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 why a movie theater shows a preview
of an upcoming movie that is 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.
• http//www.aviationairportdevelopmentlaw.com/2009/
  10/articles/faa-1/it-is-official-the-faa-rescinds-
  slot-auction-rule/

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)
Marthas options Marthas options
Dont Confess Confess
Peters Options Dont Confess M 2 years P 2 years M 1 year P 10 years
Peters Options Confess M 10 years P 1 year M 6 years P 6 years
Confess (1lt2)
What is Peters best option if Martha doesnt
confess?
Confess (6lt10)
What is Peters best option if Martha confess?
19
Example 1 Prisoners Dilemma
Marthas options Marthas options
Dont Confess Confess
Peters Options Dont Confess M 2 years P 2 years M 1 year P 10 years
Peters Options Confess M 10 years P 1 year M 6 years P 6 years
Confess (1lt2)
What is Marthas best option if Peter doesnt
confess?
Confess (6lt10)
What is Marthas best option if Peter Confesses?
20
Example 1 Prisoners Dilemma
Marthas options Marthas options
Dont Confess Confess
Peters Options Dont Confess 2 years , 2 years 10 years , 1 year
Peters Options Confess 1 year , 10 years 6 years , 6 years
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
Example 2 Price Setting Game
Firm Bs options Firm Bs options
Low Price High Price
Firm As Options Low Price 0 , 0 50 , -10
Firm As Options High Price -10 , 50 10 , 10
Is there a dominant strategy for Firm B?
Low Price
Is there a dominant strategy for Firm A?
Low Price
22
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.)

23
Example 1 Nash?
Marthas options Marthas options
Dont Confess Confess
Peters Options Dont Confess 2 years , 2 years 10 years , 1 year
Peters Options Confess 1 year , 10 years 6 years , 6 years
Nash Equilibrium (Confess, Confess)
24
Example 2 Nash?
Firm Bs options Firm Bs options
Low Price High Price
Firm As Options Low Price 0 , 0 50 , -10
Firm As Options High Price -10 , 50 10 , 10
Nash Equilibrium (Low Price, Low Price)
25
Chump, Chump, Chump
http//videosift.com/video/Game-Theory-in-British-
Game-Show-is-Tense?loadcomm1
26
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.

27
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.

28
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 Braesss paradox, after another
  mathematician, Dietrich Braess, who found that
  adding extra capacity to a network can sometimes
  reduce its overall efficiency.

29
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.

30
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.

31
Example 3 Bush and Kerry
Kerrys options Kerrys options
Ohio Florida
Bushs Options Ohio 21 , 79 24 , 76
Bushs Options Florida 16 , 84 21 , 79
Bushs dominant strategy is to visit Ohio.
.3.7
.4.6
.2.8
.3.7
Nash Equilibrium (Ohio, Ohio)
32
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.
Burger Kings options Burger Kings options
Enter Skaneateles Dont Enter Skaneateles
McDonalds Options Enter Skaneateles PBK -40 PM -30 PBK 0 PM 50
McDonalds Options Dont Enter Skaneateles PBK 40 PM 0 PBK 0 PM 0
NO
Is there a dominant strategy for BK?
NO
Is there a dominant strategy for McD?
33
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
34
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
Workers options Workers options
Work Shirk
Managers Options Monitor W 1 M -1 W -1 M 1
Managers Options Dont Monitor W -1 M 1 W 1 M -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?
35
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.

36
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.

37
Example 6 Mixed Strategy
Workers options Workers options
Work Shirk
Managers Options Monitor W 1 M -1 W -1 M 1
Managers Options Dont Monitor W -1 M 1 W 1 M -1
1-PW
PW
PM
1-PM
38
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.
39
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 .
40
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 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.
41
Example 6A What if costs of Monitoring decreases
and Changes the Payoffs for Manager
Workers options Workers options
Work Shirk
Managers Options Monitor W 1 M -1 W -1 M 1
Managers Options Dont Monitor W -1 M 1 W 1 M -1
1.5
-.5
42
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 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.
43
Example 7

• A Beautiful Mind
• http//www.youtube.com/watch?vCemLiSI5ox8

44
Example 7 A Beautiful Mind
Other Students Options
Pursue Blond Pursue Brunnette 1 Pursue Brunnette 2
John Nashs Pursue Blond 0 , 0 100 , 50 100 , 50
Options Pursue Brunnette 1 50 , 100 0 , 0 50 , 50
Pursue Brunnette 2 50 , 100 50 , 50 0 , 0
Nash Equilibria (Pursue Blond, Pursue Brunnette
1) (Pursue Blond, Pursue Brunnette 2)
(Pursue Brunnette 1, Pursue Blond)
(Pursue Brunnette 2, Pursue Blond)
45
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.)

46
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?
47
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.
48
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)

49
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)
50
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
51
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.
52
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.)
53
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.

54
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.'"

55
US Postal Service
Example 10
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)
56
Slide from Oligopoly Lecture
Example 11

Firm 1s Profits 6020-2020800
Firm 2s Profits 6020-2020800
AVCATC
If firms collude on Q120 and Q220
57
Slide from Oligopoly Lecture
Example 11

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.
58
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)

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 2s Strategy Rely on a Non-credible
Threat?
Depends on Game unlikely to be credible even if
infinitely repeated game
59
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?

60
Cournot Equilibrium Q126.67 and Q226.67
IN 2ND PERIOD!!!!

r1(Q2)
26.67
r2(Q1)
26.67
61
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
62
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.
63
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.
64
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.
65
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.
66
Example 12 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.
67
Example 12 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.
68
Example 12 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.
69
Example 12 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.
70
Example 13 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).
71
Example 13 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.
72
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).

73
Example 14 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 pgt60. 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.
74
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)
• East Lansing Public Schools allocating a certain
  amount of money for capital expenditures and a
  certain amount for operating expenditures

75
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.
76
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.