’s expected payoff from monitoring

PW*(-1)+(1- PW)*(1) = 1-2*PW

Manager’s expected payoff from not monitoring

PW*(1)+(1- PW)*(-1) = -1+2*PW

Manager’s expected payoff the same from monitoring and not monitoring if PW=.5. Therefore, the manager’s best response is to either monitor or not monitor or randomize between monitoring or not monitoring .

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Nash Equilibrium of Example 6

Worker works with probability .5 and shirks with probability .5 (i.e., PW=.5)

Manager monitors with probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5)Neither the Worker nor the Manager can increase their expected payoff by playing some other strategy (expected payoff for both is zero). They are both playing a best response to the other player’s strategy.

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Example 6A: What if costs of Monitoring decreases and Changes the Payoffs for Manager

Worker’s options

Work Shirk

Manager’s Options

Monitor W: 1M: -1

W: -1M: 1

Don’t Monitor W: -1M: 1

W: 1M: -1

-.5 1.5

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Nash Equilibrium of Example 6A where cost of monitoring decreased Worker works with probability .625 and

shirks with probability .375 (i.e., PW=.625) Same as in Ex. 5, Manager monitors with

probability .5 and doesn’t monitor with probability .5 (i.e., PM=.5)

The decrease in monitoring costs does not change the probability that the manager monitors. However, it increases the probability that the worker works.

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Example 7

A Beautiful Mind

http://www.youtube.com/watch?v=CemLiSI5ox8

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Example 7: A Beautiful Mind

Nash Equilibria: (Pursue Blond, Pursue Brunnette 1) (Pursue Blond, Pursue Brunnette 2) (Pursue Brunnette 1, Pursue Blond) (Pursue Brunnette 2, Pursue Blond)

Other Student’s Options

Pursue

Blond

Pursue

Brunnette 1

Pursue

Brunnette 2

John Nash’s

Pursue

Blond0 , 0 100 , 50 100 , 50

Options Pursue

Brunnette 150 , 100 0 , 0 50 , 50

Pursue

Brunnette 250 , 100 50 , 50 0 , 0

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Sequential/Multi-Stage Games

Extensive form game: A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoffs resulting from alternative strategies.

(Often used to depict games with sequential play.)

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Potential Entrant

Don’t Enter Enter

Incumbent Firm

Price War Share Market (Hard) (Soft)

Potential Entrant: -1 +5Incumbent: +1 +5

Potential Entrant: 0 Incumbent: +10

Example 8

What are the Nash Equilibria?

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Nash Equilibria

1. (Potential Entrant Enter,

Incumbent Firm Shares Market)

2. (Potential Entrant Don’t Enter, Incumbent Firm Price War)

Is one of the Nash Equilibrium more likely to occur? Why?

Perhaps (Enter, Share Market) because it doesn’t rely on a non-credible threat.

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Subgame Perfect Equilibrium

A condition describing a set of strategies that constitutes a Nash Equilibrium and allows no player to improve his own payoff at any stage of the game by changing strategies.

(Basically eliminates all Nash Equilibria that rely on a non-credible threat – like Don’t Enter, Price War in Prior Game)

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Potential Entrant

Don’t Enter Enter

Incumbent Firm

Price War Share Market (Hard) (Soft)

Potential Entrant: -1 +5Incumbent: +1 +5

Potential Entrant: 0 Incumbent: +10

Example 8

What is the Subgame Perfect Equilibrium?

(Enter, Share Market)

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Big Ten Burrito

Enter Don’t Enter

Chipotle Chipotle

Enter Don’t Enter Don’tEnter Enter

BTB: -25 +40 0 0Chip: -50 0 +70 0

Example 9

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Big Ten Burrito

Enter Don’t Enter

Chipotle Chipotle

Enter Don’t Enter Don’tEnter Enter

BTB: -25 +40 0 0Chip: -50 0 +70 0

Use Backward Induction to Determine Subgame Perfect Equilibrium.

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Subgame Perfect Equilibrium

Chipotle should choose Don’t Enter if BTB chooses Enter and Chipotle should choose Enter if BTB chooses Don’t Enter.

BTB should choose Enter given Chipotle’s strategy above.

Subgame Perfect Equilibrium:

(BTB chooses Enter, Chipotle chooses Don’t Enter if BTB chooses Enter and Enter if BTB chooses Don’t Enter.)

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U.S. Postal Service and AnthraxIs Mail Safer Since Anthrax Attacks?Questions Remain About Post Office Security 5 Years After 5 Died

HAMILTON, N.J., Sept. 23, 2006 Five years ago next week, American officials began to suspect that someone was sending anthrax-tainted letters through the mail. Five people eventually died and 17 other became ill as a result. The attacks remain unsolved, but there have been some security upgrades to the nation's postal system. The question remains: are we any safer? The U.S. Postal Service's Tom Day helped design the system that now tests for anthrax at all 280 mail processing centers across the country. He gave CBS News correspondent Bianca Solarzano a tour of the John K. Rafferty Hamilton Post Office Building.

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U.S. Postal Service and Anthrax (cont.)

"This was the first spot where the anthrax was coming out of the envelopes," Day said, pointing to a mail sorting machine. There has been a tunnel-like addition to the machine where letters collected from mail boxes are checked for anthrax. "If anything is escaping from an envelope at this point, we're collecting it and pulling it out through a system right here," Day said. "That, then, goes to this box which is the self contained detection system." The system's cost: $150 million per year. So, after all the improvements, is our mail safe? "I would definitely say the mail in this country is safe," Day said. "Can I give a 100 percent guarantee? The answer is 'no.'"

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US Postal Service

Buy Protector Don’t Buy Protector

Unstable Person Unstable Person

Send Don’t Send Send Don’t SendAnthrax Anth Anth Anthrax

USPS: -600 -400 -1000 0Person: -10 0 +10 0Subgame Perfect Equilibrium:

(US Postal Service Buys Protector;

Unstable Person Doesn’t Send Anthrax if USPS Buys Protector and Sends Anthrax if USPS Doesn’t Buy Protector)

Example 10

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Slide from Oligopoly Lecture

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

=AVC=ATC

Firm 1’s Profits = 60*20-20*20=800

Firm 2’s Profits = 60*20-20*20=800

Example 11

If firms collude on Q1=20 and Q2=20

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Slide from Oligopoly Lecture

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

=AVC=ATC

Firm 1’s Profits = 50*30-20*30=900

Firm 2’s Profits = 50*20-20*20=600

Example 11

Firms colluding is unlikely if they interact once because firms have incentive to cheat – in above case Firm 1 increases profits by cheating and producing 30 units.

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Slide From Oligopoly Lecture1. Repeated Interaction

Suppose Firm 1 thinks Firm 2 won’t deviate from Q2=20 if Firm 1 doesn’t deviate from collusive agreement of Q1=20 and Q2=20. In addition, Firm 1 thinks Firm 2 will produce at an output of 80 in all future periods if Firm 1 deviates from collusive agreement of Q1=20 and Q2=20.

Firm 1’s profits from not cheating

Firm 1’s profits from cheating (by producing Q1=30 Today)

Today In 1 Year In 2 Years In 3 Years In 4 Years

800 800 800 800 800 …

Today In 1 Year In 2 Years In 3 Years In 4 Years

900 0 0 0 0…

Does Firm 2’s Strategy Rely on a Non-credible Threat?

Depends on Game –unlikely to be credible even if infinitely repeated game

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What if Firms interact for 2 periods as Cournot Competitors? What is Subgame Perfect Equilibrium?

Use Backward Induction!!

In the second period, what will happen?

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Cournot Equilibrium: Q1=26.67 and Q2=26.67

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90 100Q1

Q2

r1(Q2)

r2(Q1)26.67

26.67

IN 2ND PERIOD!!!!

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Profits from Cournot Equilibrium: Q1=26.67 and Q2=26.67 so Q=Q1+Q2=53.3

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

53.33

46.66

=AVC=ATC

Firm 1 Profits=46.66*26.67-20*26.67= 713

Firm 2 Profits=46.66*26.67-20*26.67= 713

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In the 1st period, what will happen?If both firms realize that each will produce an output of 26.67 in the 2nd period (resulting in profits of $713 for each firm) no matter what occurs in the 1st period, then the equilibrium the 1st period should be for both firms to produce 26.67 and obtain profits of $713 the 1st period.

Using this logic, the Subgame Perfect Equilibrium is for each firm to produce 26.67 units of output the 1st period and 26.67 units of output the 2nd period.

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What if Firms interact for 1000 periods as Cournot Competitors? What is Subgame Perfect Equilibrium?

Using similar logic as when the firms interact 2 periods, the Subgame Perfect Equilibrium is for each firm to produce 26.67 units of output each period.

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Do you really expect this type of outcome if the firms interact 1000 periods?Laboratory experiments suggest that when facing a player a finite number of times, the players will “collude” for a number of periods. Many of these experiments involve a prisoners dilemma game being played a finite number of times.

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In the real world, how do firms (and individuals) and individuals address the finite period problem?

Attempt to build in uncertainty associated with when the final period occurs.

Attempt to “change game”.

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Dan Conlin

Invest in Firm Don’t InvestSpecific Knowledge

Dan Conlin: wI-CI wDI

Marsh&McClennan: 200-wI 150-wDI

Example 12: The Hold-Up Problem

Dan Conlin and M&M negotiate salary

Dan Conlin and M&M negotiate salary

Let wI and wDI denote Dan’s wage if he invests and doesn’t invest in the firm specific knowledge, respectively. Let the cost of investing for Dan be CI and let CI=30. Dan Conlin is worth 200 to M&M if he invests and is worth 150 if he doesn’t.

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Dan Conlin

Invest in Firm Don’t InvestSpecific Knowledge

Dan Conlin: wI-CI wDI

Marsh&McClennan: 200-wI 150-wDI

Example 12: The Hold-Up Problem

Dan Conlin and M&M negotiate salary

Dan Conlin and M&M negotiate salary

Assume that Dan’s best “outside option” is a wage of 100 whether or not he invests in the firm specific knowledge and that the outcome of the negotiations are such that Dan and M&M split the surplus. This means that wI=150 and wDI=125.

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Dan Conlin

Invest in Firm Don’t InvestSpecific Knowledge

Dan Conlin: wI-CI=150-30 wDI=125Marsh&McClennan: 200-wI=200-150 150-wDI=150-125

Example 12: The Hold-Up Problem

Dan Conlin and M&M negotiate salary

Dan Conlin and M&M negotiate salary

Subgame Perfect Equilibrium outcome has Dan Conlin not investing in the firm specific knowledge and receiving a wage of 125 even though the cost of the knowledge is 30 and it increases his value to the firm by 50.

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Dan Conlin

Invest in Firm Don’t InvestSpecific Knowledge

Dan Conlin: wI-CI=150-30 wDI=125Marsh&McClennan: 200-wI=200-150 150-wDI=150-125

Example 12: The Hold-Up Problem

Dan Conlin and M&M negotiate salary

Dan Conlin and M&M negotiate salary

What would you expect to happen in this case?

Dan Conlin and M&M would divide cost of obtaining the knowledge.

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Dan Conlin

Invest in Don’t InvestGeneral Knowledge

Dan Conlin: wI-CI=160-30 wDI =125Marsh&McClennan: 200-wI=200-160 150-wDI=150-125

Example 13: General Knowledge Investment

Dan Conlin and M&M negotiate salary

Dan Conlin and M&M negotiate salary

Assume the game is as in the “hold-up” problem but that Dan’s best “outside option” is a wage of 120 if he invests in general knowledge and 100 if he does not. This means that wI=160 and wDI=125 (assuming split surplus when negotiate).

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Dan Conlin

Invest in Don’t InvestGeneral Knowledge

Dan Conlin: wI-CI=160-30 wDI =125Marsh&McClennan: 200-wI=200-160 150-wDI=150-125

Example 13: General Knowledge Investment

Dan Conlin and M&M negotiate salary

Dan Conlin and M&M negotiate salary

Subgame Perfect Equilibrium outcome has Dan Conlin investing in the general knowledge and receiving a wage of 160.

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Example 14: Hold-up Problem (same idea as the Fisher Auto-body / GM situation)

Suppose there are two players: a computer chip maker (MIPS) and a computer manufacturer (Silicon Graphics). Initially, MIPS decides whether or not to customize its chip (the quantity of which is normalized to one) for a specific manufacturing purpose of Silicon Graphics. The customization costs $75 to MIPS, but adds value of $100 to the chip only when it is used by Silicon Graphics . The value of customization is partially lost when the chip is sold to an alternative buyer, who is willing to pay $60. If MIPS decides not to customize the chip, it can sell a standardized chip to Silicon Graphics at a price of zero and Silicon Graphics earns a payoff of zero from using the chip. If MIPS customizes the chip, the two players enter into a bargaining game where Silicon Graphics makes a take-it-or-leave-it price offer to MIPS. In response to this, MIPS can either accept the offer (in which case the game ends) or reject it (in which case MIPS approaches an alternative buyer who pays $60).

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MIPS

Customize Don’t Customize

MIPS: p-75 60-75= -15Silicon Graphics: 100-p 0

Example 14: Hold-Up Problem

Subgame Perfect Equilibrium – MIPS accepts price p if p>60. Silicone Graphics offers a price p=60. MIPS does not customize. The outcome of this game is that MIPS does not customize even though there is a surplus of $25 to be gained.

MIPS

Silicone Graphics

Accept Reject

Offer Price p

0 : MIPS

0 : Silicon Graphics

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Is the Hold-Up Problem Applicable to other Situations?1. Upstream Firm Investing in Specific Capital to produce

input for Downstream Firm.

Coal Mines located next to Power Plants.

2. An academic buying a house before getting tenure or a big promotion.

3. Taxing of Oil and Gas Lines by local jurisdictions.

4. Multinational firms operating in foreign countries (Foreign Direct Investment)

5. East Lansing Public Schools allocating a certain amount of money for capital expenditures and a certain amount for operating expenditures

YES

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Using Game Theory to Devise Strategies in Oligopolies that Increase ProfitsExamples:1. Price Matching- advertise a price and promise to

match any lower price offered by a competitor.

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

D

MCQ

Bertrand Oligopoly

In the end, you would expect both firms to set a price of $20 (equal to MC) and have zero profits.

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Using Game Theory to Devise Strategies in Oligopolies that Increase ProfitsExamples:1. Price Matching- advertise a price and promise

to match an lower price offered by a competitor. In Bertrand example, perhaps each firm would set a price of $60 and say will match.

2. Induce Brand Loyalty – frequent flyer program3. Randomized pricing – inhibits consumers

learning as to who offers lower price and reduces ability of competitors to undercut price.