317-F · 317F cos — cos — Sin Sin cos cos cos i + cos —2 i + SlnY— J —2 i + sin —Zi + sin nC
COS 444 Internet Auctions: Theory and Practice
description
Transcript of COS 444 Internet Auctions: Theory and Practice
![Page 2: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/2.jpg)
week 6 2
Conditional expectation
Intuitively clear, very useful in auction theory
If x is a random variable with cdf H, and A is an event, define the conditional expectation of x given A:
A
xxdHAprob
AxE )(}{
1]|[
Not defined when prob{ A } = 0.
![Page 3: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/3.jpg)
week 6 3
Two quick examples of conditional expectation
1) Suppose x is uniformly distributed on [0,1]. What is the expected value of x given that it is less than a constant c ≤ 1?
2) Given two independent draws x1 and x2 , what is the expected value of x1, given that x1≤ x2?
![Page 4: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/4.jpg)
week 6 4
Interpretation of FP equil.
Now take a look once more at the equilibrium bidding function for a first-price IPV auction:
1
0
1
)(
)()(
n
v n
fp vF
ydFyvb
I claim this has the form of a conditional expectation. What is the event A?
![Page 5: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/5.jpg)
week 6 5
Interpretation of FP equil.
Claim: event A = you win! = {Y1,(n-1) ≤ v }, where Y1,(n-1) = highest of (n-1) independent draws
To check this•
• Let y = Y1,(n-1) , the “next highest value”. The cdf of y = Y1,(n-1) is F(v)n-1 so the integral in the expected value of y given that you win is
1)(}{ nvFAprob
v n
AydFyydHy
0
1)()(
![Page 6: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/6.jpg)
week 6 6
Interpretation of FP equil.
Therefore,
]wins|[
]|[)(
)1,(1
)1,(1)1,(1
vYE
vYYEvb
n
nnfp
That is, in equilibrium, bid the expected next-highest value conditioned on your winning. …Intuition?
![Page 7: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/7.jpg)
week 6 7
Stronger revenue equivalence
)(
)(}wins{
wins]|[}{)( )1,(1
vP
vbvprob
vYEwinsvprobvP
fp
fp
nsp
Let Psp(v) be the expected payment in equilibrium of a bidder in a SP auction (and similarly for FP).
So SP and FP are revenue equivalent for each v !
![Page 8: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/8.jpg)
week 6 8
Graphical interpretation
• Once again, by parts:
v nn
fpsp dyyFvFPP0
11 )(
![Page 9: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/9.jpg)
week 6 9
Bidder preference revelation
Theorem:Theorem: Suppose there exists a symmetric Bayesian equilibrium in an IPV auction, and assume high bidder wins. Then this equilibrium bidding function is monotonically nondecreasing.
*Thanks to Dilip Abreu for showing me this elegant proof.
![Page 10: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/10.jpg)
week 6 10
Bidder preference revelation
• Proof: Proof: Bid as if your value is z when it’s actually v. Let
w(z) = prob. of winning as fctn. of z
p(z) = exp. payment as fctn. of z
For convenience, let w=w(v), w΄=w(v΄), p=p(v), p΄=p(v΄), for any v, v΄ .
![Page 11: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/11.jpg)
week 6 11
Bidder preference revelation
• From the definition of equilibrium, the expected surplus satisfies:
v·w – p ≥ v·w΄ – p΄ v΄·w΄ – p΄ ≥ v΄·w – p for every v,v΄ . Add: (v – v΄ )·(w – w΄) ≥ 0 .So v > v΄ → w ≥ w΄ → b(v) ≥ b(v΄). □
![Page 12: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/12.jpg)
week 6 12
Example of an IPV auction with no symmetric Bayesian equil.:third-price (see Krishna 02, p. 34)
![Page 13: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/13.jpg)
week 6 13
Riley & Samuelson 1981:Optimal Auctions
• Elegant, landmark paper, constructs the benchmark theory for optimal IPV auctions with reserves
• Paradoxically, gets more powerful results more easily by generalizing
![Page 14: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/14.jpg)
week 6 14
![Page 15: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/15.jpg)
week 6 15
all-pay
3rd-price
war of attr.
losers weepav. others
Santa Claus
last-price
![Page 16: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/16.jpg)
week 6 16
![Page 17: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/17.jpg)
week 6 17
Riley & Samuelson’s class Ars
1. One seller, one indivisible object2. Reserve b0 (open reserve, starting bid)3. n bidders, with valuations vi i=1,…,n4. Values iid according to cdf F, which is strictly
increasing, differentiable, with support [0,1] ( so f > 0 )
5. There is a symmetric equilibrium bidding function b(v) which is strictly increasing (we know by preference revelation it must be nondecreasing)
6. Highest acceptable bid wins7. Rules are anonymous
![Page 18: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/18.jpg)
week 6 18
Abstracting away…
• Bid as if value = z, and denote expected payment of bidder by P(z). Then the expected surplus is
• For an equilibrium, this must be max at z=v, so differentiate and set to 0:
)()( 11 zPzFv n
0)()( 1 xPxFdx
dx n
![Page 19: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/19.jpg)
week 6 19
We need a boundary condition…
• Denote by v* the value at which it becomes profitable to bid positively, called the entry value:
• Now integrate d.e. from v* to our value v1:
1*** )()( nvFvvP
1
*
1*1 )()()(
v
v
nxdFxvPvP
![Page 20: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/20.jpg)
week 6 20
Once more, integrate by parts…
• And use the boundary condition:
• A truly remarkable result! Why?
1
*
11111 )()()(
v
v
nn dxxFvFvvP
![Page 21: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/21.jpg)
week 6 21
Once more, integrate by parts…
• And use the boundary condition:
• A truly remarkable result! Why?
Where is the auction form? FP? SP? Third-price? All-pay? …
1
*
11111 )()()(
v
v
nn dxxFvFvvP
![Page 22: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/22.jpg)
week 6 22
Revenue Equivalence Theorem
Theorem:Theorem: In equilibrium the expected revenue in an (optimal) Riley & Samuelson auction depends only on the entry value v* and not on the form of the auction. □
![Page 23: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/23.jpg)
week 6 23
Marginal revenue, or virtual valuation
• Let’s put some work into this expected revenue:
• And integrate by parts (of course, what else?)…
v
v
n
v
nrs vdFdxxFvFnR
**
)(])([ 11 1
![Page 24: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/24.jpg)
week 6 24
Marginal revenue, or virtual valuation
nvdF
vrs vf
vFvR )(][
1
* )(
)(1
1
*
)()(vrs
nvdFvMRR
)(
)(1)(where
vf
vFvvMR
![Page 25: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/25.jpg)
week 6 25
Interpretation of marginal revenue
Because F(v)n is the cdf of the highest, winning value, we can interpret this as saying:
The expected revenue of an (optimal) Riley & Samuelson auction is the expected marginal revenue of the winner.
![Page 26: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/26.jpg)
week 6 26
Hazard rate
Let the failure time of a device be distributed with pdf f(t) and cdf F(t).
Define the “survival function” = R(t) = prob. of no failure before time t = prob. of survival till time t.
Since F(t) = prob. of failure before t ,
R(t) = 1 – F(t).
![Page 27: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/27.jpg)
week 6 27
Hazard rate
The conditional prob. of failure in the interval (t, t+Δt ] , given survival up to time t , is
The “Hazard Rate” is the limit of this divided by Δt as Δt → 0 :
)(
)()(
tR
ttRtR
F
f
R
RHR
1
![Page 28: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/28.jpg)
week 6 28
Hazard rate
Thus, the marginal revenue, which is key to finding the expected revenue in a Riley-Samuelson auction, is
1/HR is the “Inverse Hazard Rate”
HRv
f
FvMR
11
![Page 29: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/29.jpg)
week 6 29
Why call it “marginal revenue”?
Consider a monopolist seller who makes a take-it-or-leave-it offer to a single seller at a price p. The buyer has value distribution F, so the prob. of her accepting the offer is 1–F(p). Think of this as the buyer’s demand curve. She buys, on the average, quantity q = 1–F(p) at price p. Or, what is the same thing, the seller offers price
p(q) = F-1(1– q) to sell quantity q.
after Krishna 02, BR 89
![Page 30: COS 444 Internet Auctions: Theory and Practice](https://reader036.fdocuments.us/reader036/viewer/2022070404/56813b85550346895da4b134/html5/thumbnails/30.jpg)
week 6 30
Why call it “marginal revenue”?
The revenue function of the seller is therefore q·p(q) = q F-1(1-q) , the revenue derived from selling quantity q. The derivative of this wrt q is by definition the marginal revenue of the monopolist :
F-1(1-q) = p, so this is
□
))1(()1(
11
qFF
qqF
)(
)(1)(
pf
pFppMR