Lecture 2
Auctions with Private Values Under
Incomplete Information
Gian Luigi Albano, Ph.D.
Head of R&D
Italian Public Procurement Agency - Consip S.p.A.
…Moving on: Auctions with Incomplete Information
What if each bidder knew her/his valuation of the object BUT not her/his
competitors’?
We have to define a new class of games, namely games with incomplete
information
A game with incomplete information is a game in which, at the first point
in time when the players can begin to play their moves in the game, some
players
l
already
l d have
h
private information
f
about
b
the
h game that
h other
h players
l
do not know
The initial private information that a player has at this point in time is
called the type of the player
Games with incomplete information can be modeled in extensive form
by using a historical chance node to describe the random determination
of the players’ types
Harsanyi (1967-68) argued that a generalization of the strategic form,
called the Bayesian form,
form is needed to represent games with incomplete
information
The Bayesian form gives us a way of representing games with incomplete
information which is almost as simple as the strategic form, but which
does not require us to pretend that players choose their strategies before
learning any private information
To define a Bayesian game, we must specify a set of players N and,
for each player i in N, we must specify a set of possible actions Ci, a
set of possible types Ti, a probability function pi, and a utility
function ui. We let
C = X Ci, T = X Ti , T−ii =
i∈N
i∈N
XT
j ∈N−i
j
j.
The probability function pi in the Bayesian game must be a function
from Ti into Δ(T-i), the set of probability distributions over T-i. That is,
for anyy p
possible ti in Ti, the p
probabilityy function must specify
p
y a
probability distribution pi (• | ti) over the set T-i, representing what
player i would believe about the other players’ types if his own type
were ti.
For any player i in N, the utility function ui in the Bayesian game
must be a function from C × T into the real numbers ℜ.
ℜ
These structures together define a Bayesian game Γb, so we may
write
Γ = (N,(C
N (Ci )i∈NN ,(T
(T)
(pi )i∈NN ,(u
(ui )i∈NN ).
i )i∈N
N ,(p
b
We assume each player i knows the entire structure of the game (as
defined above) and his own actual type in T, and this fact is
common knowledge among all the players in N.
A strategy for player i in the Bayesian game Γb is defined to be a
function from his set of types Ti into his set of actions Ci.
We say that beliefs (pi)i∈N in a Bayesian game are consistent iff there
is some common prior distribution over the set of type profiles t such
that each player’s beliefs given his type are just the conditional
probability distribution that can be computed from the prior
distribution by Bayes’ formula. That is, beliefs are consistent iff
there exists some probability distribution P in Δ(T) such that
P(t)
pi (t−i | ti ) =
, ∀t ∈ T, ∀i ∈ N.
∑P(s−ii,ti )
s−i ∈T−i
Most of the g
games that have been studied in applied
pp
game theory,
g
y,
and all of the examples we will be studying, have beliefs that are
consistent with a common prior in this sense
Bayesian Equilibrium
For a Bayesian game with incomplete information, Harsanyi (1967-68)
defined a Bayesian equilibrium to be any Nash equilibrium of the
type-agent representation in strategic form. That is, a Bayesian
equilibrium specifies an action or a randomized strategy for each type
of each player, such that each type of each player would be
maximizing
g his own expected
p
utility,
y, when he knows his own g
given
type but does not know other players’ types
A randomized-strategy profile for the Bayesian game Γb is any σ
such that
σ = ((σ i (c i | t i )) c
i
∈C i
)
t i ∈T i ,ii ∈ N
,
σ i (c i | t i ) ≥ 0, ∀c i ∈ C i , ∀t i ∈ Ti , ∀i ∈ N,
(
∑ σ (c
i
c i ∈C i
i
| t i ) = 1,
1 ∀t
∀ i ∈ Ti , ∀i
∀ ∈ N.
A Bayesian
i
equilibrium
ii i
of the game Γb is any randomizedstrategy profile σ such that, for every player i in N and every
type ti in Ti
σ i (• | t i ) ∈
arg max
τ i ∈ Δ (C i )
⎛
⎞
∑ pi (t− i | t i ) ∑ ⎜⎜ ∏ σ j (c j | t j )⎟⎟τ i (c i )ui (c, t).
t − i ∈T − i
c ∈C ⎝ j ∈ N − i
⎠
Example 1: One-sided incomplete information
t2 = 2a
t2 = 2b
C2
C2
C1
x2
y2
C1
x2
y2
x1
12
1,2
01
0,1
x1
13
1,3
04
0,4
y1
0,4
1,3
y1
0,1
1,2
Two-player, one-sided incomplete information game:
C1={x1,yy1},
} C2={x2, y2},
} T1={1},
={1} T2={2a,
={2a 2b},
2b} p1(2a | 1)=.6,
1)= 6 p1=(2b |
1)=.4. Utility payoffs (u1, u2) depend upon the actions and player 2’s
type as in the tables above
Framework under Incomplete Information
Bidder i’s value, i=1,…,N, is an independent draw from a smooth and
differentiable c.d.f F(·) with strictly positive density f(·) defined on
[0,1]
v˜ i distrib. according to F (⋅).
Preliminaries on Order
Statistics
(v˜1,..., v˜ N ) i.i.d. random variables.
vṽ (1:N
ṽ1,..., vṽ N ]
(1 N ) ≡ max[ v
Fv˜ (1:N ) ( x) ≡ Pr( v˜(1:N ) ≤ x) = Pr( v˜1 ≤ x, v˜ 2 ≤ x,..., v˜ N ≤ x) = F N ( x).
v˜( 2:N ) = second - order statistics.
Fv˜ ( 2:N ) (x) ≡ Pr(v˜(2:N ) ≤ x) = Pr(at least (N -1) of ( v˜1,...v˜ N ) take a value at most of x) =
=
N
∑ Pr(exactly r of (v˜ ,..., v˜
1
N
) take a value at most of x) =
r= N −1
⎛N ⎞
r
N −r
N −1
N
= ∑ ⎜ ⎟{F (x)} {1− F (x)} = N {F (x)} {1− F (x)} + {F (x)} .
r
r= N −1⎝ ⎠
N
We can then compute:
f v˜(1:N ) (x) = N{F(x)}
N 1
N−1
f (x); f v˜( 2:N ) (x) = N (N −1){F(x)}
N 2
N−2
{1− F(x)} f (x).
Bidding Strategy in the First-Price (Sealed-Bid) Auction
Assume:
• Risk neutrality on both sides of the market;
• Seller’s reserve price = entry cost = participation cost = 0.
A bidding strategy for bidder i is a mapping
βi :[0,1]→[0,+∞
:[01]→[0 +∞),
)
from his set of possible values, [0,1], to his set of possible actions
[0,+∞).
Given a profile of strategies
β1(⋅),...,βN (⋅)
we can define a “probability-of-winning function” for bidder i,
Qi (b) = Pr[i wins ∨ each
eachjj ≠ i bids according toβ j (⋅)].
( )]
Bidd i’s
Bidder
i’ expected
t d profit
fit writes
it
Π i (v i ,b) = (v i − b)Qi (b).
In a symmetric equilibrium
β i(⋅) = β(⋅),∀i.
The (unique) symmetric bidding function for any type-v is
β ((v)) =
*
v
∫
0
⎡ g(y) ⎤
y⎢
y
⎥dy,
G( ) ⎦
⎣G(v)
((1))
where G(y) ≡ FN-1(y), g(y) = G’(y).
Interpretation
A bidder’s
bidd ’ equilibrium
ilib i
bid is
i equall to the
h expectation
i off the
h maximum
i
of his competitors’ values conditional on that value being less than his
own:
β * (v ) = E [~y | ~y ≤ v ]
g( ~
y ) is the density function of ~
y , the maximum of N − 1 values.
Pr (y˜ ≤ v ) = G (v ) = {F (v)}
N −1
g(v)
is the conditional density of
G(v)
.
y˜ , given {y˜ ≤ v }.
Underbidding and Competition
β * (v)
( ) < v.
This can be seen more clearly by integrating by parts the
expression off the
h equilibrium
lb
b dd
bidding
f
function
and
d by
b replacing
l
G(v) with FN-1(v). Thus we get
⎡ F (u ) ⎤
β (v ) = v − ∫ ⎢
⎥
F
(
v
)
⎦
0 ⎣
v
*
Since
N −1
du
F (u )
< 1 ⇒ the second term vanishes when N → ∞
F (v )
Revenue Equivalence
q
Define
~I
P ≡ equilibriu
q
m sale p
price in a FPA,,
~
P II ≡ equilibriu m sale price in a SPA
These are two random variables with the same expectation,
expectation but they
are NOT the same random variables
Theorem
In the FPA (with zero reserve price),
(i) the strategy β*(·) defined in (1) is a symmetric equilibrium,
and
(ii) if β(·) is any symmetric equilibrium, then β(v)= β*(v), for all
v>0
Technique
we start the p
proof byy making
g some “inspired
p
guesses” and then we
g
go back to rigorously prove the guesses (actually, not all of them)
Guess A: All types
yp of bidder submit bids
Guess B: β(v) > β(z), for all v > z
Guess C: β(·) is differentiable
Many functions other than β*(·) satisfy guesses A-C. We show that the
only possible equilibrium satisfying them is β*(·).
Lemma 1
If β(·) is a symmetric equilibrium of the FPA satisfying guesses A-C, then
β(·)= β*(·)
Proof:
(i)
Use A-C to show that ∏(b,v) is differentiable in b;
(ii) Since β(·) is optimal for type v, then b = β(v) satisfies ∏b(β(v) ,v) = 0;
((iii)) Since the last p
point holds for all v,, a differential equation
q
is obtained
whose solution β*(·)
From A-C, β(·) has an inverse function ϕ(·) ≡ β-1(·) that satisfies
β(ϕ(b))=b for all numbers b in the range of b.
Value v = ϕ(b) is the value that a bidder must have in order to bid b
when using strategy β(·)
b
β (v)
b'
ϕ (b ' )
v
Because β(·) is strictly increasing, the probability that bidder i wins
g b is:
when bidding
Q(b) = {F (ϕ (b))}
N −1
= G(ϕ (b))
The expected profit of a type v who bids b when other bidders use
β(·)
( ) is
i
Π(b, v) = (v − b)G(ϕ (b))
According to guess C, the inverse function ϕ(·) is differentiable
Π b ( b , v ) = − G (ϕ ( b )) + (v − b )g (ϕ ( b ))ϕ ' ( b )
Since β(·) is an equilibrium, β(v) is optimal for type v, that is, b=β(·)
maximizes ∏(b ,v) and the FOC ∏b(β(v) ,v) = 0 must hold
− G (ϕ ( β ( v ))) + (v − β ( v ) )g (ϕ ( β ( v ))) ϕ ' ( β ( v )) = 0
1
U ϕ (β (v))
Use
( )) = v and
d ϕ (β (v))
( )) = '
to get
β (v)
'
(
v − β ( v ) )g ( v )
− G (v ) +
=0
β ' (v )
G ( v ) β ' ( v ) + g ( v ) β ( v ) = vg ( v )
1 4 4 42 4 4 43
d
[G ( v ) β ( v ) ]
d
dv
So we can integrate from any v0 to v to get
G ( v ) β ( v ) − G ( v0 ) β ( v0 ) =
v
∫ yg ( y ) dyy
v0
From guess A all types bid so we can take v0 → 0. Since
lim v 0 →0 G(v 0 )β (v 0 ) = 0,
we finally obtain
β (v ) =
v
∫
v0
g( y)
y
dy = β * ( v )
G (v )
Q .E .D.
L
Lemma
1 provides
id
only
l a necessary condition
diti
f β(·)
for
( ) = β*(·)
( ) to
t be
b an
equilibrium. In Lemma 2 it is proven that β*(·) satisfies a “pseudoconcavity”
co
cav ty co
condition
d t o tthat
at iss a kind
do
of seco
second-order
d o de co
condition.
dto .
Definition: A differentiable function f: ℜN→ ℜ is pseudo-concave if for all x
and
d y in ℜN, (y-x)∇f(x)
f
≤ 0 implies
l
f
f(x)
≤ f(y).
f
f(·)) is
Pseudo-concavityy is a sufficient ((second-order)) condition because if f(
pseudo-concave and ∇f(x*) = 0, then x* maximizes f(·).
The g
graph
p of
a p
pseudo-concave function f: ℜ→ ℜ is single-peaked
g p
(although the peak can be at plus/minus infinity)
Lemma 2
β*((·)) is a symmetric
y
equilibrium
q
off the FPA.
Proof:
The expected profit of a type-v bidder who bids b when other bidders
use β*(·) is
Π(b,v) = (v −b)G(ϕ* (b)),
where ϕ*(·) is the inverse of β*(·). We must prove that b = β*(v)
maximizes type-v’s expected profit. We will prove it by showing that
∏(·,v) is pseudo-concave, i.e., that
Π b (b,v)
(b ) ≥ 0 for
f b < β (v)
( )
*
Π b ((b,v)) ≤ 0 for b > β ((v).
)
*
This in turn implies that ∏(b ,v) is maximized at b = β*(v) since ∏(b ,v)
is continuous in b.
The key point is to show that ∏bv > 0.
By differentiating ∏b with respect to v we obtain:
Π bv (b, v ) = g (ϕ (b))ϕ (b) > 0 for all v ∈ (0,1) and b ∈ (0, β * (1))
*
*'
Ch
Choose
b ∈ [0,
b̂
[0 β * (v)].
( )] Let
L t vˆ bbe th
the type
t
who
h is
i supposedd to
t bid
so that bˆ = β * (vˆ ).
Since bˆ < β * (v), vˆ < v. Because Π bv > 0, then
Π b (bˆ, vˆ ) = 0 and then
Π b (bˆ, v ) ≥ 0
Q . E . D.
b̂,
b
Food for Thought
• “Buying bids” auctions
• Two-animal
Two animal war of attrition with incomplete information
• Takeovers with “winner’s embarassment”