
Myerson: Bayesian Single-item Auction
problem:
◦ Single item for sale (can be extended to ``service”)
◦ 𝑛 bidders
◦ Distribution 𝐹 = 𝐹1 × 𝐹2 × ⋯ × 𝐹𝑛 from which bidder
values are drawn
◦ VCG on virtual values
1




What about Myerson when bidders are
correlated? (Not a product distribution)
Answer: Can beat Myerson revenue (by cheating
somewhat)
Exp. Utility to agent if value = 10 is zero
Exp. Utility to agent if value = 100 is 30
2





Run VCG AND charge extra payments:
P(b) – extra payment if OTHER GUY bids b
𝑝 10 = −30, 𝑝 100 = 60
Both bid 100: both get utility -60
Both bid 10: both get utility +30
Revenue = TOTAL “surplus”
Better than
Reserve price of 100
3

On Profit-Maximizing Envy-Free Pricing

Algorithmic Pricing via Virtual Valuations

Pricing Randomized Allocations

Approximate Revenue Maximization with Multiple
Items
◦ Guruswami, Hartline, Karlin, Kempe, Kenyon, McSherry
◦ SODA 05
◦ Chawla, Hartline, Kleinberg
◦ arXiv 2008
◦ Briest, Chawla, Kleinberg, Weinberg
◦ arXiv, 2009
◦ Hart, Nisan
◦ arXiv 2012
4

Myerson: Bayesian Single-item Auction
problem:
◦ Single item for sale (can be extended to ``service”)
◦ 𝑛 bidders
◦ Distribution 𝐹 = 𝐹1 × 𝐹2 × ⋯ × 𝐹𝑛 from which bidder
values are drawn
◦ VCG on virtual values

Bayesian Unit-demand Pricing Problem:
◦ Single unit-demand bidder
◦ 𝑛 items for sale
◦ Distribution 𝐹 = 𝐹1 × 𝐹2 × ⋯ × 𝐹𝑛 from which the
bidder valuation for items is drawn
5

How do n bidders single item relate to single
bidder n items?
6


For 𝒏 = 𝟏 the Bayesian Single item Auction
and the Bayesian Unit-demand Pricing
problem are the same problem.
Offer the item at a price of 𝑝1 = 𝜙1−1 (0) where
𝜙1 is the virtual valuation function for
distribution 𝐹 = 𝐹1
1 ¡ Fi (vi )
Ái (vi ) = vi ¡
:
f i (vi )
(For regular distributions,
Use ironed virtual values if
irregular)
7


Theorem: For any price vector p, the revenue of
Myerson (when bidder 𝑖 value of the single item
comes from 𝐹𝑖 ) is ≥ the revenue when the single
bidder gets value for item 𝑖 from 𝐹𝑖
Given price vector p, new mechanism M:
◦ Allocate the item to the bidder 𝑖 with 𝑣𝑖 ≥ 𝑝𝑖 that
maximizes 𝑣𝑖 − 𝑝𝑖
◦ Have bidder 𝑖 pay the critical price (the lowest bid at
which bidder 𝑖 would win.


The allocation function 𝑎𝑖 is monotone and so this is
a truthful mechanism.
Myerson is optimal amongst all Bayes Nash IC
mechanisms,
◦ → Myerson gets more revenue than M
8
◦ Allocate the item to the bidder 𝑖 with 𝑣𝑖 ≥ 𝑝𝑖 that maximizes 𝑣𝑖 − 𝑝𝑖
◦ Have bidder 𝑖 pay the critical price (the lowest bid at which bidder 𝑖
would win.

The minimum bid for bidder 𝑖 to win is 𝑝𝑖 +
max(𝑣𝑗 − 𝑝𝑗 , 0) ≥ 𝑝𝑖 This is the revenue to M
𝑖≠𝑗


when 𝑖 wins
The revenue from a single bidder who buys item
𝑖 with value 𝑣𝑖 at a price of 𝑝𝑖 is 𝑝𝑖
QED
9


Let 𝜒 𝑝 = 𝑖 𝐹𝑖 𝑝𝑖 - the probability that no
item is sold at pricing vector 𝑝
Use prices 𝑝𝑖 = 𝜙𝑖−1 (𝑣), where 𝑣 is chosen as
follows:
◦ Choose 𝑣 such that 𝑝𝑖 = 𝜙𝑖−1 (𝑣) and 𝜒 𝑝 = 0.5

Theorem (Chawla, Hartline, B. Kleinberg): Gives
no less than Myerson revenue / 3
◦ Algorithmic Pricing via Virtual Valuations
10
11
The following two problems are equivalent:
◦ Given n items and m “value vectors” for m bidders 𝑆 =
𝑣11 , 𝑣12 , … , 𝑣1𝑛 , 𝑣21 , 𝑣22 , … 𝑣2𝑛 , … , 𝑣𝑚1 , 𝑣𝑚2 , …
compute the revenue-optimal item pricing
◦ Given a single bidder whose valuations are chosen
uniformly at random from 𝑆, compute the revenueoptimal item pricing
12

Reduction from vertex cover on graphs of
maximum degree at most B (APX hard for B≥3).
◦ Given (connected) Graph G with n vertices construct n
items and m+n bidders (or bidder types)
◦ For every edge 𝑒 = (𝑖, 𝑗) add bidder with value 1 for
items 𝑖, 𝑗 and zero otherwise
◦ Additionally, for every item there is a bidder with
value 2 for the item.
◦ The optimal pricing gives 𝑚 + 2𝑛 − 𝑘 profit, where 𝑘 is
the smallest vertex cover of 𝐺
13
◦ Given (connected) Graph G with n vertices construct n items and m+n bidders
(or bidder types)
◦ For every edge 𝑒 = (𝑖, 𝑗) add bidder with value 1 for items 𝑖, 𝑗 and zero
otherwise
◦ Additionally, for every item there is a bidder with value 2 for the item.
◦ The optimal pricing gives 𝑚 + 2𝑛 − 𝑘 profit, where 𝑘 is the smallest vertex
cover of 𝐺


Let S be vertex cover, charge 1 for items in
S, 2 otherwise
If 𝑖, 𝑗 ∈ 𝐸 and price(i)=price(j)=2 then no
profit from bidder 𝑖, 𝑗 - reducing price of
(say) 𝑖 to one keeps profit unchange.
14

Pricing Randomized Allocations
◦ Patrick Briest, Shuchi Chawla, Robert Kleinberg, S.
Matthew Weinberg
◦ Remark: When speaking, say “Bobby Kleinberg”, when
writing write “Robert Kleinberg”.
15





2 items uniformly value
uniformly distributed in
[a,b]
Optimal item pricing sets
𝑝∗ = 𝜙 −1 0
Offer lottery equal prob
item 1 or item 2, at cost
𝑝∗ − δ
Lose here
Gain here
16


Buy-one model: Consumer can only buy one
option
Buy-many model: Consumer can buy any number
of lotteries and get independent sample from
each (will discard multiple copies)
17



Polytime algorithm to compute optimal lottery
pricing
For item pricing (also known as envy-free unit
demand pricing) the optimal item pricing is
APX-hard (earlier today)
There is no finite ratio between optimal lottery
revenue and optimal item-pricing revenue (for
𝑛 ≥ 4)
18




𝜆 = 𝜙, 𝑝 where 𝜙 = (𝜙1 , 𝜙2 , … , 𝜙𝑛 ), and
≤ 1, 𝑝 ≥ 0 price of lottery
Λ – collection of lotteries
Bidder type is 𝑣 = (𝑣1 , 𝑣2 , … , 𝑣𝑛 )
Utility of picking lottery 𝜆 is
u(v; ¸ ) = (



P
n
i = 1 Ái vi )
𝑖 𝜙𝑖
¡ p
Utility maximizing lotteries Λ 𝑣 ⊆ Λ
Max payment for utility maximizing lottery:
p+ (v; ¤ ) = maxf pj(Á; p) 2 ¤(v)g:
Z
Profit of Λ is
¼(¤ ) =
p+ (v; ¤ )dD
19
𝑣𝑗 = 𝑣𝑗1 , 𝑣𝑗2 , … , 𝑣𝑗𝑛 values of
type j bidders
𝜇𝑗 - prob of type j bidders
𝑥𝑗 = (𝑥𝑗1 , 𝑥𝑗2 , … , 𝑥𝑗𝑛 ) – lottery
designed for type j
𝑧𝑗 - price for lottery 𝑗
Feasible
Affordable
Type j prefer
lottery j
𝑥𝑗𝑖 ≥ 0, 𝑧𝑗 ≥ 0
20


𝑟𝐿∗ 𝐷 - maximal expected revenue for single
bidder when offered optimal revenue
maximizing lotteries, bidder values from (joint)
distribution 𝐷
𝑟 ∗ 𝐷 - maximal expected revenue for single
bidder when offered optimal revenue
maximizing item prices, bidder values from
(joint) distribution 𝐷
21

Theorem: 𝑟𝐿∗ 𝐷 = 𝑟 ∗ 𝐷
¤
=
f (Á0 ; p0 ); (Á1 ; p1 ); : : : ; (Ám ; pm )g;
0 =
Á0 < Á1 < ¢¢¢< Ám ;
0 =
p0 < p1 < ¢¢¢< pm :
Assume optimal
WLOG
Valuation 𝑣 chooses to purchase (𝜙𝑗 , 𝑝𝑗 )
Áj v ¡ pj
¸
Áj ¡ 1 v ¡ pj + 1
Áj v ¡ pj
¸
v
2
Áj + 1 v ¡ pj + 1
·
¸
pj ¡ pj ¡ 1 pj + 1 ¡ pj
;
Áj ¡ Áj ¡ 1 Áj + 1 ¡ Áj
22
Valuation 𝑣 chooses to purchase (𝜙𝑗 , 𝑝𝑗 )
Áj v ¡ pj
¸
Áj ¡ 1 v ¡ pj + 1
Áj v ¡ pj
¸
Áj + 1 v ¡ pj + 1
·
¸
pj ¡ pj ¡ 1 pj + 1 ¡ pj
;
Áj ¡ Áj ¡ 1 Áj + 1 ¡ Áj
v
2
23

Correlated distributions, unit demand, finite
distribution over types:
◦ Deterministic, item pricing
 We saw: APX hardness
 Lower bound of log(𝑚) (m- number of items) or log 𝑛 number of different agent types – not standard
complexity assumptions Briest
◦ We saw: Lotteries: optimal solvable Briest et al

Product distributions:
◦ 2 approximation Chawla Hatline Malec Sivan
◦ PTAS – Cai and Daskalakis
24


𝑛 bidders, 1 item
1 bidder, n items
25


Gambler
◦ 𝑛 games
◦ Each game has a payoff drawn from an independent
distribution
◦ After seeing payoff of i th game, can continue to next
game or take payoff and leave
◦ Each payoff comes from a distribution
◦ Optimal gambler stategy: backwards induction
Theorem: There is a threshold t such that if
the gambler takes the first prize that exceeds
t then the gambler gets ½ of the maximal
payoff
26
Choose 𝑡 such that (exists?)
Yn
Fi (t) = 1=2:
i= 1
Benchmark (highest possible profit for gambler):
For 𝑥𝑖 ≥ 0
𝐸 𝑥𝑖 ≤ 𝑥𝑖
27
1
= 𝑡
2

Profit of gambler

𝜖𝑖 - event that for all 𝑗 ≠ 𝑖 : 𝑣𝑗 < 𝑡
Y
Prob(² i ) =
Y
Fj (t) ¸
j6
=i
+ “extras”
Fj (t) = 1=2:
j
E[max(vi ¡ t; 0)j² i ] = E[max(vi ¡ t; 0)]
28
Benchmark
Gambler profit ≥
𝑡
2
1
+ 2 𝐸[max(𝑣𝑖
− 𝑡, 0)] ≥
𝐸 max 𝑣𝑖
𝑖
2
29
Find t such that 𝑡 = 𝐸 max 𝑣𝑖 − 𝑡, 0
𝑡 increases and is continuous
𝐸[max 𝑣𝑖 − 𝑡, 0 ] decreases and is continuous
30

Many bidders, one item, valuation for bidder 𝑖
from 𝐹𝑖
◦ Vickrey 2nd price auction maximzes social welfare
(max 𝑣𝑖 )
𝑖
◦ Myerson maximize virtual social welfare maximizes
revenue

Many items, one bidder, value for item 𝑖 from 𝐹𝑖
31

Many bidders, 1 item:
◦ There is an anonymous (identical) posted price that
1
gives a approximation to welfare. Directly from
2
Prophet inequality.
◦ Who gets the item? Could be anyone that wants it.
There is no restriction on tie breaking.

Many items, one bidder:
◦ There is a single posted price (for all items) that
1
gives a approximation to social welfare.
2
32

Many bidders, 1 item:
1
◦ There is a single virtual price 𝑡 that gives a
2
approximation to welfare. Directly from Prophet
inequality and that the revenue is the expected
virtual welfare.
¡ 1
Á
◦ This means non-anonymous prices i (t)
◦ Who gets the item? Could be anyone that wants it.
There is no restriction on tie breaking.

Many items, one bidder:
◦ Use same prices for
1
2
approximation
33