CS294 P29 Algorithmic Game Theory
November 1, 2011
Lecture 10: Profit Maximization
Lecturer: Christos Papadimitriou
Scribe: Di Wang
Lecture given by George Pierrakos. In this lecture we will cover
1. Characterization of 1D-domain.
2. Myerson’s optimal mechanism design.
3. Approximation in Bayesian mechanism design.
10.1
Characterization
Consider the case where we have n bidders and 1 item. The bidders have private values for the item
v1 , v2 , . . . , vn , and the mechanism is given by the allocation and payment functions.
xi : (v1 , . . . , vn ) → [0, 1]
pi : (v1 , . . . , vn ) → R+
We want the mechanism to satisfy the following constraints
• Incentive Compatibility (IC):xi (vi )vi − pi (vi ) ≥ xi (bi )vi − pi (bi ) for any bi .
• Individual Rationality (IR): xi (vi )vi − pi (vi ) ≥ 0
�
•
i xi ≤ 1 (probability of allocating the item ≤ 1)
Theorem 1. (Characterization) A mechanism is IC+IR iff
1. xi (vi ) is increasing in vi
�v
2. pi (vi ) = xi (vi )vi − 0 i xi (z)dz
10-1
10-2
Lecture 10: Profit Maximization
Proof. The only if part. Prove by picture, consider the various areas in the plot
pi (vi ) = A1 + A2, pi (bi ) = A2, ui (bi ) = A3 + A4, ui (vi ) = A3 + A4 + A5.
IC is satisfied since ui (vi ) ≥ ui (bi ), IR is also satisfied.
The if part. Using IC two times with bidder i’s true value being vi and bi respectively, we can write
xi (vi )vi − pi (vi ) ≥ xi (bi )vi − pi (bi )
xi (bi )bi − pi (bi ) ≥ xi (vi )bi − pi (vi )
Adding the above two equations together, we get
(xi (vi ) − xi (bi ))(vi − bi ) ≥ 0
Thus we know xi (vi ) is increasing with vi .
ui (bi ) = vi xi (bi ) − pi (bi )
Take derivative of the utility with respect to the bid bi , we get
u�i (bi ) = vi x�i (bi ) − p�i (bi )
IC suggests that whatever bidder i� s true value is, his utility should be maximized at that point, thus
p�i (z) = zx�i (z)
If the true value of bidder i is vi , we have
� vi
�
p�i (z)dz =
∀z
vi
zx�i (z)dz
0
0
� vi
pi (vi ) − pi (0) = [zxi (z)]v0i −
xi (z)dz
0
� vi
pi (vi ) = vi xi (vi ) −
xi (z)dz + pi (0)
0
IR suggests pi (0) = 0, thus we get pi (vi ) = vi xi (vi ) −
� vi
0
xi (z)dz.
Notice when the image of xi is {0, 1}, the above theorem states that for each bidder i there exists a threshold
ti such that the bidder gets the item iff his value is at least ti , and the price he pays when he gets the item
is ti .
10.2
Myerson’s mechanism (Optimal mechanism design)
For simplicity we will focus on deterministic mechanisms. Denote
�
Social Welfare (SW) =
vi xi (v)
i
Revenue (Rev) =
�
i
pi (v)
Lecture 10: Profit Maximization
10-3
It’s an ill-defined problem to maximize profit without any knowledge about users’ true utilities. Instead,
we maximize expected profit assuming the value of bidder i is drawn from known distribution with density
function fi , and the fi ’s are independent.
�
max E[
pi ]
=
�
i
v∈T
�
i
pi (v)f1 (v1 )f2 (v2 ) · · · fn (vn )dv
Example 1. Consider the auction with 2 bidders and 1 item. v1 ,v2 are independent, uniform over [0, 1].
Vickrey: E[
�
pi ] = E[min(v1 , v2 )] =
i
1
3
Vickrey( 12 ) (Vickrey with reserve price 12 ): run a normal Vickrey auction as if there is a third bidder who
bids 12 , and if it turns out that the third bidder wins, the auctioneer keeps the item.





E[p1 + p2 |v1 , v2 ≥ 12 ]Pr[v1 , v2 ≥ 12 ] = 23 14 = 16
E[p1 + p2 |v1 > 12 > v2 ∨ v2 > 12 > v2 ]Pr[v1 >
E[p1 + p2 |v1 , v2 < 12 ] = 0
Adding the three cases together we have E[p1 + p2 ] =
than Vickrey in terms of profit maximization.
5
12
1
2
> v2 ] =
11
22
> 13 . With a reserve price of
1
2,
> v2 ∨ v 2 >
1
2
=
1
4
we can do better
Consider the general situation with n bidders and 1 item
v1 ∼ f1 , . . . , vn ∼ fn (independent)
Rev(v) =
��
i
v−i
f−i (v−i )(
�
vi
fi (vi )pi (v)dvi )dv−i
SW takes a similar form except pi (v) is replaced by vi xi (vi ).
�
��
SW (v) =
f−i (v−i )( fi (vi )vi xi (vi )dvi )dv−i
i
v−i
vi
High level idea of Myerson’s mechanism: Reduce expected Rev maximization to expected SW maximization
in some ”virtual” space.
Virtual valuation: φi (vi )
Goal: Find φi (vi ) s.t. Rev(v) = SW (φ(v)) It suffices for the virtual valuations to satisfy
�
�
fi (vi )pi (v)dvi =
fi (vi )xi (v)φi (vi )dvi
∀i, v−i
vi
vi
From our Characteriaztion theorem (for deterministic mechanisms), we for know each bidder there is a
threshold zi (fixing v−i ). Then the above condition becomes
�
1
fi (vi )zi dvi =
zi
�
1
fi (vi )φi (vi )dvi
zi
∀i, v−i
10-4
Lecture 10: Profit Maximization
Take derivative w.r.t zi on both sides, we get
�
1
zi
fi (vi )dvi − zi fi (zi ) = −fi (zi )φi (zi )
φi (zi ) = zi −
�1
Virtual valuation:
φi (vi ) = vi −
zi
fi (vi )dvi
fi (zi )
1 − Fi (vi )
fi (vi )
Myerson’s Auction: Input f1 , . . . , fn , b1 , . . . , bn
1. Compute φ1 (b1 ), . . . , φn (bn )
2. Get the allocation and price function (x, p) from VCG with bids φ1 (b1 ), . . . , φn (bn )
3. Output (x� , p� ) ← (x, φ−1 (p))
Example 2. (Revisit) n bidders, vi ∼ u[0, 1] independent. VCG on φi (bi ) is just the second price auction,
except we don’t want to pay the winner any money. Thus we only consider bidders with virtual bids at least
1
i (vi )
0, i.e. vi − 1−F
fi (vi ) ≥ 0 ⇒ vi ≥ 2 . Essentially Myerson’s auction in this case is Vickrey with reserve price
1
2.
Corollary 1. When bidders are i.i.d according to any distribution, the optimal aunction is Vickrey (with
reserve price).
The reason is that φi ’s are all the same, and the φ in step 2 cancels with φ−1 in step 3.
Is Myerson’s auction truthful? We know that VCG is IC, and we want Myerson to be IC. What will
make that work is for φi to be increasing. φi is indeead increasing for regular distributions.
Even when φi is not increasing, we can further reduce it to another virtual space.
Ironing φi (not increasing) → φ̂i (increasing)
Some high-level intuition. Write the expected revenue with price z as
Ri (z) = z
�
1
fi (t)dt
z
Then it’s easy to check
φi (vi ) =
∂
[− ∂z
Ri (z)]z=vi
fi (vi )
i (z)
We can rewrite Ri () as a function of 1 − Fi (z), since ∂1−F
= −fi (z), when we take the derivative w.r.t z,
∂z
the chain rule will kill the fi (vi ) in the denominator. Thus
φi = Ri� when we write R as a function of Fi (z)
φi is increasing when Ri is convex, thus we can let R̂i be the convex hull of Ri , and φ̂i =R̂� .
Ironed SW ≥ Revenue. Equality holds if xi constant across ironed region.
Lecture 10: Profit Maximization
10-5
10.3
Aprroximation
10.3.1
Many items?
Don’t know optimal solution. For 1 bidder, n items with unit-demand, [Chawla, Hartline, Kleinberg EC07]
gives 2-approximation, and [Cai Daskalakis FOCS11] gives PTAS for some distributions.
10.3.2
Simple versus Optimal auctions
[Hartline, Roughgarden EC09] What if you run V ickreym (when the distributions of bidders’ values not
i.i.d)
mi = φ−1
i (0)
1. Reject all bidders with value vi < mi
2. Run Vickrey on remaining bidders.
Theorem 2. V ickreym is 2-approximation if users’ valuations are regular
10.3.3
Optimal auction with correlated valuations?
n bidders, (v1 , . . . , vn ) ∼ f , 1 item
[Ronen, Saberi FOCS02] gives
1
2
of optimal revenue for any n bidders
[Papadimitriou Pierrakos STOC11] Optimally for n = 2 bidders, NP-hard to get a PTAS for n ≥ 3
10.3.4
Prior-freeness?
Define benchmark for revenue [Hartline, Roughgarden STOC08]