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Game Theory

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Title: Game Theory


1
Game Theory
  • Chapter 10

2
Applications of Game Theory
  • National Defense Terrorism and Cold War
  • Movie Release Dates and Program Scheduling
  • Auctions http//en.wikipedia.org/wiki/Spectrum_auc
    tion
  • http//en.wikipedia.org/wiki/United_States_2008_w
    ireless_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

3
Game Theory and Terrorism
  • Game 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.

4
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

5
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.

6
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.

7
Applications of Game Theory
  • National Defense Terrorism and Cold War
  • Movie Release Dates and Program Scheduling
  • Auctions http//en.wikipedia.org/wiki/Spectrum_auc
    tion
  • http//en.wikipedia.org/wiki/United_States_2008_w
    ireless_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

8
Greys Anatomy vs. The Donald
  • NBC delays 'Apprentice' premiere
  • By 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,

9
Greys Anatomy vs. The Donald
  • 'Grey' move has NBC red Peacock shifts
    'Apprentice' back
  • By 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.

10
Game Theory and Movie Release Dates
  • The Imperfect Science of Release Dates
  • New 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.

11
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 a movie theater shows a preview of an
upcoming movie to be released in 2 years?
12
Applications of Game Theory
  • National Defense Terrorism and Cold War
  • Movie Release Dates and Program Scheduling
  • Auctions http//en.wikipedia.org/wiki/Spectrum_auc
    tion
  • http//en.wikipedia.org/wiki/United_States_2008_w
    ireless_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

13
FAA Auctions
  • Blame 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.

14
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.

15
Applications of Game Theory
  • National Defense Terrorism and Cold War
  • Movie Release Dates and Program Scheduling
  • Auctions http//en.wikipedia.org/wiki/Spectrum_auc
    tion
  • http//en.wikipedia.org/wiki/United_States_2008_w
    ireless_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

16
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 players
    move (ex. Stackelberg Oligopoly).

17
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.

18
Example 1 Prisoners Dilemma(Normal Form of
Simultaneous Move Game)
Confess (1What is Peters best option if Martha doesnt
confess?
Confess (6What is Peters best option if Martha confess?
19
Example 1 Prisoners Dilemma
Confess (1What is Marthas best option if Peter doesnt
confess?
Confess (6What is Marthas best option if Peter Confesses?
20
Example 1 Prisoners Dilemma
First Payoff in each Box is Row Players Payoff
.
Dominant Strategy A strategy that results in
the highest payoff to a player regardless of the
opponents action.
21
Game Theory and Altruism
  • Mathematics and faith explain altruism
  • The Boston Globe, September 27, 2008 Saturday
  • If evolution is all about survival of the
    fittest, then why have humans evolved a sense of
    altruism and cooperation? The seeming
    contradiction has engaged theologians,
    scientists, and even comic book writers (think
    the Incredible Hulk) who've probed human duality
    and how its good half sometimes empowers
    selflessness to override self-interest
  • The British biologist and atheist Richard
    Dawkins believes that altruism in modern humans
    is essentially an evolutionary oops, albeit a
    beneficial one. It paid off in prehistory, when
    people lived in clans and protecting others meant
    the survival of their own gene pools now that
    we've expanded into large cities, our instinct to
    help others still kicks in, even though those we
    aid may have no relation to us.

22
Game Theory and Altruism
  • On the other hand, Francis Collins, former
    director of the National Human Genome Research
    Institute and a Christian, sees in our
    willingness to work with others the handprint of
    God.
  • Then there is Harvard's Martin Nowak. A
    mathematician and biologist, he agrees with
    Dawkins's explanation of how we evolved to be
    good Samaritans. Yet as a Catholic, he rejects
    Dawkins's notion that believing in evolution
    precludes belief in a God who included altruism
    in evolution's bequest to us. Needless to say, he
    also rejects Dawkins's disdain for believers as
    scientifically illiterate yahoos. This
    Vienna-born mathematician says that if you do the
    math, you'll find that cooperation is more than
    just a nice leftover from humanity's infancy
    it's a winning strategy for living, a way to
    thrive.

23
Game Theory and Altruism
  • For the past three years, with Sarah Coakley,
    formerly of Harvard Divinity School and now at
    Cambridge University in England, Nowak pursued a
    study project, the title of which - "The
    Evolution and Theology of Cooperation"- gives a
    clue to its partnership between science and
    religion. Nowak said his work demonstrated the
    mathematical probability that being cooperative,
    generous, and forgiving produces better results
    for people than looking out for Number One.
  • As part of his demonstration, Nowak devised
    repeated rounds of an exercise from game theory
    called the prisoner's dilemma. The math is
    complex to laypeople, but the basic premise of
    the game is straightforward Two prisoners held
    separately are given their options If both stay
    silent, each gets six months in jail. If both
    implicate the other, they each get five years. If
    one turns traitor and the other stays mum, the
    gabby prisoner goes free, but the other gets 10
    years. Neither knows what the other will do.
  • In isolation, each thinks Finking on the other
    guy could bring me freedom, but it could also
    bring us both five years. Cooperating with each
    other, by both of us clamming up, guarantees a
    short, six-month sentence. Mathematically
    speaking, Nowak said, cooperation is the best bet.

24
Example 2 Price Setting Game
Is there a dominant strategy for Firm B?
Low Price
Is there a dominant strategy for Firm A?
Low Price
25
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 players strategy. (Every player is doing
    the best they possibly can given the other
    players strategy.)

26
Example 1 Nash?
Nash Equilibrium (Confess, Confess)
27
Example 2 Nash?
Nash Equilibrium (Low Price, Low Price)
28
Traffic and Nash Equilibrium
  • Queuing conundrums Traffic jams
  • The Economist, September 13, 2008
  • Strange as it might seem, closing roads can cut
    delays
  • DRIVERS 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.

29
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.

30
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.

31
Game Theory and Politics
  • Game 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.

32
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.

33
Example 3 Bush and Kerry
Bushs dominant strategy is to visit Ohio.
.3.7
.4.6
.2.8
.3.7
Nash Equilibrium (Ohio, Ohio)
34
EXAMPLE 4 Entry into a fast food market
Is there a Nash Equilibrium(ia)?
Yes, there are 2 (Enter, Dont Enter) and
(Dont Enter, Enter). Implies, no need for a
dominant strategy to have NE.
NO
Is there a dominant strategy for BK?
NO
Is there a dominant strategy for McD?
35
Example 5 Cournot Example from Last Class
Nash Equilibrium is Q126.67 and Q226.6

r1(Q2)
Do Firms have a dominant Strategy?
No, output that maximizes profits depends on
output of other firm.
26.67
r2(Q1)
26.67
36
EXAMPLE 6 Monitoring Workers
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
NO
Is there a dominant strategy for the worker?
NO
Is there a dominant strategy for the manager?
37
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.

38
Calculating Mixed Strategy EXAMPLE 5 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.

39
Example 6 Mixed Strategy
1-PW
PW
PM
1-PM
40
Manager selects PM to make Worker indifferent
between working and shirking (i.e., same expected
payoff)
  • Workers expected payoff from working
  • PM(1)(1- PM)(-1) -12PM
  • Workers expected payoff from shirking
  • PM(-1)(1- PM)(1) 1-2PM

Workers expected payoff the same from working
and shirking if PM.5. This expected payoff is 0
(-12.50 and 1-2.50). Therefore, workers
best response is to either work or shirk or
randomize between working and shirking.
41
Worker selects PW to make Manager indifferent
between monitoring and not monitoring.
  • Managers expected payoff from monitoring
  • PW(-1)(1- PW)(1) 1-2PW
  • Managers expected payoff from not monitoring
  • PW(1)(1- PW)(-1) -12PW

Managers expected payoff the same from
monitoring and not monitoring if PW.5.
Therefore, the managers best response is to
either monitor or not monitor or randomize
between monitoring or not monitoring .
42
Nash Equilibrium of Example 5
  • Worker works with probability .5 and shirks with
    probability .5 (i.e., PW.5)
  • Manager monitors with probability .5 and doesnt
    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
players strategy.
43
Example 6A What if costs of Monitoring decreases
and Changes the Payoffs for Manager
1.5
-.5
44
Nash Equilibrium of Example 5A 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 doesnt 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.
45
Example 7
  • A beautiful mind

46
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)
47
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.)

48
Potential Entrant
Example 8
Dont Enter Enter
Incumbent Firm
Potential Entrant 0 Incumbent
10
Price War Share Market
(Hard) (Soft)
Potential Entrant -1 5 Incumbent
1 5
What are the Nash Equilibria?
49
Nash Equilibria
  • (Potential Entrant Enter,
  • Incumbent Firm Shares Market)
  • (Potential Entrant Dont Enter, Incumbent Firm
    Price War)

Is one of the Nash Equilibrium more likely to
occur? Why?
Perhaps (Enter, Share Market) because it doesnt
rely on a non-credible threat.
50
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 Dont Enter,
    Price War in Prior Game)

51
Potential Entrant
Example 8
Dont Enter Enter
Incumbent Firm
Potential Entrant 0 Incumbent
10
Price War Share Market
(Hard) (Soft)
Potential Entrant -1 5 Incumbent
1 5
What is the Subgame Perfect Equilibrium?
(Enter, Share Market)
52
Big Ten Burrito
Example 9
Enter Dont Enter
Chipotle Chipotle
Enter Dont Enter
Dont Enter Enter
BTB -25 40 0 0 Chip -50
0 70 0
53
Big Ten Burrito
Enter Dont Enter
Chipotle Chipotle
Enter Dont Enter
Dont Enter Enter
BTB -25 40 0 0 Chip -50
0 70 0
Use Backward Induction to Determine Subgame
Perfect Equilibrium.
54
Subgame Perfect Equilibrium
Chipotle should choose Dont Enter if BTB chooses
Enter and Chipotle should choose Enter if BTB
chooses Dont Enter. BTB should choose Enter
given Chipotles strategy above.

Subgame Perfect Equilibrium (BTB chooses Enter,
Chipotle chooses Dont Enter if BTB chooses Enter
and Enter if BTB chooses Dont Enter.)
55
U.S. Postal Service and Anthrax
  • Is 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.

56
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.'"

57
US Postal Service
Example 11
Buy Protector Dont Buy Protector
Unstable Person Unstable Person
Send Dont Send Send Dont
Send Anthrax Anth Anth Anthrax
USPS -600 -400 -1000 0 Person
-10 0 10 0
Subgame Perfect Equilibrium (US Postal Service
Buys Protector Unstable Person Doesnt Send
Anthrax if USPS Buys Protector and Sends Anthrax
if USPS Doesnt Buy Protector)
58
Slide from Oligopoly Lecture
Example 12

Firm 1s Profits 6020-2020800
Firm 2s Profits 6020-2020800
AVCATC
If firms collude on Q120 and Q220
59
Slide from Oligopoly Lecture
Example 12

Firm 1s Profits 5030-2030900
Firm 2s Profits 5020-2020600
AVCATC
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.
60
Slide From Oligopoly Lecture
  • Repeated Interaction
  • Suppose Firm 1 thinks Firm 2 wont deviate from
    Q220 if Firm 1 doesnt deviate from collusive
    agreement of Q120 and Q220. 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 Q120 and Q220.
  • Firm 1s profits from not cheating
  • Firm 1s profits from cheating (by producing
    Q130 Today)



Does Firm 2s Strategy Rely on a Non-credible
Threat?
Depends on Game unlikely to be credible even if
infinitely repeated game
61
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?

62
Cournot Equilibrium Q126.67 and Q226.67
IN 2ND PERIOD!!!!

r1(Q2)
26.67
r2(Q1)
26.67
63
Profits from Cournot Equilibrium Q126.67 and
Q226.67 so QQ1Q253.3

Firm 1 Profits46.6626.67-2026.67 713
Firm 2 Profits46.6626.67-2026.67 713
46.66
AVCATC
53.33
64
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.
65
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.
66
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.
67
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.
68
Example 13 The Hold-Up Problem
Dan Conlin
Invest in Firm Dont
Invest Specific Knowledge

Dan Conlin and MM negotiate salary
Dan Conlin and MM negotiate salary
Dan Conlin wI-CI
wDI MarshMcClennan 200-wI
150-wDI
Let wI and wDI denote Dans wage if he invests
and doesnt invest in the firm specific
knowledge, respectively. Let the cost of
investing for Dan be CI and let CI30. Dan Conlin
is worth 200 to MM if he invests and is worth
150 if he doesnt.
69
Example 13 The Hold-Up Problem
Dan Conlin
Invest in Firm Dont
Invest Specific Knowledge

Dan Conlin and MM negotiate salary
Dan Conlin and MM negotiate salary
Dan Conlin wI-CI
wDI MarshMcClennan 200-wI
150-wDI
Assume that Dans 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 MM split the
surplus. This means that wI150 and wDI125.
70
Example 13 The Hold-Up Problem
Dan Conlin
Invest in Firm Dont
Invest Specific Knowledge

Dan Conlin and MM negotiate salary
Dan Conlin and MM negotiate salary
Dan Conlin wI-CI150-30
wDI125 MarshMcClennan
200-wI 200-150 150-wDI150-125
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.
71
Example 13 The Hold-Up Problem
Dan Conlin
Invest in Firm Dont
Invest Specific Knowledge

Dan Conlin and MM negotiate salary
Dan Conlin and MM negotiate salary
Dan Conlin wI-CI150-30
wDI125 MarshMcClennan
200-wI 200-150 150-wDI150-125
What would you expect to happen in this case?
Dan Conlin and MM would divide cost of obtaining
the knowledge.
72
Example 14 General Knowledge Investment
Dan Conlin
Invest in Dont Invest General Knowledge

Dan Conlin and MM negotiate salary
Dan Conlin and MM negotiate salary
Dan Conlin wI-CI160-30 wDI
125 MarshMcClennan 200-wI 200-160
150-wDI150-125
Assume the game is as in the hold-up problem
but that Dans best outside option is a wage of
120 if he invests in general knowledge and 100 if
he does not. This means that wI160 and wDI125
(assuming split surplus when negotiate).
73
Example 14 General Knowledge Investment
Dan Conlin
Invest in Dont Invest General Knowledge

Dan Conlin and MM negotiate salary
Dan Conlin and MM negotiate salary
Dan Conlin wI-CI160-30 wDI
125 MarshMcClennan 200-wI 200-160
150-wDI150-125
Subgame Perfect Equilibrium outcome has Dan
Conlin investing in the general knowledge and
receiving a wage of 160.
74
Example 15 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).

75
Example 15 Hold-Up Problem
MIPS
Customize Dont Customize
Silicone Graphics
0 MIPS 0 Silicon Graphics
Offer Price p

MIPS
Accept Reject
MIPS p-75 60-75 -15 Silicon
Graphics 100-p 0
Subgame Perfect Equilibrium MIPS accepts price
p if p60. Silicone Graphics offers a price
p60. 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.
76
Is the Hold-Up Problem Applicable to other
Situations?
YES
  • Upstream Firm Investing in Specific Capital to
    produce input for Downstream Firm.
  • Coal Mines located next to Power Plants.
  • An academic buying a house before getting tenure
    or a big promotion.
  • Taxing of Oil and Gas Lines by local
    jurisdictions.
  • Multinational firms operating in foreign
    countries (Foreign Direct Investment)

77
Using Game Theory to Devise Strategies in
Oligopolies that Increase Profits
  • Examples
  • Price Matching- advertise a price and promise to
    match any lower price offered by a competitor.

Bertrand Oligopoly In the end, you would expect
both firms to set a price of 20 (equal to MC)
and have zero profits.
78
Using Game Theory to Devise Strategies in
Oligopolies that Increase Profits
  • Examples
  • 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.
  • Induce Brand Loyalty frequent flyer program
  • Randomized pricing inhibits consumers learning
    as to who offers lower price and reduces ability
    of competitors to undercut price.
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