Sequential Rationality

1
Chapter 5

• Sequential Rationality

2
Example 1 should we offer the position?

• A candidate to the computer science department
  has
• already written 11 research papers, and the
• department would like to decide on whether to
• make her a job offer based on the quality
• of the papers.
• There are 11 committee members who are each given
  one paper to read in order to make a
  recommendation.
• Initially, each paper may be "good" or "bad" with
  equal probabilities, and the department has
  chosen to make an offer to a candidate if he has
  a majority of "good" papers.
• Committee member values the correct
  recommendation of the committee, at 1000 to him,
  but values the time he needs to spend on reading
  the paper at 400.

3
Example 1 should we offer the position?

• A simple mechanism asks all the committee
  members, simultaneously, for their
  recommendations. The strategy tuple where all
  agents choose to read the papers and report
  truthfully is not an equilibrium.
• Consider the perspective of agent 1 assuming
  all agents
• replied (truthfully, or not), then agent 1 can
  alter the outcome
• only if the other 10 replies split evenly
  between 0 and 1
• which has a probability of approximately 0.25.
• Therefore, by guessing, and assuming all other
  agents compute,
• he will gain 0.25 X 500 0.75 X 1000 875.
• 25 of the time the right decision is made (as
  he has a 50 chance of guessing right) and 75 of
  the time he gets the right decision as the others
  make the correct decision.
• However, by computing an agent gains
  1000-400600, and so player 1 has no incentive
  to compute (the same for all 11 agents).
• .25600 .75600 600

4
Example 1 should we offer the position?

• This elicitation mechanism will also fail if
  only agent 1 has the above cost and all other
  agents have zero costs (the same analysis will
  hold for agent 1).
• If however agents 2,3,.,11 are asked first for
  their recommendations, and agent 1 is approached
  only if there is a tie among the ten
  recommendations, then all agents will have enough
  incentive to invest the effort as 100 of the
  time you are asked you gain 600 as compared to
  500 for guessing!
• This illustrates the power of sequential
  mechanisms.

This motivates the careful discussion of
sequential elicitation mechanisms, i.e. the
construction of mechanisms that approach agents
in a well designed sequence. Goal is to change
the way the game is played so there is no
temptation to shirk! This is the positive outcome
of the study! We arent trying to teach you to
be a shirker, but to devise mechanisms to
de-incentivize shirking.
5

• Subgame perfection ensures that the players
  continue to play rationally as the game
  progresses.
• We have a problem with imperfect information,
  however, as we might not have subgames at all.
• sequential equilibrium a pair (?,µ) where ? is
  a behavior strategy profile and µ is a system of
  beliefs consistent with ? such that no player can
  gain by deviating from ? at any member of the
  information set.

6
5.1 Market for Lemons

• Two player game in which one player has more
  information than the other.
• Like my example of forgetting which cards had
  been played, but in this case, one player just
  knows more.
• Selling cars.
• ph is the reservation price (least he is willing
  to accept) for a high quality car.
• pl is the reservation price (least he is willing
  to accept) for a low quality car.

7

• Similarly, the buyer has similar reservation
  prices
• H highest price he is willing to pay for high
  quality car.
• L highest price he is willing to pay for a low
  quality car.
• Look at sequential game when a seller puts a car
  on the market. Nature reveals the quality of the
  car to the seller, but the buyer doesnt know it.
• Seller must decide whether to ask ph or pl.
• Buyer doesnt know price.

8
buy
(p-ph, H-p)
2
not
X
(0,0)
ph
1
(p-pl, L-p)
buy
2
not
pl
Y
(0,0)
G
(p-ph, H-p)
Nature
ph
buy
2
B
not
M
(0,0)
1
pl
(p-pl, L-p)
buy
2
not
(0,0)
N
9

• Note, there are no subgames.
• What should player 2 do?
• If price is close to ph, could assume the car is
  of high value. If close to pl, could assume the
  car is of low value.
• The seller knows this. Both would agree to split
  the profit in half, so gains would be equal.
• BUTWhy not just charge a high price for every
  car!
• So now the buyer doesnt know what to do.
• Could assume has equal likelihood of being high
  or low (uniform distribution)

10

• His expected value is
• ½(H-p) ½(L-p)
• Since he needs a postive expected value
• ½(H-p) ½(L-p) gt0
• p ? ½ (HL)
• If offered a high price, he believes he is at
  node X with probability ½ and at node Y with
  probability ½.
• But if he gets such a low offer, he knows he
  isnt at X or Y, but at N.

11
Consider a case by case analysis

• Case 1 ½(HL) ? ph
• Both types of cars could really be for sale, so
  buyer will buy at this price.
• Case 2 ½(HL) lt ph
• No high quality car can be sold for that price,
  so you are at node N. Buyer insists on somewhere
  between pl and L, as he knows he is looking at a
  low quality car.
• Equilbrium which is driven by consistent
  beliefs

12
(No Transcript)
13

• Lets take another look. utilities (seller,
  buyer)
• Lots of prices could be offered not just ph or
  pl.
• Use line to represent infinite number of nodes

(p-ph, .5(HL)-p)
Like before only assume equally likely you have
Good as Bad car
buy
X
not
price, p
(0,0)
Good
Nature
(p-pl, .5(HL)-p)
Bad
buy
Y
price, p
not
(0,0)
14

• Case 1 ½(HL) gtph
• u1(p,s) p-ph (if s(X) buy)
• p-pl (if s(Y) buy)
• 0 (if s(X) not buy)
• 0 (if s(Y) not buy)
• u2(p,s) ½(HL)-p if s buy
• 0 if s not buy
• Optimal stategy
• p ½(HL) s buy if ½(HL) ? p
• not buy if ½(HL)
  ltp

15

• Case 2 ph gt ½(HL)
• Seller knows that buyer will never buy a car at a
  price greater than ½(HL) so only low-quality
  cars are on the market.
• u1(p,s) p-pl (if buy)
• 0 (if not buy)
• u2(p,s) L-p (if buy)
• 0 (if not buy)
• p L
• S buy if L gt p
• not buy if L lt p

16

• Notice no subgames
• role of beliefs is critical.

17
Lets look again using different valuesExample
of adverse selection the market for lemons
You want to buy a used car. There are two types
of cars good cars and lemons.
18
Example of adverse selection the market for
lemons
Suppose the seller always knows what type of car
they are selling. What happens in the market
depends on whether buyers can also tell the type
of car.
19
The market for lemons case 1 symmetric
information (Both can tell a lemon)
20
The market for lemons case 2 asymmetric
information
Suppose the buyer cannot tell a good car from a
lemon before they buy. Will you buy if the price
is at least 8,000? NO! At this price, every
seller will want to sell. But this means that if
you buy a car it has a 60 chance of being a
lemon (worth 6,000 to you) and a 40 chance of
being good (worth 10,000 to the buyer). So the
expected value of a car to the buyer is 7,600.
So you will not pay more than 8,000 for a car
with an expected value of 7,600!
21
The market for lemons case 2 asymmetric
information
Suppose the buyer cannot tell a good car from a
lemon before they buy. Will you buy if the price
is between 6,000 and 8,000? NO! At this price,
only the sellers of lemons will want to sell.
Every car being offered is a lemon and you will
not pay more than 6000. So we expect that the
market will have a price of between 3,000 and
6,000 with only lemons sold
22
The market for lemons case 2 asymmetric
information
23
Adverse selection

• So the problem of adverse selection can lead to
  the complete collapse of the market for good cars
• A similar problem faces insurance companies and
  the market for loanable funds.
• Adverse selection can also lead to statistical
  discrimination.

24
Responses to adverse selection

• Note that the problem of adverse selection harms
  the un-informed parties and some of the informed
  parties.
• In the lemons example, it meant that buyers could
  not buy good cars. But also sellers of good cars
  could not get a reasonable price for their cars.
• Un-informed buyers may try to overcome the
  information asymmetry by searching for more info
• Informed sellers may try to overcome the problem
  by
• Warranties
• Signaling

25

• Example job-market model of bilateral
  uncertainty uncertainty on both sides.
• Workers are uncertain about what job descriptions
  advertised by firms really mean
• Firms are uncertain about the qualifications of
  workers before they are interviewed. Both types
  of uncertainty can be resolved, but both
  processes are costly. Intermediaries (recruiters)
  can perform the job matching but only at the cost
  of transforming the firms objectives between the
  parties.

26

• Each branch has an associated probability

firms info sets Knows what fits, but cant tell
if good person
good person, fits
employee info sets Knows if hes good or bad but
cant tell it hes a fit.
bad person, fits
good person, doesnt fit
bad person, doesnt fit
27
(No Transcript)
28
Information and market failureCan you answer
these questions?

• Why does a new car lose about one quarter of its
  value when you drive it away from the dealer?
  (cant convince others it is a good car)
• Why do manufacturers of products that almost
  never break down still offer warranties? (need to
  convince others the product is good)
• Why is car insurance more expensive for all
  younger drivers? (paying more of the actual cost
  of converage)
• Why do insurance companies make you pay the first
  part of any claim (the deductible)? (No
  temptation to let a flood water set so you get
  all new furniture. No temptation to be careless.)

29
Information and market failure

• There are two basic types of information
  asymmetry that can lead to market failure
• Adverse selection one party to a deal has
  private information that affects the value of the
  deal
• Moral Hazard one party to a deal has to take an
  action that cannot be perfectly monitored by the
  other party. The action affects the value of the
  deal. Called a moral hazard as there may be an
  incentive to do something immoral (dishonest)
  like shirk at your job.

30

• Moral hazard can be present any time two parties
  come into agreement with one another. Each party
  in a contract may have the opportunity to gain
  from acting contrary to the principles laid
  out by the agreement.
• For example, when a salesperson is paid a flat
  salary with no commissions for his/her sales,
  there is a danger that the salesperson may not
  try very hard to sell because the wage stays the
  same regardless of how much or how little the
  owner benefits from the salesperson's work.

31

• Sometimes people do better than break even when
  misfortune strikes, and this possibility has
  greatly interested economists.
• If the misfortune costs a person 1000, but
  insurance will pay 2000, the insured person has
  no incentive to avoid the misfortune and may act
  to bring it on.
• For example, if you have full replacement costs
  on your house insurance, you may be happy when
  grape juice ruins your 10 year old carpet.
  Obviously, deliberately throwing grape juice to
  get insurance reimubursement is illegal.
• This tendency of insurance to change behavior
  falls under the label moral hazard.

32
Adverse Selection
Asymmetric information is feature of many markets
- some market participants have information
that the others do not have 1) The hiring
process a worker might know more about his
ability than the firm does - the idea is
that there are several types of workers -
some are more productive than others are 2)
Insurance insurance companies do not observe
individual characteristics such as
driving skills 3) Project financing
entrepreneurs might have more information about
projects than potential
lenders 4) Used cars sellers know more about
the cars quality than buyers Adverse selection
is often a feature in these settings - it
arises when an informed individuals decisions
depend on his privately held information in a way
that adversely affects uninformed market
participants
.
33
Warranties
Lets return to our car market example. Remember
that buyers are willing to pay up to 10,000 for
a good car but only up to 6,000 for a lemon. The
problem is which is which? Suppose that good
cars never break down. However, a lemon breaks
down often (that is why it is a lemon). Say
lemons break down 80 of the time. Fixing a
broken down car is expensive about 5,000.
Suppose now that a car seller offers you the
following deal Buy the car for 9,000. If it
breaks down, the seller will not only fix your
car for you but also pay you 3,000 in cash as
compensation Should you buy the car? But, have
you ever been offered that good of a warranty?
34
5.2 Beliefs

• beliefs are in important in finding solution to a
  game without subgames.
• beliefs must be consistent with the way game is
  played.
• For example, in the Star Trek/Game Theory book
  gift example you needed to know the likelihood
  a gift would be offered given the type of book.
  The game structure is key to deciding which
  beliefs you need to formulate.
• System of beliefs assigns probability
  distribution to nodes in the information set.
  (What node do I think I am at)
• Use µ to represent that probability
• Behavior strategy ? for a player is the
  probability he will take each edge. (mixed
  strategy what edge will I take)
• completely mixed strategy at every node, every
  choice is taken with positive probability

35
Example 5.2

• Player 1 plays a with .1
• Player 2 plays T with .1
• Player 1 plays L and L with .1

(4,2)
.001
.1
L
R
.1
.9
T
(0,3)
.009
E
.1
X
B
L
.9
(1,7)
.1
.009
F
a
R
.9
(3,0)
(2,6)
.081
O
L
.1
G
b
.1
T
.9
.9
R
(2,4)
.081
Y
B
.9
(3,5)
.081
L
.1
H
R
.9
(4,3)
.729
36

• Note, than in general, the probability you are at
  E is .01.
• The conditional probability of p(EX) .1 as
  once you know you are at X, the probability of E
  is greater.
• In the book example, the probability of giving a
  gift could be different depending on what the
  book is. A person might be MUCH more likely to
  give Star Wars as a gift than Game Theory.

37
5.3 Bayes Consistent

• A system of beliefs µ is said to be Bayes
  consistent with respects to a mixed behavior
  profile ? if µ can be generated by ?.
• In other words, you beliefs about probabilities
  make a behavior profile reasonable.
• For example, if I1E,F and I2G,H
• if µ(EI1) µ(GI2) 0 and
• µ(FI1) µ(HI2) 1
• µ(X) 0, µ(Y) 1

38
Example 5.2 This means

• O-gt Y -gt H -gt (4,3) is a good plan.

(4,2)
.1
L
R
.1
.9
T
(0,3)
E
.1
B
X
L
.9
(1,7)
.1
F
a
R
.9
(3,0)
(2,6)
O
L
.1
G
b
.1
T
.9
.9
R
(2,4)
Y
B
.9
(3,5)
L
.1
H
R
.9
(4,3)
39
5.4 Expected Payoff

• We compute expected payoff in the regular way
  multiply the payoff by the probability you get
  it.

40
(4,2)
1
.6
Example 5.6
N
(0,3)
.4
E(I2NF) .2E(N) .8E(F)
.2
2
(1,7)
.6
.8
X
F
.4
(2,6)
The value player 1 uses for E(X) use HIS beliefs
and his strategy.
(3,0)
1
(2,4)
G
(3,5)
Y
H
(4,3)
41
5.5 Sequential Equilibrium

• Sequential Equilbrium a pair (?,µ) where ? is a
  strategy profile and µ is a system of beliefs
  consistent with ? such that no player can gain by
  deviating from ?.
• Note that what I believe may not be exactly the
  case, as I am not privy to the other persons
  strategy. My strategy must be consistent with
  what I believe to be true.

42
Kreps-Wilson

• Every sequential game with imperfect information
  has a sequential equilibrium.
• I interpret that to mean, that given my
  understanding that is the stable thing to do.
  It might not be the right thing, but given what I
  know, I can do no better.

43
(4,2)
1
L 1
Example 5.10
N
(0,3)
R 0
T 1/2
2
(1,7)
.
B 1/2
X
F
(2,6)
a 0
(3,0)
L 1/6
1
(2,4)
R 5/6
G
b 1
(3,5)
Y
H
(4,3)
Can show equilibrium by seeing if can gain with
other strategy.
44
Warranties
A GOOD SELLER CAN MAKE SUCH AN OFFER
Dont buy
(0,0)
Breakdown under warranty (zero chance)
Buyer
Offer deal
Good seller
(-7,000, ?)
Buy
Dontoffer deal
No breakdown (100)
(0,0)
(1000,?)
45
Warranties
But the buyer can infer this the seller of a
lemon would not offer the deal.
Dont buy
(0,0)
Buyer
Offer deal
Breakdown (80)
Lemon seller
(-2,000, ?)
Buy
Dontoffer deal
No breakdown (20)
(0,0)
(6,000, ?)
46
Warranties
Dont buy
(0,0)
Buyer
Offer deal
Breakdown (80)
Lemon seller
Expected payoff for lemon seller if buyer
accepts offer is -400. So if the lemon
seller will not offer deal!
Buy
Dontoffer deal
No breakdown (20)
(0,0)
47
Warranties

• So the warranty works
• Only the good sellers will offer the warranty
• Buyers can buy the car with the warranty, sure
  that they are buying a good car (and will never
  need to use the warranty)
• But it is not worth while for the sellers of
  lemons to offer the warranty their cars break
  down and the warranty costs more than the
  increased price that they receive for their cars

48
Signaling

• The warranty is an example of a signal that the
  good seller can send to the buyer.
• In our example here the signal had no cost to the
  good seller, but was expensive for the bad
  seller. This is not generally the case.
• This is a Good Car sign is ineffective every
  type of seller will use it, and it will provide
  no new info
• For a signal to work it requires three features
• It must be less costly to the good type than to
  the bad type.
• Even given the cost, it must be better for the
  good type to distinguish themselves than be
  mistaken for a bad type.
• The cost to the bad type must be high enough so
  that they do not want to pretend to be a good
  type

49
Other examples the early career rat race

• Suppose there are ordinary and talented workers
• Your boss can observe the quality of your work
  but not how difficult you found the task
• If everyone spends the same time, the talented
  workers will be recognised and gain promotion
• So the ordinary workers work harder to try and
  appear to be talented
• So to distinguish themselves, the talented
  workers also have to work hard

50
Other examples the early career rat race

• What will be the outcome?
• Could get a separating equilibrium. This is where
  the signal works. The talented workers work too
  hard but are recognised. The ordinary workers
  just give up.
• Or could get a pooling equilibrium. In this
  situation, the ordinary workers work hard and
  talented workers work normally. The boss
  interprets ordinary performance as a sure sign
  of lack of talent. But the boss cannot infer
  anything from exceptional work because everyone
  is doing it!

51
Signaling

• Signaling can overcome information problems
• But it can be costly to the good type who is
  trying to distinguish themselves
• Choosing the wrong signal just means copying by
  the bad type. Despite the cost of the signal
  there is no gain in information
• So it is important to carefully choose your
  signaling strategy. It needs to be low cost to
  you and high cost to others, so that it will be
  believed and cannot be jammed by the bad
  types.

52
Moral hazard

• While Adverse selection is about private
  information, moral hazard is where one party can
  take a hidden action that affects the value of a
  transaction
• For example
• Precaution with car and household insurance
• Employees
• Outsourcing of services
• Regulation

53
Example of moral hazard
You can work hard or slack. Working hard adds
1000 per week to firm value but has a personal
cost of 100. Slacking has no personal cost but
only adds 300 to firm value. If you do not work
for this firm you can get a job elsewhere for
400 per week
Work hard
Hire you at w per week
(1000 - w, w - 100)
You
Firm
Slack
Dont hire you
(300 - w, w)
(0, 400)
54
Example of moral hazard
If the firm just offers you a fixed wage then by
roll back it is better for you to slack
Work hard
Hire you at w per week
(1000 - w, w - 100)
You
Firm
Slack
Dont hire you
(300 - w, w)
(0, 400)
55
Example of moral hazard
Knowing this, the firm will not want to pay you a
wage of more than 300. But given such a low wage
you are better off working elsewhere Notice that
this is inefficient. If you could commit to work
hard, there is 500 extra value that can be
shared (1000 - 400 (outside wage) - 100 (cost
of hard work))
Work hard
Hire you at w per week
(1000 - w, w - 100)
You
Firm
Slack
Dont hire you
(300 - w, w)
(0, 400)
56
Example of moral hazard
This problem can be overcome if the firm can
either observe your effort OR if the firm can
observe your individual added value perfectly. In
this case, the firm can offer you a contingent
wage wh if work hard and ws if slack.
Hire you at wh and ws per week
Work hard
(1000 - wh, wh - 100)
You
Firm
Slack
Dont hire you
(300 ws, ws)
(0, 400)
57
Example of moral hazard

• How does the firm calculate the best wh and ws to
  offer so that you will work hard?
• The firm faces an individual rationality
  constraint. You must receive at least as much by
  working hard for the firm as if you decided not
  to work for the firm
• If you do not work for the firm you receive 400
  per week
• If you work hard for the firm you receive wh -
  100.
• So you will only accept a wage contract that
  leads you to work hard if overall you receive at
  least 400 (your outside offer).
• This means that wh gt 500

58
Example of moral hazard

• The firm also faces an incentive compatibility
  constraint. You must prefer to work hard for the
  firm than slack
• If you work hard for the firm you receive wh -
  100. If you slack you receive ws.
• This means that the firm must set wh gt ws 100

59
Example of moral hazard

• So the firm faces two constraints when setting
  your (contingent) wage contract
• wh gt 500
• wh gt ws 100
• What is the profit maximising wage contract for
  the firm?
• wh 501, ws lt 401

60
Example of moral hazard
Notice that now it is better for you to work hard
than to slack.
Hire you at 501 if work hard and 100 if slack
Work hard
(499, 401)
You
Firm
Slack
Dont hire you
(200, 100)
(0, 400)
61
Example of moral hazard
Notice that now it is better for you to work hard
than to slack. So the firm now knows that if you
accept the contract then you will work hard.
Hire you at 501 if work hard and 100 if slack
Work hard
(499, 401)
You
Firm
Slack
Dont hire you
(200, 100)
(0, 400)
62
Example of moral hazard
And the firm makes as much profit as possible
given that it must beat your outside option.
Hire you at 501 if work hard and 100 if slack
(499, 401)
Firm
Dont hire you
(0, 400)
63
Example of moral hazard

• Notice that the firm could also achieve an
  outcome where you work hard by paying you an
  output based contract (e.g. the firm pays you
  50 of your value added so you receive 500 if
  work hard and 150 if slack).
• More generally, moral hazard analysis forms the
  basis of incentive contracting
• Issues of observability
• Issues of risk sharing

64
Signaling game set-up

• Imagine there are two types of people in the
  world, but that type is private information know
  only to individual
• people who good at business but not at art
  (business people)
• people who are very talented at art but not
  business (artists)
• Employers want to hire business people not
  artists
• Businesses pay very well such that artists would
  like to have business jobs
• How should businesses find business people?

65
How do businesses find business people?

• One solution is to ask people, are you a
  business person or an artist?
• What will business people say?
• What will artists say?
• Is there a signal business people can send?
• What if wearing a suit signals that one is a
  business person? What will artists do?
• Is there a credible signal business people can
  send that businesses will believe?

66
Credible signals

• A signal is credible if it is costly enough such
  that artists will not want to invest in signaling
• One potential credible signal is going to
  business school
• Interestingly, the signal works even if business
  school does not affect business peoples
  productivity

67
Signaling game set-up

• ½ the people in the world are business people and
  ½ are artists
• Business people are worth 5 to businesses, while
  artists are worth 4.
• (But wont get paid this much as then there would
  be no profit to business.)
• There are only enough business jobs for ½ the
  people in the world
• Firms pay 3 to anyone they hire, regardless of
  type (since this is unobservable)
• Business school is free however, it costs 1 of
  effort from business people and 4 of effort from
  artists (artists dislike business school)
• Interesting assumption School does not change
  the productive capacity of the workers

68
Signaling game Starting point is at
center.Utility (individual, company)
Business people
3,2
2,2
H
H
B School
No B School
company pays 3 but gets 5 value, so profit is
2
50
0,0
-1,0
N
N
-1 individual is paid nothing and loses
investment of 1 in education
Nature
H
H
3,1
-1,1
50
B School
No B School
N
N
0,0
-4,0
Artists
69
Equilibrium

• Business people go to business school, artists do
  not and firms only hire business school graduates
• This is the only equilibrium in this game, no one
  can do better by changing their strategy given
  what other players do
• Note that if everyone goes to school the expected
  value for artists is 2½ (etc. etc.)
• If they hire an artist, he gets same utility with
  or without an education. If they dont hire an
  artist, lose with education, but same is true of
  business person. Difference is that companies
  remove the hiring option for those without a
  business degree.
• Note the role of business school in this game
• Business people dont learn anything useful in
  business school in this set-up
• However business school is still a socially
  useful institution since it allows business
  people to send credible signals to potential
  employers

70
Incentive Schemes

• Salary and bonus contracts can compensate for
  information asymmetry
• Often, this is unreasonable
• Employees unwilling to assume risks
• Contracts must be perfectly balanced
• May be better to settle for low effort
• Today
• The flip side are bonuses going to good
  employees or just lucky ones?
• Signaling screening

71
Leakages

• IBM Variable Pay
• Bonus of 10 of annual earnings if
• annual objectives are met in key areas
• Internal Memo
• We observe, across divisions, performance in
  line with expectations through about March.
  Performance declines consistently in later
  months.

72
Leakages

• If bonus is tied to
• Increases over last year
• Reduce this years growth
• Output / Quantity
• Reduce quality
• Average customer satisfaction
• Reduce number of service calls
• How would you rate teacher evaluations?
• If score is tied to percent of happy, stop
  unhappy from replying.
• If score is only on how happy, give everyone As
  and require no work.
• Homework helps students to learn, but gives
  poorer evaluations and takes a ton of instructor
  time.

73
Example Incentives Market Conditions

• Patent races over high-profit pharmaceuticals
  worth up to 2 billion
• Resource devotion ranges from twenty to sixty
  hours per employee, with staff of fifty per
  project (low level to high level)
• Project time frame 6 months

74
Market Conditions

• Independent labs contracted
• Average cost of labor 16/hour
• Chance of success
• Minimally 1
• Maximally 2.5

75
Cost Calculations

• Extra cost to lab of high effort
• 40 hours / week / employee
• x 25 weeks time frame
• x 16 / hour _
• 16,000 / employee

76
To entice high effort

• Costs
• 16,000 per employee in costs
• Benefits
• 1½ extra chance of success (2½ - 1)
• Incentive compatibility
• .015 x bonus gt 16,000
• bonus gt 1.1M

77
To entice high effort

• Bonus per employee must be greater than 1.1
  million
• Fifty employees, so total bonus must be greater
  than 55 million
• Final conclusion 75 million bonus to be safe

78
Extra Profit if it Works

• Value of extra chance of success
• 0.015 x 2B 30M
• Cost of bonus
• 0.025 x 75M 2M
• Benefit of plan
• 30M 2M 28M

79
Problem

• Ignoring individual incentives
• Analysis assuming that entire group works hard
  or does not
• Quick Dirty Check
• If fifty people working hard increases chance of
  success by 1.5, each person, on average,
  increases chance by only 1.5/50 0.03
• Each person earns a bonus of 75M/50 1.5M

80
Conclusion

• A persons value of extra time
• 1.5M x 0.03 450
• A persons cost of extra time
• 16,000
• NOT EVEN CLOSE!

81
Signally example Auto Insurance

• Half of the population are high risk drivers and
  half are low risk drivers
• High risk drivers
• 90 chance of accident
• Low risk drivers
• 10 chance of accident
• Accidents cost 10,000

82
Pooling

• An insurance company can offer a single insurance
  contract
• Expected cost of accidents
• (½ .9 ½ .1 )10,000 5,000
• Offer 5,000 premium contract
• The company is trying to pool
  high and low risk drivers
• Will it succeed?

83
Self-Selection

• High risk drivers
• Dont buy insurance (.9)(-10,000) -9K
• Buy insurance -5K
• High risk drivers buy insurance
• Low-risk drivers
• Dont buy insurance (.1)(-10,000) -1K
• Buy insurance -5K
• Low risk drivers do not buy insurance
• Only high risk drivers self-select into the
  contract to buy insurance

84
Adverse Selection

• Expected cost of accidents in population
• (½ .9 ½ .1 )10,000 5,000
• Expected cost of among the insured
• .9 (10,000) 9,000
• Insurance company loss 4,000
• Cannot ignore this adverse selection
• If only going to have high risk drivers, might as
  well charge more (9,000)

85
Screening

• Offer two contracts, so that the customers
  self-select
• One contract offers full insurance with a premium
  of 9,000
• Another contract offers a deductible, and a lower
  premium

86
How to Screen

• Want to know an unobservable trait
• Identify an action that is more costly for bad
  types than good types
• Ask the person (are you good?)
• But attach a cost to the answer
• Cost
• high enough so bad types dont lie
• Low enough so good types dont lie

87
Screening

• Education as a signaling
  and screening device
• Is there value to education?
• Good types less hardship cost

88
Example MBAs

• How long should an MBA program be?
• Two types of workers
• High and low quality
• NPV of salary
• high quality worker 1.6M
• low quality worker 1.0M
• Disutility per MBA class
• high quality worker 5,000
• low quality worker 20,000

89
High Quality Workers

• If I get an MBA
• Signal I am a high quality worker
• Receive 1,600,000 - 5,000 N
• If I dont get an MBA
• Signal I am a low quality worker
• Receive 1,000,000
• 1,600,000 5,000 N gt 1,000,000
• 600,000 gt 5,000N
• 120 gt N

90
Low Quality Workers

• If I get an MBA
• Signal I am a high quality worker
• Receive 1,600,000 - 20,000 N
• If I dont get an MBA
• Signal I am a low quality worker
• Receive 1,000,000
• 1,600,000 20,000 N lt 1,000,000
• 600,000 lt 20,000N
• 30 lt N

91
Hiding from Signals

• Suppose students can take a course pass/fail or
  for a letter grade.
• An A student should signal her abilities by
  taking the course for a letter grade separating
  herself from the population of Bs and Cs.
• This leaves Bs and Cs taking the course
  pass/fail. Now, B students have incentive to take
  the course for a letter grade to separate from
  Cs.
• Ultimately, only C students take the course
  pass/fail.
• If employers are rational will know how to read
  pass/fail grades. C students cannot hide!

92
(No Transcript)
93
Bayesian games

• A game has incomplete information when players
  know different things about payoffs (or other
  relevant information)
• Remember information is imperfect if players
  know different things about (prior) moves.
• Applications include competition between firms
  with private information about costs and
  technology, auctions where each potential buyer
  may attach a different valuation to the item,
  negotiations with uncertainty about the other
  partys preferences or objectives, etc. in
  short, any real economic situation!
• The basic trick that lets us handle such games
  is to reduce incomplete to imperfect information
  by adding a chance player whose move chooses the
  payoffs.

94

• With a simple but brilliant trick Harsanyi
  allows us to turn any game of incomplete
  information into a much more manageable game of
  imperfect information.
• The Harsayni trick is used in a game (especially
  an extensive-form game) in which at least one
  player at the time of making a decision does not
  know what moves or choices were made previously
  by other players.

95
Suppose that two players, Callum and Rowena, are
playing the following game
96

• But whereas Rowena has complete information, i.e.
  knows every entry in each cell in the above
  matrix, Callum knows only his own payoffs, i.e.,

97

• At this point Callum follows Harsanyis
  suggestion first, to consider all the Rowenas
  payoffs that he considers likely (i.e., with a
  positive probability). For simplicity, lets
  suppose that, according to Callum, Rowena can be
  of only two types. If shes of type 1, then her
  payoffs are as in the following matrix

98
If instead shes of type 2, then the payoffs
matrix looks like this
99

• Second, Callum attaches to each type a subjective
  probability. For example, Callum may
• assume that Rowena is Type 1 with a probability
  of 2/3.
• Third, Callum formulates an extensive-form game
  with imperfect information whereby at the initial
  node Nature chooses Rowenas type with Callums
  probabilities and informs Rowena (but not Callum)
  of her true type, i.e., Callum draws the
  following tree

100
(No Transcript)
101

• Fourth, Callum computes the normal form of the
  above game and locates any Nash equilibria. These
  equilibria are called Bayesian Nash Equilibria
• A Bayesian Nash equilibrium is a NE of the game
  where Nature moves first, chooses players types
  from a distribution with probability p(t ) and
  reveals ti to player i.
• Rowena still doesnt know Callums type, but she
  can compute a strategy based on what she knows.

102
Another example

• Friend or foe player 2 does not know whether
  player 1 is friendly or not
• Which dilemma player 2 does not know which of
  the following Prisoners Dilemmas is being played

103
Bayesian Game Example

• Two players, two types. If they are the same
  type, they play Prisoners Dilemma. If they are
  of different types, they play Battle of the
  Sexes. The joint distribution of types is given
  by p, and the resulting game matrix is shown
  below.

104
Best replies

• Compute best replies for each type of each
  player, e.g.
• if type 1 of player 1 plays Top expected PO is
• If he plays Bottom instead ( 0), payoff is
• Simplifying and comparing, we get

• To use, apply appropriate prior probability. Ex
  independent, equally likely types, all ps ¼,
  and cut off values above are 1 for each
  strategy of pl. 1 and 2/3 for each strategy of
  player 2 each type of pl. 1 plays B unless
  opposite type of player 2 plays L with
  probability 1.
• Only pure strategy equilibria (0,0,0,0),
  (1,0,1,0), (0,1,0,1), (1,1,1,1).

105
Sequential rationality

• Due to presence of information sets, need to
  redefine rationality by moving from a strategy b
  to an assessment (b, m) consisting of a strategy
  b and a system of beliefs m.
• Informal definitions
• (b, m) is sequentially rational if for every
  information set Ii, bi(Ii) is a best reply given
  the beliefs m.
• (b, m) is strategically consistent if m is
  derived from b via Bayes Rule wherever
  applicable.
• (b, m) is structurally consistent if m at every
  information set is derived from some strategy b
  via Bayes Rule.
• (b, m) satisfies common beliefs if all players
  share the same belief about the cause of every
  unexpected event.
• The outcome of an assessment conditional on an
  information set I is a distribution O(b, mI)
  over the set Z of terminal histories. It assumes
  independence (multiply probabilities) which is
  supported by perfect recall specifically, if h
  is in Z