Incomplete Information

1
Incomplete Information
This lecture shows how to analyze games that have
more complicated information sets than complete
information games. This requires us to apply
rules, already learned in previous lectures, to
more complicated settings.
2
Incomplete information games

• All games are either complete information games
  or incomplete information games.
• Therefore simultaneous move and perfect
  information games are not games of incomplete
  information.
• A game has incomplete information if one player,
  when making her move, does not know information
  that would affect her choice that was known by
  another player when he had moved at an earlier
  point in the game, that is at a predecessor node.
• Read Chapter 12, Incomplete Information Games
  in Strategic Play.

3
Defining incomplete information games

• Suppose one player called Alpha moves before
  another player Beta does. That is Alphas
  decision node precedes Betas.
• If Alpha is more informed than Beta about an
  event that occurred before either of them had the
  opportunity to move, then they are playing a game
  of incomplete information.
• Alpha might wish to signal to Beta his knowledge
  through his move, or he might wish to confuse
  Beta, depending on their strategic
  complementarities.

4
What kind of information?

• Incomplete information can arise from two
  sources
• Knowledge about a players past move
• For example Gamma moves first, followed by Alpha
  and then Beta. Alpha sees Gammas move, Beta sees
  Alphas move, but Beta does not see Gammas move.
• Knowledge about an earlier move by nature
• Change the name of Gamma to nature in the above
  example, and let natures move be a random
  variable.

5
Incomplete information induced by earlier moves
of other players
First we shall analyze two games in which one
player knows more about a second players
previous move than a third player.
6
Predators
This game features a corrupt government, a poorly
run state enterprise and an opportunistic foreign
investor wrestle for mineral and oil wealth.
7
Why Predators is anincomplete information game

• When the foreign investor decides near the end of
  the game whether to withdraw or commit, he does
  not know whether the state enterprise has is
  trying steal his assets or not.
• However if the government moves at all, it moves
  before the investors makes his final decision,
  and in that case knows that the state enterprise
  has indeed decided to try stealing the foreigners
  assets.

8
Governments seize assets given the opportunity
A dominant strategy for the government in this
game is to seize foreign assets when presented
with the opportunity to do so.
9
The reduced game upon anticipating the
governments choice

• We now reduce the game by conditioning on the
  governments dominant strategy.
• If the government seizes its assets, the foreign
  firm should withdraw if it can.
• If its assets were left intact, it should commit.

10
A further reduction of the predator game

• Folding back the final decisions of the foreign
  investor, the further reduction yields a
  simultaneous move game.
• Note the state owned enterprise has a dominant
  strategy to propose a joint venture.

11
Quality control

• Manufacturers do not consistently produce
  flawless products despite legions of consultants
  who have advised them against this policy.
• Retailers help guard against flawed products by
  returning some of the defective items sent, and
  lending their brand to the ones they retail.
• Consumers cannot judge product quality as well as
  retailers and producers, since each one
  experiences only a tiny fraction of the end
  product.
• What is an acceptable defect rate, how often
  should retailers return defective items, and what
  are the implications for consumer demand?

12
Total quality management
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If the consumer always buys . .

• To solve this game we first remark that the
  consumer plays a mixed strategy in equilibrium.
  This claim can be established by contradicting
  the alternative hypothesis that she plays a pure
  strategy.
• Suppose she always buys the product.
• Then the retailer would never return a defective
  one, and the producer would specialize in
  producing defective products.
• Thus the consumer only purchases defective
  products, and this is not a best response to the
  strategies of the manufacturer and the retailer.

14
If the consumer never buys . . .

• But if the customer never buys the product, the
  retailer would always return defective ones.
• In this case the manufacturer specializes in
  produced flawless products.
• It now follows that the strategy of not buying is
  not a best response
• Therefore the consumer follows a mixed strategy.

15
Defining the probabilities in the TQM problem

• Let q denote the probability that the retailer
  offers a defective product item sale.
• Let r denote the probability the shopper buys the
  item.
• Let p be the probability a producer produces a
  flawless item.
• Both probabilities are strictly positive.
  receiving both kinds of products is strictly
  positive.
• If the shopper mixes between buying and not
  buying the product, then she must be indifferent
  between making either choice.

16
A schematic
17
Solving for r, the probability of buying

• If 0 between offering a defective product and
  returning it.
• In that case
• 3r - 2(1 - r) -1
• ? 3r 2 2r -1
• ? 5r 1
• ? r 0.2

18
How to solve for p and q

• Once we substitute for r 0.2 in the shoppers
  decision, we are left with the diagram
• q is chosen so that the producer is indifferent
  between production methods
• p is chosen so that the shopper is indifferent
  between buying and not buying.

19
Solving q,the probability of offering the product

• The producer will only mix between defective
  and flawless items if the benefit from both are
  equated
• 6r (1 - r)q - 3(1 - q) 3r (1- r)
• ? 2q 3 3q 1.4
• ? 5q 4.4
• ? q 0.88

20
Solving for p, the probability of producing a
flawless product

• Investigating the cases above shows that in a
  mixed strategy equilibrium r 0.2 and q 0.88.
• Since the shopper is indifferent between buying
  the item versus leaving it on the shelf, there
  are no expected benefits of acquiring the item
• 9p - 10(1 - p)q 0 ? (9 10q)p 10q
• ? p 44/89

21
Offering a partial refund
We now modify the game slightly. If the customer
buys a defective product, she receives partial
compensation.
22
A different outcome

• In this case the manufacturer has a weakly
  dominant strategy of specializing in the
  production of flawless goods.
• Recognizing this, the shopper picks a pure
  strategy of buying.
• Realizing that the shopper will buy everything
  she is offered, the retailer never returns its
  merchandise to the manufacturer (and indeed there
  is never any reason too).

23
Lecture summary

• We defined games of incomplete information as the
  complement of complete information games.
• Then we explained how games of incomplete
  information might arise.
• The tools used to solve simultaneous move and
  perfect information games can be combined to
  solve games of incomplete information.
• We analyzed two examples of incomplete
  information generated within a three player game.

24
Incomplete Information due to Nature
This lecture considers four games where nature is
the source of the incomplete information, and
discusses the costs and benefits of making
players more informed.
25
Starting a new business

• Seeking outside funding for a new business is
  tricky because the entrepreneur seeking support
  typically knows much more about the probability
  of success than the angel or the venture
  capitalist.

• In this game the entrepreneur reaches one of
  three information nodes before he decides whether
  to approach a potential business partner or not,
  but the partner does not know which node the
  entrepreneur reached.

26
Folding back from natures nodes
The analysis of this game can be somewhat
simplified by investigating the reduced game
27
The strategic form

• There are 8 strategies for the entrepreneur,
  listed as rows in the matrix and 2 strategies for
  the potential partner (the columns).
• Calculating the expected payoffs for each
  strategy pair we obtain the strategic form for
  this game.

28
Dominated strategies for the entrepreneur

• Inspecting the entrepreneurs strategies we see
  that the fourth and the eighth are dominated by
  the second and the sixth respectively.
• Similarly mixing the second with the sixth
  strategies rule out the third and fifth.

29
Dominance solvable

• Once strategy eight is ruled out, Decline is a
  dominant strategy for the potential partner.
• The entrepreneur goes solo, unless there is
  nothing to be had.

30
Cost of incomplete information

• The expected value of this game, the sum of the
  expected values to each player, from playing the
  solution is
• 0.1 8 0.5 1.6 1.6
• In a perfect information game, where the partner
  is as equally informed as the entrepreneur, it is
  easy to prove the expected value of the game is
• 0.1 (20 10) 0.5 1.6 3.8
• Moreover the expected value of maximizing the sum
  to both players is
• 0.1 (20 10) 0.5 (8 - 2) 6

31
The alternative of corporate research

• This solution to this game illustrates the
  problem that practically every independent
  inventor faces in funding his research for
  commercial development.
• One alternative is for large companies to conduct
  research in-house, where scientists and inventors
  are paid a fixed wage to develop new products
  with company equipment and support, the profits
  from their discoveries being distributed as
  dividends to shareholders.
• This approach creates two other problems
• Workers resign from the company start their own
  firms just before they announce major discoveries
  
• Researchers on wages lack motivation.

32
Aircraft Mechanic

• Hiring expert help, when you are the residual
  claimant, is also fraught with costs associated
  with incomplete information.
• We examine this question from the viewpoint of a
  companys ultimate status symbol, its executive
  jet fleet.

33
Background log on Aircraft Mechanic

• The flight crew notice the turbines emit a new
  noise when they start to turn over, although it
  disappears before executive management boards, so
  the passengers are unaware of any differences.
• Executive personnel decide whether to have the
  plane checked into unscheduled maintenance, at a
  cost of 200.
• If the problem is severe there will be an
  additional cost of 800 in parts plus 400 labor.
  If left unchecked, and the problem is severe, an
  extra 800 will later be spent in parts.
• If the plane is checked, the (outsourced)
  mechanics advise personnel whether the noise
  signals whether the engines require new parts or
  not.

34
Aircraft mechanic reduced

• Folding back the resolution of uncertainty that
  occurs if personnel ignores the noise, we see the
  expected loss to the firm from taking no action
  is 1000, and the expected payout to the mechanic
  is 200.
• Should personnel have the plane checked?

35
Aircraft mechanic Strategic form for subgame

• If personnel has the plane checked, the following
  subgame ensues, which we present in strategic
  form.
• Note the strategy Authorize is dominated by
  Ignore.

36
Best replies in thereduced strategic form

• Only two cells do not have at least one arrow
  leading out.
• They are the only pure strategy Nash equilibrium.
• Both involve the Personnel choosing ignore.
• In both equilibrium the expected payoffs are the
  same.

37
Cost of incomplete information

• The expected cost to Personnel from having the
  plane checked is 1200, because the mechanics
  advice is sought and then ignored.
• Therefore the solution to the game is not to have
  the plane checked.
• An alternative arrangement to owning a corporate
  fleet is to outsource executive air travel, by
  leasing, partial sharing, or relying on an
  air-taxi service that owns and maintains its own
  fleet.
• In the latter case the executive air-taxi service
  would solve a perfect information game with its
  own mechanics using backwards induction.

38
Air-taxi service

• The solution to this game yields the Air taxi
  service losses of 800 and benefits for the
  mechanic of 400.
• Personnel should be willing to pay Air taxi up to
  200 rent to rid itself of its incomplete
  information problem, while the mechanic should be
  willing to take a wage cut of 200 to get the
  extra business.

39
Product development race
Often companies do not know precisely how much
competition they will face before launching a new
product
40
Strategic form ofproduct development race
Both firms have a dominant strategy to advertise
the product, which determines the unique solution
to this game.
41
An industry newsletter
Now suppose a newsletter is produced to keep
firms abreast of the latest developments. The
extensive form becomes
42
Sub-games

• There are three proper sub-games beginning at
  nodes 2, 3, and 4.
• If Thompson is the only firm to develop the
  product, it should advertise rather than choose a
  low price, and similarly for Smith.

• The sub-game starting node 4, when both firms
  develop the product, illustrates the prisoners
  dilemma. The unique solution is for both firms to
  charge the low price.

43
The strategic cost of better information

• This example shows that more information about an
  industry could sometimes hurt it.
• Additional information helps firms to identify
  situations where their positions are opposed to
  each other, and induce competition that might
  lead to the detriment of all firms.

44
Industrial espionage

• What happens if only one firm can acquire the
  information about its rival, with the new payoffs
  shown?

• In this version of the game thompson has
  implanted a spy in smith who cannot steal the
  technology, but keeps thompson abreast of news
  about smiths product development.
• How should Smith behave? How does this compare
  with the solution(s) where there is no spy?

45
Medical malpractice

• One problem health insurance providers face is
  fraudulent behavior by doctors who prescribe
  treatment for healthy clients.
• Consider the following extensive form game

46
Strategic form of medical malpractice

• In the strategic form of the game, we see that
  ignore is a dominated strategy.
• Furthermore the best replies indicate that the
  game has a unique mixed strategy Nash equilibrium.

• Solving, the patient takes the treatment with
  probability 5/6, while the doctor prescribes
  treatment 1/4 of the time to healthy patients.
• Thus healthy patients receive the treatment with
  probability 5/24, about 20 of the time.

47
Diagnostic test

• Is it profitable to administer a test that
  verifies whether someone is ill or not?

• Solving the perfect information game, the doctor
  will only prescribe treatment to sick patients,
  who will always take it.

48
Gains from testing

• The solution to the perfect information game
  yields an expected benefit of 84 to the patient
  and 6 to the doctor.
• In the malpractice game, the expected benefit to
  the patient is
• (382 50 1584 544)/64 74
• while the expected benefit to the doctor is
• (32 - 38 156 514)/ 64 5.3
• Together the patient and doctor are willing to
  pay up to 10.6 to administer the diagnostic test.

49
Lecture summary

• We analyzed several games of incomplete
  information that are induced by chance events,
  that is beyond the control of the players.
• We showed that firm boundaries and its core
  competencies are partly defined by its the
  strategic implications of its knowledge base.
• We also proved that in a strategic context the
  old adage that more information is better is
  fallacious.
• Finally we proved that if ethical principles are
  widely held and acknowledged profitability and
  consumer satisfaction is increased. In our
  example a diagnostic test preventing medical
  malpractice served the same purpose.