Transcript By: Amir Ronen, Presented By: Oren Mizrahi Matan Protter

By: Amir Ronen,
Department of CS
Stanford University
Presented By:
Oren Mizrahi
Matan Protter
Issues on border of economics & computation, 2002
•We will discuss the issue of revenue maximization,
also known as optimal auction design.
•It is a subject of long and intensive research in
microeconomics.
•We will look for an approximation.
• [ n ] = { 0 , 1 , 2 , .. , n}
• Wi = {1, 1 + ε , 1 + 2 ε , … , 2 , 2 + ε , … , h } : The possible
types (valuations ) of each agent.
•Φ = A distribution over the type space.
• Rm = The revenue of the auction m = The expected payment
An Auction: A pair of function (k,p) such that:
• K : W [n] is an allocation algorithm determining who wins
the object (a zero – no winner).
• P : W R is a payment function determining how much the
winner must pay.
A Valid Auction: An auction the satisfies both:
• Individual Rationality (IR): The profit of a truth telling agent
is always non – negative: p(w) ≤ wk(w).
• Incentive Compatibility (IC): Truth-telling is a dominant
strategy for each agent.
C – Approximation: An Auction m is a C-approximation over Φ
1
R

R.
if for every valid auction v’,
m
c
If c=1, the auction is optimal.
An Algorithm with the following charecaristics:
Input:
• One item to sell.
• A probability distribution over the type space.
• Constant C.
Output:
• An auction.
Restrictions:
• Auction is a C-approximation optimal auction.
• Both Algorithm and auction are polytime.
Suppose Alice wishes to sell a house to either Bob1 or Bob2, for
prices in the range [0,100].
Let’s look at a few simple connections:
• Independent Valuations: Both v1 and v2 are uniform in [0,100].
Good: Second price auction.
Better: Second price auction with reserve price 50.
• Correlation: v1 is uniform in [0,100]. v2 = 2v1.
Bob1 is always rejected.
Optimal: P = twice the lower bid.
• Anti - Correlation: v1 is uniform in [0,100]. v2 = 100 - v1.
Optimal: P = The maximum of (w,100-w) where w is the
lower bid.
The 1 – lookahead auction computes, based on
declarations from the non-highest bidders, a price p1:
p1  p1 (w2 ,..., wn )
That maximizes it’s revenue from agent1 (according
to 1 ).
1
1
w

p
If
than agent1 wins, and pays p1.
Otherwise, nobody wins.
Theorem: the 1-lookahead auction is a 2-approximation.
Sketch Of Proof:
• It satisfies IR and IC, therefore a valid auction.
• It’s a 2-approximation auction:
splitting R ' to two cases:
R '1
and R'2 , and showing that :
R  R '1 and R  R '2
• The approximation ratio of 2 is tight.
Agent2’s type is fixed to 1.
v1 is determined acording to:


Pr v1  k 

1
h
1
k h
1
h
k 1  
The optimal revenue is about 2.
Our auction generates a revenue of about 1.
When we have a polytime algorithm that can compute, given a
price k and valuations (v2,…,vn), the probability:

Pr v1  k (v 2 ,..., v n )

We can simply try for all possible k’s and choose the one that
maximizes:

k  Pr v1  k (v 2 ,..., v n )

If h is large, we can, for some α, try only the cases:
(v2, α·v2, α2·v2,…,h), and we will get a α-approximation of the
optimal price.
Vickrey Auction With Reserved Price:
Let r  0 . It is the following the auction:
If v1 < r, all agents are rejected.
Otherwise, agent1 wins and pays max(v2,r).
Their exists a price r, such that the Vickrey auction with
reserved price r is a 2log(h) approximation.
Proof:
Given a distribution d, v d  is the expectation of v1.
1
Look at intervals [2i,2i+1). (log(h) such intervals).
Ii is the interval that contributes most to v d  .
1
Take r = 2i.
The revenue:
1
1
1
Rd  
 v d  
 ROPT d 
2 log h 
2 log h 
Let  be the conditional distribution


 v1 ,..., v k v k 1 ,..., v n

The K-lookahead auction is the optimal auction on agents (1,…,k)
according to  .
Obviously, at least a 2 – approximation.
The approximation ratio is tight!
Three agents, k = 2.
Agent3’s type is always 1.
Agent2’s type is uniformly drawn from 1  j    where
j  1,2,..., log h
The probability of the type of agent1 is determined by
agent2’s type. If v 2  1  j   ,then v1  2 j 1 with
probability 2
1
j 1
, and v  1   j  1  
1
Our auction’s revenue is around
1
with probability 1 
1
log h 
.
j
2
A better auction: Asks agent1 for
. If v1  2 j , sells to
agent3 for the price 1. Revenue – around 2.
1
2 j 1
.
Theorem: If (v1,…,vn) are independent, the k-lookahead auction is
a
k 1
k
-approximation.
Sketch Of Proof:
Fix the (n-k) lowest valuations (agents k+1,…,n).
Aopt is the optimal auction, R is our revenue, Ropt the optimal revenue.
k 1
mk 1 the optimal revenue from agents (k+1,…,n). mk 1  v
For j  k , mj is the contribution of agent j to Ropt.
Ropt   m j
j
Case I: for all j  k , mk 1  m j .
Case II: Not all j  k , mk 1  m j .
Let
ĵ denote the agent with minimal mj: mk 1  m ˆj
Pretend he declared vk+1, and run Aopt on it.
If any of the (n-k) won, sell to agent ĵ for v k+1.
Now, mk 1  m ˆj .
Because the distributions are independent, the distributions of the
other agents don’t change.
k
R  R opt  mk 1  R opt  m ˆj   m j 
R opt
k 1
j  ˆj
• We showed a simple 2-approximation. (1 – lookahead auction).
•It can be computed in polytime if there are polytime algorithms
computing the distribution Φ.
• We showed an improvement of that auction – to improve the
k 1
approximation ratio to k
, but only under the assumption
that the valuations are independent.
• Same techniques can be used to show bounds for weakly connected
valuations.
• Finding an auction which does better than 2-approximation on general
distributions (or proving it’s impossible).