Economics 3012
Strategic Behavior
Andy McLennan
October 13, 2006
Lecture 10
Topics
• Bayesian Games
– An Example
– General Description
– Problem Set 8
– Bayesian Updating of Beliefs
– Additional Examples
– Auction Theory
1
Introduction to Bayesian Games
• In many important situations the players
have private information:
– A firm may not know whether a
competitor has high or low costs of
production.
– In an auction, the bidders typically do
not know each others’ valuations.
– In bargaining, there may be incomplete
information about the strength of the
players’ bargaining positions.
– In warfare, the strength and disposition
of enemy forces is often unknown.
– A doctor deciding whether to
recommend surgery has information not
possessed by the patient.
2
Example: Exercise 282.1
• Two players each have to decide whether to
fight or yield.
• Player 2 may be either strong or weak:
Strong
f
F
Y
Weak
y
f
(−1, 1) (1, 0)
(0, 1) (0, 0)
!
F
Y
y
!
(1, −1) (1, 0)
.
(0, 1) (0, 0)
• The player’s information:
– Player 1 does not know whether player
2 is strong or weak, but believes the
probability of strength is a.
– Player 2 knows whether she is strong or
weak, and she knows player 1’s beliefs.
3
Analysis:
• It is natural to think that this game has
three players:
– Player 1.
– The strong player 2.
– The weak player 2.
• For the strong player 2 fighting is a
dominant strategy.
• Player 1’s payoff does not depend on
whether the weak player 2 chooses to fight.
– Therefore player 1 will fight if a <
and yield if a > 12 .
1
2
• The weak player 2 prefers to fight if player
1 yields and yield if player 1 fights, so the
weak player 2 will yield if a < 12 and fight if
a > 21 .
4
The General Structure
A Bayesian game has the following timeline:
• A state of the world is randomly chosen.
• Each player receives a signal that depends
on the state of the world.
• Based on the signal she received, each
player forms a belief about the
probabilities of the states.
• Each player then chooses an action.
• Each player receives a payoff that depends
on the state and the profile of actions.
A Nash equilibrium is a specification of an
action for each type of each player with the
property that each type is maximizing expected
payoff, given her belief and the actions chosen
by all types of the other agents.
5
Problem Set 8
Exercise 277.1: Problem Statement: In this
variant of the Bach vs. Stravinsky each player
has two equally likely types.
• The first type of player i, denoted yi ,
wishes to go out with the other person.
Her payoff is:
– 2 if they meet at her favorite concert,
– 1 if they meet at the other concert, and
– 0 if they don’t meet.
• The second type ni prefers not to meet the
other person. Her utility is:
– 0 if they meet,
– 2 if she is alone at her favorite concert,
and
– 1 if she is alone at the other concert.
The problem is to give charts like Figure 275.1
with expected utility after each own action and
each combination of actions for the two types
of the other agent.
6
Analysis: These charts obey the following
general description: the expected payoff is the
quality of the concert you attend multiplied by
the probability that you get what you want in
terms of being with the other person. For type
n1 of player 1 the chart is as follows:
BB
B
S
0
1
BS
SB
1
1
1
2
1
2
SS
2
0
!
.
For type y2 of player 2 the chart is:
B
S
BB
BS
SB
1
0
1
2
1
2
1
1
SS
0
2
!
.
For type n2 of player 2 the chart is:
B
S
BB
BS
SB
0
2
1
2
1
2
1
1
7
SS
1
0
!
.
Exercise 282.2: Problem Statement:
• Two individuals each receive a ticket with
an integer between 1 and m.
– The two numbers are independently and
identically distributed.
– Each number between 1 and m has a
positive probability.
• Each player then indicates whether they
wish to exchange their ticket for the one
the other agent received.
– If both wish to exchange, the exchange
takes place.
– Otherwise each player keeps the ticket
she started with.
• For each agent the payoff is the number on
the ticket she holds at the end.
Model this as a Bayesian game, and show that
in equilibrium neither individual is willing to
exchange any tickets except the least valuable.
8
Analysis: To display this situation as a
Bayesian game we describe each of the items in
Definition 279.1:
• The set of players is {1, 2}.
• The set of states is
{(v1 , v2 )|v1 , v2 ∈ {1, . . . , m} }.
• The set of actions for player i is {Ti , Ni }
where Ti is being willing to exchange and
Ni is not being willing to exchange.
• The set of signals agent i can receive is
{vi : vi ∈ {1, . . . , m} }.
• Let pk be the probability that an agent i
receives a ticket with vi = k. If agent i
receives a ticket with vi = k, then
(regardless of k) agent i assigns probability
pvj to each state (vi , vj ).
• Agent i’s payoff ui ((a1 , a2 ), (v1 , v2 )) is vj if
(a1 , a2 ) = (T1 , T2 ) and vi otherwise.
9
Let v i be the maximum value of a ticket that
agent i is (with some positive probability)
willing to trade.
• If v i > 1, then:
– Agent j would certainly be willing to
trade when vj = 1.
– Agent j would never be willing to trade
if vj > v i .
– When agent i has the ticket
v i ,Subsidizing purchases of health
insurance leads to fewer people being
uncovered both directly, because people
pay a smaller fraction of the cost, and
indirectly, by improving the average
health of those covered, so that the
average expense to the insurer is lower,
and the cost of coverage is
lower.choosing Ti has a positive
probability of losing money, but it can
never gain money.
– Therefore in equilibrium the probability
10
of choosing Ti when vi = v i is zero.
– This contradicts the definition of v i .
• Therefore it must be the case that v i = 1.
This example is closely related to the no trade
theorem: if one player sells a share of stock to
another, and both are motivated only by
speculation, then one of the two is not properly
taking account of the information contained in
the fact that the other is willing to trade.
11
Exercise 282.3: Problem Statement:
• The value of the target firm under current
management is x ∈ {0, 1, . . . , 100}.
– The target firm knows x.
– The acquiring firm believes that each of
the 101 possible values is equally likely.
• The acquirer makes an offer of y.
• The target either accepts or rejects.
– If the target accepts, then the acquirer’s
payoff is 32 x − y and the target’s payoff
is y.
– If the target rejects the offer the
acquirer’s payoff is 0 and the target’s
payoff is x.
Model this situation as a Bayesian game, find
the Nash equilibria, and explain why the logic
of the equilibrium is called adverse selection.
12
Analysis: To display this situation as a
Bayesian game we describe each of the items in
Definition 279.1:
• The set of players is {A, T }.
• The set of states is the set
X = {0, 1, . . . , 100}
of possible values of the target.
• For the acquirer the set of actions is the set
of possible bids.
– The problem does not specify this
explicitly; we will allow the acquirer to
bid any nonnegative number.
• The set of actions for the target is the set
of functions from bids to decisions
f : [0, ∞) → {α, ρ}
where α and ρ are acceptance and
rejection.
13
• The set of signals for the acquirer is a set
with a single element.
• The set of signals for the target is X.
• The belief of the acquirer is that each
1
element of X has probability 101
. The
target is perfectly informed about the state.
• The payoff for the acquirer is
3 x − y, f (y) = α,
uA (x, y, f ) = 2
0,
f (y) = ρ.
The payoff for the responder is
y, f (y) = α,
uR (x, y, f ) =
x, f (y) = ρ.
14
• In a Nash equilibrium the target will accept
a bid y if y > x and reject y if y < x.
• Suppose that the acquirer offers an integer
value of y.
– The most favorable assumption for the
acquirer is that this is accepted
whenever y ≤ x.
• Then the acquirer’s expected payoff is
y
X
1 3
101 ( 2 x
− y) =
1 3 y(y+1)
101 ( 2
2
x=0
− (y + 1)y
1
= − 404
y(y + 1).
• This is less than 0 unless y = 0, so 0 is the
equilibrium offer.
The phrase adverse selection refers to a
situation in which the agents who choose some
action are “self-selected” and are consequently
those with an unfavorable type.
15
Bayesian Updating of Beliefs
An example:
• Consider a medical test for a disease with
the following properties:
– If you have the disease, the test always
gives a positive result.
– If you don’t have the disease, the test
gives a negative result 99% of the time
and a false positive 1% of the time.
• Suppose you take the test and get a
positive result. What is the probability
that you have the disease?
• Answer: it depends.
– Suppose that out of one hundred
thousand people, 100 have the disease.
If they all take the test, there will be
approximately 100 true positives and
999 false positives.
– Conditional on testing positive, the
probability of disease is roughly 9%.
16
Suppose that E and F are events of interest.
• For example:
– E is the event “disease;”
– F is the event “positive test result.”
The conditional probability of E given F is
Pr(E|F ) =
Pr(E and F )
.
Pr(F )
• Bayes’ rule is a formula for computing this
when we are given Pr(E), Pr(F |E), and
Pr(F |not E).
• Specifically,
Pr(E and F ) = Pr(E)Pr(F |E) and
Pr(F ) = Pr(E)Pr(F |E)+Pr(not E)Pr(F |not E).
• Therefore
Pr(E|F ) =
Pr(E)Pr(F |E)
.
Pr(E)Pr(F |E) + Pr(not E)Pr(F |not E)
17
• If Pr(E) = 0.001, Pr(F |E) = 1, and
Pr(F |not E) = 0.01, then
Pr(E|F ) =
0.001 × 1
= 0.092.
0.001 × 1 + 0.999 × 0.01
Another Example
• Suppose E is the event “rain later” and F
is the event “cloudy now.” Assume that:
– Pr(E) = 0.4;
– Pr(F |E) = 0.75;
– Pr(F |not E) = 0.5.
• We compute that conditional on it being
cloudy now, the probability of rain later is:
0.4 × 0.75
Pr(E|F ) =
= 0.5.
0.4 × 0.75 + 0.6 × 0.5
• Exercise: compute the probability of rain
later conditional on it being sunny now.
18
Example: Exercise 284.1
• The set of states is Ω = {α, β, γ}.
9
– The probability of α is 13
, the
3
probability of β is 13
, and the
1
.
probability of γ is 13
• Both players choose between left and right,
with the payoff matrices depending on the
state as shown below.
α
ℓ
L
R
β
r
ℓ
(2, 2) (0, 0)
(3, 0) (1, 1)
!
L
R
(2, 2) (0, 0)
(0, 0) (1, 1)
γ
ℓ
L
R
r
!
(2, 2) (0, 0)
.
(0, 0) (1, 1)
19
r
!
• Player 1 receives signal s1 if α occurs.
Otherwise player 1 receives s2 , in which
case she believes that β has probability 43
and γ has probability 14 .
– Her beliefs are derived from Bayes’ rule
as follows:
Pr(β|s2 ) =
Pr(β)Pr(s2 |β)
Pr(β)Pr(s2 |β) + Pr(not β)Pr(s2 |not β)
=
3
13
3
13
·1
10
13
·1+
·
1
10
=
3
.
4
• Player 2 receives signal t1 if α or β occurs, in
which case she believes that α has probability
and β has probability 14 .
3
4
– Her beliefs are derived from Bayes’ rule as
follows:
Pr(α|t1 ) =
Pr(α)Pr(t1 |α)
Pr(α)Pr(t1 |α) + Pr(not α)Pr(t1 |not α)
=
9
13
9
13
·1
·1+
4
13
·
3
4
=
3
.
4
• If γ occurs player 2 receives signal t2 .
20
Analysis:
• For player 1 with signal s1 playing R is a
dominant strategy.
• When player 2 has signal t1 , she believes
that the probability that player 1 will
choose R is at least 43 , so she will choose r.
• When player 1 has signal s2 , she believes
that the probability that player 2 will
choose r is at least 43 , so she will choose R.
• When player 2 has signal t2 she will choose
r because she expect player 1 to choose R.
21
Auction Theory
• The theory of auctions is richer and more
realistic if we suppose that the bidders are
uncertain about each others’ valuations.
• We will study two models of the
relationship between agents’ signals and
valuations.
– In the independent private values model :
∗ the agents valuations are
independently distributed;
∗ each agent knows her own valuation.
– In the common values model there is an
underlying value that is the same for all
agents, and each agent’s signal is a
noisy measurement of that value.
– For each of these models it is possible to
study both first and second price sealed
bid auctions.
22
Probability Distributions
• A random variable is a number ṽ whose
value is random.
– A cumulative distribution function on
an interval [a, b] is a function
F : [a, b] → [0, 1]
that is weakly increasing with F (b) = 1.
– We say that ṽ is distributed according to
F if
Pr(ṽ ≤ t) = F (t)
for all t with a ≤ t ≤ b.
– The description of the distribution of ṽ
given by F is “complete” in a sense that
is technically deep. Some sense of what
is meant by this is given by the
observation that
Pr(s < ṽ ≤ t) = F (t) − F (s)
for all s, t with a ≤ s ≤ t ≤ b.
23
Jointly Distributed Random Variables
• A cumulative distribution function on an
rectangle [a1 , b1 ] × [a2 , b2 ] is a weakly
increasing function
G : [a1 , b1 ] × [a2 , b2 ] → [0, 1]
with G(b1 , b2 ) = 1.
• We say that a pair of random variables
(ṽ1 , ṽ2 ) is jointly distributed according to G
if
Pr(ṽ1 ≤ t1 and ṽ2 ≤ t2 ) = G(t1 , t2 )
for all t1 , t2 with a1 ≤ t1 ≤ b1 and
a2 ≤ t2 ≤ b2 .
– We have
Pr(s1 < ṽ1 ≤ t1 and s2 < ṽ2 ≤ t2 ) =
G(t1 , t2 )−G(s1 , t2 )−G(t1 , s2 )+G(s1 , s2 ).
24
Independence
• If F1 is a cumulative distribution function
on [a1 , b1 ] and F2 is a cumulative
distribution function on [a2 , b2 ], then the
function
F1 × F2 : [a1 , b1 ] × [a2 , b2 ] → [0, 1]
that takes (t1 , t2 ) to F1 (t1 )F2 (t2 ) is a
cumulative distibution function on
[a1 , b1 ] × [a2 , b2 ].
• If ṽ1 is distributed according to F1 and ṽ2
is distributed accorging to F2 , then we say
that ṽ1 and ṽ2 are independently
distributed, or independent, if (ṽ1 , ṽ2 ) is
jointly distributed according to F1 × F2 .
– This will be the case if the process
generating ṽ1 is unrelated to the process
generating ṽ2 .
– When ṽ1 and ṽ2 are independently
distributed, learning the value of ṽ1 does
not convey any information about ṽ2 .
25
Independent Private Values
We describe an auction as a Bayesian game:
• The players are the bidders 1, . . . , n.
• The set of states is the set Ω = [v, v]n of
profiles (v1 , . . . , vn ) of valuations.
• For each bidder the set of possible actions
is [0, ∞).
• When (v1 , . . . , vn ) is the state, vi is bidder
i’s signal. That is, bidder i knows her own
valuation, and that is her only information.
• Regardless of vi , bidder i believes that
(v1 , . . . , vi−1 , vi+1 , . . . , vn )
is jointly distributed according to
F n−1 : [v, v]n−1 → [0, 1]
where each agent’s valuation is distributed
according to the cumulative distribution
function F : [v, v] → [0, 1].
26
• The payoff to bidder i is:
– 0 if bi is less than some other bid, and
– (vi − P (b))/m when m bids tie for
highest and bi is one of them.
• Here P (b) is the payment made by the
winning bidder.
– In a first price auction P (b) is the
winning bid.
– In a second price auction P (b) is the
second highest bid.
27
Second Price Auctions
Even with ignorance about the other agents’
valuations, it remains the case that in a second
price auction, bidding your own valuation is a
weakly dominant strategy.
• As before, there are many other Nash
equilibria.
• Example:
– Assume that v = 0, v = 1, and F (v) = v
for all v between 0 and 1.
∗ In this circumstance we say that each
vi is uniformly distributed on the
interval [0, 1].
– A somewhat lengthy calculation shows
that if each agent bids here value, then
the expected revenue raised by the
auction is
n−1
.
n+1
28
A First Price Auction
First price auctions are more complicated than
second price auctions because there is no
weakly dominant strategy. Instead, each agent
faces a tradeoff:
• A high bid gives a low surplus. (The
surplus is the difference between the value
of the object and its price.)
• A low bid gives rise to a low probability of
winning.
For the most part attention is restricted to
equilibria with two properties:
• Symmetry: There is a single function
b : [v, v] → [0, ∞) that governs the bidding:
with valuation vi , agent i bids bi = b(vi ).
• Monotonicity: b is a strictly increasing
function.
29
• We will also assume that b is continuous
and differentiable. Let b = b(v) and
b = b(v). There is an inverse function
b−1 : [b, b] → [v, v]
such that b−1 (b(v)) = v for all v.
Suppose that bidder i has valuation vi and bids
bi .
• The surplus if she wins is vi − bi .
• The probability that she wins is the
probability that all other agents bid less
than bi .
– Since the other agents valuations are
independent, this is the probability that
any particular agent j bids less than bi
raised to the power n − 1.
– The probability that bj = b(vj ) < bi is
the probability that vj < b−1 (bi ), which
is F (b−1 (bi )).
– Thus the probability that bi is a
winning bid is F (b−1 (bi ))n−1 .
30
If agent i wishes to maximize expected surplus
(in which case we say that agent i is “risk
neutral”) then agent i should choose bi to
maximize
F (b−1 (bi ))n−1 (vi − bi ).
• The first order condition for this
maximization problem is
F (b−1 (bi ))n−2 f (b−1 (bi ))
(vi −bi )
0 = (n−1)
db
−1
(bi ))
dv (b
−F (b−1 (bi ))n−1 .
– Here f is the derivative of F .
– This simplifies to
−1
(n−1)f (b−1 (bi ))(vi −bi ) = F (b−1 (bi )) db
(b
(bi )).
dv
• In equilibrium this must hold when
bi = b(vi ), in which case the equation
above becomes:
(n − 1)f (vi )(vi − b(vi )) = F (vi ) db
dv (vi ).
31
• General theorems concerning differential
equations imply that there is a unique
function b satisfying this equation and the
“terminal condition” b(v) = v.
• An example: suppose that v = 0, v = 1,
and F (v) = v for all v between 0 and 1.
– Then the derivative of F is 1 at each
point, so the differential equation
becomes
(n − 1)(v − b(v)) = v db
dv (v).
– This equation is satisfied when
b(v) =
n−1
n v.
– A somewhat lengthy calculation shows
that in this equilibrium the expected
revenue is
n−1
.
n+1
32
The Revenue Equivalence Theorem
• In the 1960’s it was observed that the first
price sealed bid auction and the second
price sealed bid auction gave the same
revenue for many other distributions of
valuations in addition to the uniform
distribution.
– This was an unresolved puzzle for
several years.
– Eventually it was proved that this holds
quite generally. The result is called the
revenue equivalence theorem.
33
• The intuition underlying the proof of the
revenue equivalence theorem has the
following components:
– The first price and second price
auctions are both efficient, meaning
that the winner is always the agent with
the highest valuation, so they give rise
to the same relationship between
valuation and probability of winning.
– In both auctions we may think of the
agents as trading off probability of
winning against expected payment.
– Using the first order conditions for
optimization, one can show that
equilibrium implies that the two
auctions give rise to the same tradeoff,
hence the same relationship between
valuation and expected payment.
∗ For details, see the Appendix to
Chapter 9 of Osborne.
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