EC992 - Advanced Microeconomics. Lecture 5:
Bayesian Games and Auctions
Ludovic Renou
Academic year: 2013-204
Games with Incomplete Information
Bayesian games model strategic situations in which players are
uncertain about the situation of other players.
DEFINITION: A Bayesian game consists of
•
a finite set N (the set of players)
•
a set Ω (the set of states)
•
for each player i ∈ N a nonempty set Ai (the set of actions
available to player i)
•
for each player i ∈ N a nonempty set Ti (the set of signals
or types of player i) and a signal function τi : Ω → Ti
•
for each player i ∈ N a probability measure pi on Ω (the prior
belief of player i)
•
for each player i ∈ N a payoff (or utility ) function
ui : A → R (where A := ×j∈N Aj is the set of action profiles or
outcomes); preferences over lotteries on A × Ω are given by
expected utility.
This definition allows players to have different prior beliefs. They
are commonly assumed to be identical (common prior
assumption).
Frequently the state of nature is a profile of parameters that affect
players’ payoffs. However the model is more general.
Often the model is given in reduced form: player i knows his own
type, and has a belief over the types of the other players pi (t−i |ti ),
conditional on his type.
Example 1. Two players must decide between two actions. Player
2 would like to play the same action than player 2, and prefers B
to A. The preferences of player 1 are not known to player 2. Player
1 could be of one of two types. Type t1:1 has the preferences
represented in the left table. Type t1:2 has the preferences
represented on the right table. Player 2 has no private information
and hence there is only one type, t2 , of player 2. Player 2 attaches
probability p to Player 1 being of type t1:1 .
A
B
A
B
2, 1
0, 0
0, 0
1, 2
A
B
A
B
0, 1
1, 0
2, 0
0, 2
Formally, N = {1, 2}, Ω = {ω1 , ω2 }, τ1 (ω1 ) = t1:1 , τ1 (ω2 ) = t1:2 , ,
τ2 (ω1 ) = τ2 (ω2 ) = t2 , p2 (ω1 ) = p.
The following definition is most useful when, for all i, the set of
types Ti is finite.
DEFINITION:
A Bayesian equilibrium of the Bayesian game
hN, Ω, (Ai ), (Ti ), (τi ), (pi ), (ui )i is a Nash equilibrium of the game
in which each type of each player is viewed as an independent
player.
An equivalent, alternative, approach is to view the strategy of
player i as a function from i’s types to his action set, si : Ti → Ai .
DEFINITION:
A (pure strategy) Bayesian equilibrium of the Bayesian game
hN, Ω, (Ai ), (Ti ), (τi ), (pi ), (ui )i is a profile s ∗ = (s1∗ , ..., sn∗ ) of
∗ , s ∗ ) ≥ U (s ∗ , s ) for all
functions si∗ : Ti → Ai such that, Ui (s−i
i −i i
i
si : Ti → Ai , where Ui (s−i , si ) is player i’s expected utility when
the strategy profile is (s−i , si ). (This expected utility is computed
using the beliefs pi of player i concerning the.type profile t. It is
equivalent to each type ti of player i, for all i, maximizing his
expected utility conditional on his type ti ).
Example 1.
Exercise: Find the pure strategy Bayesian equilibria.
Example 2. Player 1 is an incumbent firm, player 2 is a potential
entrant. Player 1 decides whether to build a new plant (B), or not
(N), player 2 simultaneously decides whether to enter (E ) or stay
out (O). The building cost of the incumbent, player 1, is not
known to player 2. If the cost is high, payoffs are in the left table.
If the cost is low, payoffs are as in the right table. Player 2 has no
private information. The entrant, player 2 attaches probability p to
the incumbent having high cost.
E
B
N
0, −1
2, 1
O
2, 0
3, 0
E
B
N
1.5, −1
2, 1
Exercise: Find the pure strategy Bayesian equilibria.
O
3.5, 0
3, 0
Purification of mixed strategy equilibrium:. Given a mixed strategy
equilibrium, of a strategic game, one can construct a “close”
Bayesian game with pure strategy Bayesian equilibrium which is
“close” to the mixed strategy equilibrium of the original strategic
game.
Example 3. (A Bayesian version of matching pennies). The types
t1 of player 1 and t2 of player 2 are elements of the interval [0, 1].
The beliefs of player i about player j’s type are uniformly
distributed, i.e., the density of type t is f (t) = 1 and the
distribution is F (t) = t. The payoffs are shown below, where
ε > 0.
H
H
T
1 + ε(2t1 − 1), −1
−1, 1 + ε(2t2 − 1)
T
−1, 1
1, −1
Exercise: Find the pure strategy Bayesian equilibrium. Show that it
converges to the mixed strategy Nash equilibrium of matching
pennies as ε → 0.
Example 4. (Providing a public good). Two players
simultaneously decide whether to contribute to a public good C , or
not N (contributing is a 0-1 decision). Each player derives a
benefit of 1 from the public good which is provided if at least one
contributes. Players costs of contributing are ci and are privately
know. The types c1 of player 1 and c2 of player 2 are elements of
the interval [0, 2]. The beliefs of player i about player j’s type are
uniformly distributed, i.e., the density of type t is f (t) = 1/2 and
the distribution is F (t) = t/2. The payoffs are shown below.
C
C
N
1 − c1 , 1 − c2
1, 1 − c2
N
1 − c1 , 1
0, 0
Exercise: Find the pure strategy Bayesian equilibrium. (Hint:
Players contribute to providing the public good if and only if their
cost is below a threshold.)
Example 5. (A bargaining game). A buyer and a seller have
private information about their valuations for an object, vB and vS .
The (beliefs about the) valuations are uniformly distributed on
[0, 1]. If trade does not take place, payoffs of both players are
normalized to zero. If trade takes place at price p, then the buyer’s
payoff is vB − p, while the seller’s payoff is p − vS . Buyer and seller
simultaneously announce prices pB and pS . Trade only takes place
if pB ≥ pS . If trade takes place, the price is (pB + pS )/2.
There are very many Bayesian equilibria.
Consider the following class of equilibria. For any value of x ∈ [0, 1]
the buyer offers pB = x if vB ≥ x and offers pB = 0 otherwise.
The sellers asks pS = x if vS ≤ x and asks pS = 1 otherwise.
There is also a linear equilibrium in which:
pB (vB ) =
pS (vS ) =
2
1
vB +
3
12
2
1
vS +
3
4
Exercise: Prove this is a Bayesian equilibrium.
Trade occurs in this equilibrium if and only if vB ≥ vS + 1/4. This
turns out to be the “most efficient” equilibrium, that is the
equilibrium with the highest gains from trade. (Note: equilibrium
is not efficient, because sometimes trade does not take place even
though the buyer values the object more than the seller.)
Auction Theory
In the standard symmetric, independent, private value model, there
are N bidders for one object. Each bidder’s type x is independently
and identically drawn (i.i.d) from the increasing cumulative
distribution F (x) with support [0, 1] (this is just a normalization).
F has a continuous density function f and E [x] < ∞.
Bidders are risk neutral, so that a type x bidder’s value when
winning the object at price p is x − p. Each bidder knows his own
type, but not the types of the other bidders.
If there are interdependent values (often called common values in
the literature), then the value of the object to bidder i depends not
only on bidder i’s signal, but also on the signals of all other bidders.
In such a case the value to bidder i of winning at price p is
v (xi , x−i ) − p where v is the value of winning and x−i is the profile
of types of allPother bidders (as an example, think of
v (xi , x−i ) = N
j=1 xj this is an example of pure common values).
In a first-price sealed bid auction, bidders simultaneously submit
bids; the highest bidder wins and pay his own bid.
In a second-price sealed bid auction or Vickrey auction, bidders
simultaneously submit bids; the highest bidder wins and pays the
second highest bid.
In an open ascending auction, or English auction, the price starts
low and it is raised continuously by the auctioneer; bidders decide
when to drop out; once a bidder has dropped out, he cannot
re-enter (this is the theoretically simplest way of modeling
ascending open auctions).
In an open descending auction, or Dutch auction, the price starts
high and it is lowered continuously by the auctioneer; the first
bidder to take the current price wins the auction at the current
price.
Proposition The Dutch auction is strategically equivalent to the
sealed bid, first-price auction.
Proposition Under private values (i.e., v = xi ), but not under
interdependent values, the English auction is equivalent to the
sealed bid, second-price auction, in the sense that the optimal
strategies in the two auctions are the same. (But not strategically
equivalent.)
Proposition Under private values (i.e., v (xi , x−i ) = xi ), in a
second-price and in an English auction it is a weakly dominant
strategy for each bidder to bid b(x) = x (i.e., it is a Bayesian
equilibrium for each bidder i of type xi to bid xi ).
There are Bayesian equilibria in which players use weakly
dominated strategy for example: b(x1 ) = 1 for all x1 ∈ [0, 1] and
b(xi ) = 0 for all i 6= 1 and all xi ∈ [0, 1]. It is standard to disregard
these equilibria.
Let the order statistic Y1 be the highest of N − 1 random draws
from the distribution F (x). The distribution of Y1 is F N−1 (x); its
density is (N − 1)f (x)F N−2 (x).
Proposition Under private values (i.e., v (xi , x−i ) = xi ), it is a
symmetric equilibrium of a first-price auction to bid
b(x) = E [Y1 |Y1 < x]
Z x
y (N − 1)f (y )F N−2 (y )
dy
=
F N−1 (x)
0
Z x N−1
F
(y )
= x−
dx.
N−1
(x)
0 F
With interdependent values, we can write
v (xi , x−i ) = v (xi , Y1 , Y2 , ...) where the order statistics Y1 , Y2 ,...
are the highest, second highest,... of N − 1 random draws from the
distribution F (x).
Proposition In a second-price auction with interdependent values,
it is an equilibrium for each bidder i to bid
b(xi ) = E [v (xi , Y1 , ...)|Y1 = xi ].
Proposition Under interdependent values, it is a symmetric
equilibrium of a first-price auction for each bidder i to bid
b(xi ) = E [v (Y1 , Y1 , ...)|Y1 < xi ].
Winner curse
The winner’s curse under interdependent (or common) values.
Bidders must condition on winning. If they didn’t they would end
up overbidding. This is because winning conveys the bad news
that all other bidders have lower signals.
This is easily seen in a second-price auction, where
E [v (xi , Y1 , ...)|Y
P 1 = xi ] < E [v (xi , Y1 , ...)]. For example, if
v (xi , x−i ) = N
j=1 xj and the xj are uniformly distributed, then
E [v (xi , Y1 , ...)|Y1 = xi ] = 2xi + (N − 2)xi /2, while
E [v (xi , Y1 , ...)] = xi + (N − 1)/2. In equilibrium rational bidders
take into account that winning is bad news, and hence do not
suffer from the winner’s curse.
Proposition Under interdependent values, it is a symmetric
equilibrium of an English auction for each bidder i with signal xi to
follow the following strategy:
If nobody has dropped out yet, drop out at price
p(xi ) = v (xi , xi , ..., xi ).
If one player has already dropped out at price
pN−1 = v (xN−1 , xN−1 , ..., xN−1 ), then drop out at price
p(xi , xN−1 ) = v (xi , xi , ..., xi , xN−1 ). ...
If k players have already dropped out at prices
pN−1 = v (xN , xN , ..., xN ), ..., pN−k =
v (xN−k , xN−k , ..., xN−k , xN−k−1 , ..., xN−1 ), then drop out at price
p = v (xi , xi , ..., xi , xN−k , ..., xN−1 ).
P
Thus, for example, suppose v (xi , x−i ) = N
j=1 xj and the xj are
uniformly distributed. Suppose also that x1 > x2 > x3 ..., then the
price in a second price auction is 2x2 + (N − 2)x2 /2, while the
price in an English auction is 2x2 + x3 + x4 + .... Note: the
expected price in a second-price and in an English auction are the
same! This revenue equivalence result is true in general. With
independent signals, all efficient auctions are revenue equivalent.
Mechanism Design
Every auction can be seen as a mechanism in which bidders send
messages to a designer (their bids) and the designer commits to an
outcome decision (e.g., who gets the item) and transfer function
(who pays what).
A mechanism can be used for any general allocation, or social
decision, problem. In general a mechanism can be seen as a triple
hM, D, T i where: M = ×i={1,...,N} Mi and Mi is the set of
messages that player i can send to the designer; D : M → A, the
decision function, maps messages into the set of possible
allocations A (in the case of an auction of a single object A could
be the set of vectors of probabilities (π1 , ..., πN ), where πi is the
probability that i wins the object); T : M → RN , the transfer
function, maps messages into a transfer payment from each of the
bidders to the designer.
A direct mechanism is a mechanism in which the message space
Mi of each player is his set of types (in the auction described
before, the set [0, 1]). It is standard to omit the message space and
denote a direct mechanism by the pair hD, T i.
Revelation principle
The following is a fundamental result.
Theorem (The Revelation Principle) Given any mechanism and
a (Bayesian) equilibrium for that mechanism, there exists a direct
mechanism in which (1) it is an equilibrium for each buyer to
report her true type truthfully, and (2) the equilibrium outcomes
(decision and transfers) are the same as in the given equilibrium of
the original mechanism.
Proof (sketch) Instead of each buyer i reporting message (bid)
b = b(xi ) a direct mechanisms asks her to report xi and makes sure
the outcome is the same as in the original mechanism. Intuitively,
a direct mechanism does the equilibrium calculations in the original
mechanism directly for the bidder. If it was an equilibrium for type
xi to bid according to b(xi ) in the original mechanism (rather than,
say, bid b
b = b(z)), then it will be an equilibrium in the direct
mechanism to report the true type xi (rather than, say, z). Let < π, p > be a direct mechanism, where πi (xi , x−i ) is the
probability that i wins an object and pi (xi , x−i ) is i’s payment.
Q
Let g (x−i ) =
f (xj ). Bidder i’ s expected payoff when his type is
j6=i
xi , but he reports zi and all other bidders report truthfully is
Z
Ui (zi ; xi ) =
x
Z
x
x
[v (xi , x−i )πi (zi , x−i ) − pi (zi , x−i )] g (x−i )dx−i
...
x
Letting Ui (xi ) be type xi ’s expected payoff in the truthful
equilibrium of hπ, pi, and using a standard envelope argument
yields
Z x Z x
∂V (xi , x−i )
0
...
πi (xi , x−i )g (x−i )dx−i
Ui (xi ) =
∂xi
x
x
Bidder-payoff equivalence
Theorem. Bidders’ expected payoffs are the same in any
mechanism hπ, pi having the same outcome function π and
yielding the same payoff to the lowest type. Bidder i’ s expected
payoff is given by
Z xi Z x Z x
∂V (x, x−i )
πi (x, x−i )g (x−i )dx−i dx.
...
Ui (xi ) = u +
∂x
x
x
x
In particular, if u = 0, the outcome function π is efficient (so that
πi = 1 if xi > YKI −1 and πi = 0 if xi < YKI −1 ), and there are no
informational externalities, then the above expression reduces to
R x (I −1)
(y )dy .
Ui (xi ) = x i FK
Revenue Equivalence Theorem
As a simple consequence of bidder’s payoff equivalence, we have
the revenue equivalence theorem.
Theorem. The seller’s expected revenue is the same in any
mechanism hπ, pi having the same outcome function π and
yielding the same payoff to the lowest type. In particular, in all
efficient auctions in which the lowest type expected payoff is zero
(these include the first-price, second-price and English auction),
the seller’s expected revenue is the same.
A mechanism design application:
Example 1. (A general version of a bargaining game). A buyer
and a seller have private information about their valuations for an
object, vB and vS . The (beliefs about the) valuations are
distributed on [0, 1] according to the distribution functions FB (vB )
and FS (vS ) with positive densities fB (vB ) and fS (vS ).If trade does
not take place, payoffs of both players are normalized to zero. If
trade takes place at price p, then the buyer’s payoff is vB − p,
while the seller’s payoff is p − vS .
A mechanism specifies the probability of trade π and the payment
p from the buyer to the seller (which could be negative) as
functions of reports. A mechanism is efficient if trade takes place
whenever vB > vS . A direct mechanism is Bayesian incentive
compatible if all players telling the truth is a Bayesian equilibrium.
A mechanism is individually rational if no player’s type makes a
negative expected payoff.
Theorem (Myerson and Satterthwaite) In the bargaining
model, there is no efficient, Bayesian incentive compatible and
individual rational mechanism.