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Market Design and Analysis
Lecture 4
Lecturer: Ning Chen (陈宁)
Email: [email protected]
Last Lecture
Dominant strategy Nash equilibrium
First price auction
No
ε-Nash
Second price auction
Yes
Yes
Vickrey auction
Yes
Yes
2
Revenue Consideration

Question: For first price and second price auction,
which generates more revenue?

Second price auction may look bad - the winner’s
price may be much lower than his bid.
(An extreme once happened in a New-Zealand
government-auction: One firm bid NZ$100,000 for a
license, and paid the second-highest price of only
NZ$6)
3
Revenue Consideration

In Economics, the (revenue) analysis of auctions is
usually based on the assumption that the valuations
of the buyers are drawn from a known distribution
(e.g. [0,1]).
4
Revenue Consideration

Let’s consider a simple case – there are 2 buyers
with values v1,v2 drawn independently and uniformly
from [0,1].

For second price auction, they bid bi = vi, hence the
expected revenue is E[min(v1,v2)].

For first price auction, what do they bid? How can
we compute the expected revenue without knowing
b i?
5
Bayesian Game

A Bayesian game has, in addition to a normal game,
for each player i, a type space Ti. There is a
probability distribution P generating the type for
each player. Player i



knows his (fixed) type ti∈Ti
only knows the probability distribution of types of other
players
has a payoff function
ui(s1,…,sn; ti): S1 x …x Sn x Ti  R
6
Bayesian Game

In a Bayesian game,


a strategy for player i is a function si(ti), i.e. si: Ti  Si.
For a given type ti, si(ti) gives the strategy selected by
player i.
a strategy profile s* = (s*1,…,s*n) is called a Bayesian
Nash equilibrium if for each player i and his type ti,
s*i(ti) is the solution of
where
7
Describe Auctions as Bayesian Games

For each buyer i,

there is a privately know true value (i.e. type) vi (assume
drawn uniformly at random from [0,1])
 decide bi, a function of vi, as his submitted bid

For first price auction, his utility is
ui(b1,…,bn;vi) = vi – bi if i has largest bid and is a winner,
 ui(b1,…,bn;vi) = 0 otherwise.


For second price auction, his utility is
ui(b1,…,bn;vi) = vi – maxj≠i if i has largest bid and is a winner,
 ui(b1,…,bn;vi) = 0 otherwise.

8
Bayesian Nash of First Price Auction

Claim. Bidding bi = vi (n-1)/n for each buyer i is a Bayesian
Nash equilibrium for first price auction.

Proof. Consider any buyer i, assume all other buyers use
this strategy.
 For a bid b ≤ (n-1) / n, the probability of winning is
(b∙n/(n-1))n-1. So the expected utility is (vi-b) (b∙n/(n-1))n-1
 Derivative w.r.t b gives
– (b∙n/(n-1))n-1 + (vi-b)(n-1)bn-2(n/(n-1))n-1
which should equal zero
 Hence, –b + (vi-b)(n-1) = 0, which gives b = vi(n-1)/n.
9
Revenue Consideration

Let’s consider a simple case – there are 2 buyers
with values v1,v2 drawn independently and uniformly
from [0,1].

For second price auction, they bid bi = vi, hence the
expected revenue is E[min(v1,v2)].
For first price auction, assume they bid according to
Bayesian Nash equilibrium, i.e. bi = vi / 2. Then the
expected revenue is E[max(v1,v2)] / 2.

10
A Probability Fact

Fact. A non-negative random variable X satisfies

Proof. Let fX(z) be the density function of X. The we have
Using the fact that
we have
11
Revenue Consideration – Second Price


By uniform distribution, Pr[vi < x] = x  Pr[vi ≥ x] = 1-x.
Since v1,v2 are independent,
Pr[min(v1,v2) ≥ x] = Pr[v1 ≥ x & v2 ≥ x] = (1-x)2

Hence,
12
Revenue Consideration – First Price


E[max(v1,v2)] = 2/3.
Hence, E[max(v1,v2)] / 2 = 1/3.
13
Vickrey Auction with Reservation Price r

The Vickrey auction with reservation price r, VAr,
sells the item if any buyer bids above r. The price
the winning buyer pays is the maximum of the
second highest bid and r.

Consider the revenue obtained when


VA0.5, i.e. r = 0.5
there are 2 buyers, and their values v1,v2 are drawn
independently at random from the uniform distribution in
[0,1].
14
Vickrey Auction with Reservation Price r

Analysis:




Case 1. both v1,v2 < 0.5. No one wins and no revenue
Case 2. one bid < 0.5, one bid ≥ 0.5. Get revenue 0.5
Case 3. both v1,v2 ≥ 0.5. This is the same as the above
analysis except that the two variables are drawn
uniformly from [0.5,1]. Thus, the expected revenue is
0.5+0.5/3 = 2/3
Therefore, the expected revenue is
0 * 1/4 + 0.5 * 1/2 + 2/3 * 1/4 = 5/12
15
Single-Parameter Agents



There are n agents, each desires some item or
service.
Agent i’s value for receiving service is vi, and 0
otherwise.
The mechanism solicits sealed bids (b1,…,bn) from
agents and computes an outcome, consisting of
allocation x = (x1,…,xn)
 price / payment p = (p1,…,pn) ≥ 0


The utility of agent i is ui(bi) = vi∙xi(bi) – pi(bi).
16
Surplus and Profit

Assume there is a cost c(x) of producing the
outcome x.

The surplus (or social welfare) of an outcome is
defined by
∑i vixi - c(x)

The profit of an outcome is defined by
∑i pi - c(x)
17
Examples

Single item auction:

Combinatorial auction: There are n various items
and each agent i desires a bundle of items Si. Then
18
Truthful Mechanisms

We say a mechanism is truthful if truth-telling is a
dominant strategy for every agent. That is, for any
agent i, vi, bi and b-i,
ui(vi,b-i) ≥ ui(bi,b-i)

We assume all mechanisms are voluntary
participation, i.e. no agent has negative expected
utility for participation  pi(bi) ≤ bi∙xi(bi).
19
Characterization of Truthful Mechanisms

Theorem. A mechanism is truthful if and only if, for
any agent i and bids of other agents b-i fixed,



xi(bi) is a monotone non-decreasing
pi(bi) = bi ∙xi(bi) – ∫0bi xi(z) dz
For a fixed monotone allocation rule x(∙), the payment
rule p(∙) is also fixed. Hence, when specifying a truthful
mechanism, we need only specify a monotone allocation
rule.
20
Characterization of Truthful Mechanisms


Proof. Assume the mechanism is truthful, i.e. for any
value vi and possible lies bi,
vi ∙xi(vi) – pi(vi) ≥ vi ∙xi(bi) – pi(bi)
Consider two possible values z1 and z2 with z1 < z2, and
two scenarios:
vi = z1, bi = z2  z1 ∙xi(z1) – pi(z1) ≥ z1 ∙xi(z2) – pi(z2)
 vi = z2, bi = z1  z2 ∙xi(z2) – pi(z2) ≥ z2 ∙xi(z1) – pi(z1)



Adding them gives
z1 ∙ xi(z1) + z2 ∙ xi(z2) ≥ z1 ∙ xi(z2) + z2 ∙ xi(z1)
Hence,
(z2 – z1)∙(xi(z2) – xi(z1)) ≥ 0  xi(z2) ≥ xi(z1)
This is true for any z2 > z1, thus xi(∙) is monotone.
21
Characterization of Truthful Mechanisms



Next we will show that the payment must be
pi(bi) = bi ∙xi(bi) – ∫0bi xi(z) dz
Given ui(bi) = vi ∙xi(bi) – pi(bi), consider its partial
derivative w.r.t bi:
u’i(bi) = vi ∙x’i(bi) – p’i(bi)
Since truthfulness implies that ui(bi) is maximized at vi =
bi, we have
0 = u’i(vi) = vi ∙x’i(vi) – p’i(vi)
which holds for any value of vi. In particular, when vi = z,
p’i(z) = z ∙ x’i(z)
22
Characterization of Truthful Mechanisms

Hence,

By voluntary participation assumption, pi(0) = 0, and we
are done (for the first direction).
23
Characterization of Truthful Mechanisms



Proof. The other direction. Assume that xi(bi) is
monotone and payment is
pi(bi) = bi ∙xi(bi) – ∫0bi xi(z) dz
Consider two possible values z1 and z2 with z1 < z2.
Suppose vi = z2, we will show that agent i does not
benefit by shading his bid down to z1.
(The other case when vi = z1 and agent i does not benefit
by shading his bid up to z2 is left as an assignment.)
24
Characterization of Truthful Mechanisms

The following figure shows xi(∙) is monotone by
assumption.
25
Characterization of Truthful Mechanisms
bidding bi = vi = z2
valuation vi ∙ xi(vi)
payment pi(vi) = vi ∙xi(vi) – ∫0vi xi(z) dz
utility ui(vi) = vi ∙ xi(vi) – pi(vi)
26
Characterization of Truthful Mechanisms
bidding bi = z1
valuation vi ∙ xi(bi)
payment pi(bi) = bi ∙xi(bi) – ∫0bi xi(z) dz
utility ui(bi) = vi ∙ xi(bi) – pi(bi)
27
Characterization of Truthful Mechanisms

The difference of utility when bidding bi = vi = z2 and
bidding bi = z1 < z2. (The monotonicity of the allocation
implies this difference is always non-negative.)

Hence, the mechanism is truthful.
28
Characterization of Truthful Mechanisms

Theorem. A mechanism is truthful if and only if, for
any agent i and bids of other agents b-i fixed,


xi(bi) is a monotone non-decreasing
pi(bi) = bi ∙xi(bi) – ∫0bi xi(z) dz
29
Deterministic Truthful Mechanism

For any deterministic mechanism xi(∙)∈{0,1},
monotonicity implies that xi(∙) is a step function.
If agent i wins, his payment is
pi(vi) = vi ∙xi(vi) – ∫0vi xi(z) dz = infz {z: xi(z) = 1}
30
Maximizing Surplus




Recall that the surplus is defined by ∑i vixi – c(x).
Consider an allocation rule that maximizes the surplus,
i.e. given b, output allocation x maximizing ∑i bixi – c(x).
Theorem. The surplus maximizing allocation is
monotone.
Proof. Let S(v,x) = ∑i vixi – c(x). Let v’ be the vector of
values where v’i = vi + δ, and v’j = vj for j ≠ i. Then
S(v’,x) = ∑i v’ixi – c(x) = δ∙xi + ∑i vixi – c(x) = δ∙xi+S(v,x)

Consider any x maximizing S(v,x) with the largest xi, then x
maximizes S(v’,x) as well.
 Thus, the value of xi is monotone.
31
Virtual Valuation and Surplus

Assume that the value vi of agent i is drawn
independently at random from Fi. Let
F = F1 x F2 x … x F n

The cumulative distribution function for vi is
Fi(z) = Pr[vi < z]

The probability density function for vi is
fi(z) = F’i(z)
32
Virtual Valuation and Surplus

The virtual valuation of agent i with value vi is

Given value vi, virtual valuation Φi(vi), and allocation
x, the virtual surplus is
33
Myerson’s Optimal Mechanism

Theorem. (Myerson’81) The expected profit of a
mechanism is equal to its expected virtual surplus.

Myerson’s optimal mechanism outputs x to
maximize the virtual surplus.
Given bids b, compute the “virtual bids” b’i = Φi(bi).
 Compute the surplus maximizing allocation x’ and
corresponding payment p’.
 Output x = x’ and p, where pi= bi ∙xi(bi) – ∫0bi xi(z) dz

34
Myerson’s Optimal Mechanism

Since maximizing-surplus allocation rule is
monotone, Myerson’s optimal mechanism is truthful
if and only if Φi(vi) is monotone in vi for any i.

The condition that Φi(vi) is monotone is consistent
with the monotone hazard rate assumption, which is
a common assumption in economics. (The hazard
rate is define as f(z) / (1-F(z)).)
35
Myerson’s Optimal Mechanism

Lemma. The expected payment of an agent, as a
function of their bid bi and with all other bids fixed b-i,
satisfies

Proof. (drop the subscript i) The bid b is selected at
random from distribution F with density function f.
Hence,
36
Myerson’s Optimal Mechanism

Focusing on the 2nd term and switching the order of
integration, we have

Rename z to b and factor x(b)f(b) out
37
Myerson’s Optimal Mechanism

Proof of Myerson’s Theorem. Since the values of all
agents are independently distributed and
by linearity of expectation, we have
38
Interpretation of Myerson’s Mechanism

Single item auction:

In single-item auctions, the surplus maximizing allocation
gives the item to the agent with the highest value, unless
the highest value is less than 0. (Usually the values of all
agents are assumed to be at least 0.)
Myerson’s mechanism computes virtual surplus
maximizing allocation, given virtual valuations, which can
be negative. The mechanism, thus, allocates the item to
the agent with the largest positive virtual valuation.

39
Interpretation of Myerson’s Mechanism

Consider there are only two agents.
Agent 1 wins when Φ1(b1) ≥ max{Φ2(b2), 0}.
 This is a deterministic allocation rule. Thus the payment is
p1 = inf {b: Φ1(b) > Φ2(b2) and Φ1(b) > 0}


Suppose that F1 = F2 = F, then Φ1(z) = Φ2(z) = Φ(z)

If agent 1 wins, his payment is
p1 = inf {b: Φ(b) > Φ(b2) and Φ(b) > 0}
which is
p1 = inf {b: b > b2 and b > Φ-1(0)}
 If agent 2 wins
p2 = inf {b: b > b1 and b > Φ-1(0)}
40
Interpretation of Myerson’s Mechanism

Theorem. The optimal single-item auction for agents
with values drawn independently from F is precisely the
Vickrey auction with reservation price Φ-1(0).
 For example, when F = [0,1], then
 the cumulative distribution function F(z) = z
 the density function f(z) = F’(z) = 1
 Thus Φ(z) = 2z – 1 and the virtual valuations are
drawn uniformly from [-1,1]
 Hence, Φ-1(0) = 1/2 and the optimal mechanism for
two agents with uniform value on [0,1] is the Vickrey
auction with reservation price 1/2.
41
Summary


Characterization of truthful mechanisms.
Steps to design an optimal mechanism to maximize
the expected profit by Myerson’s optimal mechanism:

valuation distribution of each agent
 virtual valuation and surplus
 allocation rule maximizing virtual surplus
 payment rule (according to the characterization
theorem)
 simplify allocation and payment rule.
42
Revenue Equivalence Theorem

Assume that there are n agents, and the true value vi of
each agent i is drawn independently from the same
distribution F(v) on [vmin,vmax] (i.e. F(vmin)=0 and F(vmax)=1)
with density function f(v). Assume that f(v) > 0 for any
v∈[vmin,vmax].

In a Bayesian Nash equilibrium, agent i submits his bid
(a function of vi) according to the equilibrium

assume that the probability that i wins is qi(vi)
 assume that the expected payment of agent i is pi(vi)

The utility function of agent i is ui(vi) = vi∙qi(vi) – pi(vi)
43
Revenue Equivalence Theorem

Lemma. For any value v and v’,
ui(v) ≥ ui(v’) + (v – v’)∙qi(v’)

Proof. Let v be the true value of agent i, then
ui(v’) + (v – v’)∙qi(v’)
= v’qi(v’) – pi(v’) + (v – v’)∙qi(v’)
= vqi(v’) – pi(v’)
≤ ui(v)
44
Revenue Equivalence Theorem

Given the lemma, ui(v) ≥ ui(v’) + (v – v’)∙qi(v’)


when v is the true value, we have
ui(v) ≥ ui(v+dv) + (– dv)∙qi(v+dv)
when v+dv is the true value, we have
ui(v+dv) ≥ ui(v) + (dv)∙qi(v)

Hence,

Taking the limit dv  0, we have
45
Revenue Equivalence Theorem
expected
utility
the slope of the utility
function is qi(v)
ui(vmin)
46
Revenue Equivalence Theorem





Now consider any two mechanisms that have the same
ui(vmin) and the same qi(v) functions for all v and agent i.
As
they have the same utility function ui(v).
Hence, for any given value vi, agent i makes the same
expected payment in these two mechanisms.
This implies the expected payment of agent i, across all
possible values vi, is also the same.
Therefore, the two mechanisms have the same expected
revenue.
47
Revenue Equivalence Theorem

Consider any mechanisms which always



give the item to the highest-bid agent.
give an agent of lowest possible value no chance of
any positive utility, i.e. ui(vmin) = 0.
Then qi(v) = (F(v))n-1, and all these mechanisms
have the same expected revenue.
48
Revenue Equivalence Theorem

Revenue Equivalence Theorem. Assume that

there are k items and n agents, where each agent desires
one item.
 the values are independently drawn from a common
distribution [vmin,vmax], which is strictly increasing
Then any mechanism in which

the items always go to the k agents with the highest values,
 any agent with value vmin expects 0 utility,
yields the same expected revenue, and each agent i has
the same expected payment.
49
Revenue Equivalence Theorem



The condition that f(v) > 0 for any v∈[vmin,vmax] is crucial
for the theorem.
For example, there are two agents and their values are
independently and equally either 0 or 1.
Consider the following mechanism: offer a price p to two
agents simultaneously
 if only one accepts, then sell him at p
 if both accept, then choose a winner at random at price p
 if both reject, then choose a winner at random at price 0

Then ui(0) = 0 and the higher-bid always wins. But the
revenue strictly increases in p, so Revenue Equivalence
fails.
50