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.

C – Approximation: An Auction m is a C-approximation over Φ

if for every valid auction v’, .

If c=1, the auction is optimal.

Rc

Rm

1

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.

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.

• 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.

• Correlation: v1 is uniform in [0,100]. v2 = 2v1.

Bob1 is always rejected.

Optimal: P = twice the lower bid.

),...,( 211 nwwpp

111 pw

The 1 – lookahead auction computes, based on

declarations from the non-highest bidders, a price p1:

That maximizes it’s revenue from agent1 (according to ).

If than agent1 wins, and pays p1.

Otherwise, nobody wins.

Theorem: the 1-lookahead auction is a 2-approximation.

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

• It’s a 2-approximation auction:

splitting to two cases:

and , and showing that :

and

'R

1'R 2'R

1'RR 2'RR

• The approximation ratio of 2 is tight.

Sketch Of Proof:

Agent2’s type is fixed to 1.

v1 is determined acording to:

The optimal revenue is about 2.

Our auction generates a revenue of about 1.

kv1Pr h

1hk

1kh

11

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 k’s and choose the one that maximizes:

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.

),...,(Pr 21 nvvkv

),...,(Pr 21 nvvkvk

Vickrey Auction With Reserved Price:

Let . It is the following the auction:

If v1 < r, all agents are rejected.

Otherwise, agent1 wins and pays max(v2,r).

0r

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,2i+1). (log(h) such intervals).

Ii is the interval that contributes most to .

Take r = 2i.

The revenue:

dv1

dv1

dRh

dvh

dR OPTlog2

1

log2

1 1

Let be the conditional distribution

The K-lookahead auction is the optimal auction on agents (1,…,k) according to .

nkk vvvv ,...,,..., 11

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 where

The probability of the type of agent1 is determined by

agent2’s type. If ,then with

probability , and with probability .

Our auction’s revenue is around .

A better auction: Asks agent1 for . If , sells to

agent3 for the price 1. Revenue – around 2.

12

1j

j1 hj log,...,2,1

jv 12 11 2 jv

111 jv 12

11

j

hlog

11

j2 jv 21

Theorem: If (v1,…,vn) are independent, the k-lookahead auction is

a -approximation.k

k 1

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.

the optimal revenue from agents (k+1,…,n).

For , mj is the contribution of agent j to Ropt.

Case I: for all , .

11

k

k vm1km

kj

j

jopt mR

kj jk mm 1

Case II: Not all , .

Let denote the agent with minimal mj:

Pretend he declared vk+1, and run Aopt on it.

If any of the (n-k) won, sell to agent for v k+1.

Now, .

Because the distributions are independent, the distributions of the other agents don’t change.

kj jk mm 1

j jk mm ˆ1

j

jk mm ˆ1

opt

jjjjoptkopt R

k

kmmRmRR

1ˆˆ1

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

• We showed an improvement of that auction – to improve the

approximation ratio to , but only under the assumption that the valuations are independent.

k

k 1

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

• 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).