On Approximating

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On Approximating Optimal Auctions
By Amir Ronen, Department of CS Stanford
University
Presented By Oren Mizrahi Matan Protter
Issues on border of economics computation, 2002
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What Why

• 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.

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Notations

• n 0 , 1 , 2 , .. , n
• Wi 1, 1 e , 1 2 e , , 2 , 2 e , , h
  The possible types (valuations ) of each
  agent.
• F A distribution over the type space.
• Rm The revenue of the auction m The expected
  payment

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Definitions

• 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.

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Definitions - Cont.

• 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.

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Our Problem

• 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.

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Some Simple Examples
Suppose Alice wishes to sell a house to either
Bob1 or Bob2, for prices in the range
0,100. Lets 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.

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More Simple Examples

• 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.

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1-lookahead auction
The 1 lookahead auction computes, based on
declarations from the non-highest bidders, a
price p1 That maximizes its revenue from
agent1 (according to ). If
than agent1 wins, and pays p1. Otherwise, nobody
wins.
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One Short Theorem
Theorem the 1-lookahead auction is a
2-approximation.
Sketch Of Proof

• It satisfies IR and IC, therefore a valid
  auction.

• The approximation ratio of 2 is tight.

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Example Why It Is Tight
Agent2s type is fixed to 1. v1 is determined
acording to The optimal revenue is about
2. Our auction generates a revenue of about 1.
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Computing The Auction
When we have a polytime algorithm that can
compute, given a price k and valuations
(v2,,vn), the probability We can simply try
for all possible ks and choose the one that
maximizes
If h is large, we can, for some a, try only
the cases (v2, av2, a2v2,,h), and we will
get a a-approximation of the optimal price.
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Another Definition
Vickrey Auction With Reserved Price Let
. It is the following the auction If v1 lt r, all
agents are rejected. Otherwise, agent1 wins and
pays max(v2,r).
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Proposition
Their exists a price r, such that the Vickrey
auction with reserved price r is a 2log(h)
approximation.
Proof Given a distribution d, is the
expectation of v1. Look at intervals 2i,2i1).
(log(h) such intervals). Ii is the interval that
contributes most to . Take r 2i. The
revenue
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K - lookahead auction
Let be the conditional distribution 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!
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Example Why It Is Tight
Three agents, k 2. Agent3s type is always
1. Agent2s type is uniformly drawn from
where The probability of the type of agent1 is
determined by agent2s type. If ,then
with probability , and
with probability
. Our auctions revenue is around
. A better auction Asks agent1 for . If
, sells to agent3 for the price 1.
Revenue around 2.
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Another Theorem
Theorem If (v1,,vn) are independent, the
k-lookahead auction is a -approximation.
Sketch Of Proof Fix the (n-k) lowest valuations
(agents k1,,n). Aopt is the optimal auction, R
is our revenue, Ropt the optimal revenue. the
optimal revenue from agents (k1,,n). For
, mj is the contribution of agent j to
Ropt. Case I for all ,
.
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Theorem Proof- Cont.
Case II Not all ,
. Let denote the agent with minimal mj
Pretend he declared vk1, and run Aopt on it. If
any of the (n-k) won, sell to agent for v
k1. Now, . Because the
distributions are independent, the distributions
of the other agents dont change.
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Conclusions

• We showed a simple 2-approximation. (1
  lookahead auction).

• It can be computed in polytime if there are
  polytime algorithms computing the
  distribution F.

• We showed an improvement of that auction to
  improve the
• approximation ratio to , but only
  under the assumption that the valuations are
  independent.

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On To The Future

• 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 its impossible).