1. Mechanism design with correlated distributions
Michael Albert and Vincent Conitzer
and

2. Impossibility results from mechanism design with independent valuations
Myerson auction is revenue optimal for independent valuations
This is an impossibility result in disguise!
Myerson auction doesn’t always allocate the item, and it doesn’t always charge the bidders valuation
Bidder’s virtual valuation ψ(vi)= vi - (1 - Fi(vi))/fi(vi)
The bidder with the highest virtual valuation (according to his reported valuation) wins (unless all virtual valuations are below 0, in which case nobody wins)
Winner pays value of lowest bid that would have made him win
Combined with the revenue equivalence theorem, we have an impossibility result.
The impossibility result is: we can’t efficiently allocate an item and maximize revenue at the same time.
More than that, we have to give some of the utility to the bidders because they have private information.

3. Why should we care about maximizing revenue?
Auctions are one of the fundamental tools of the modern economy
In 2012 four government agencies purchased $800 million through reverse auctions (Government Office of Accountability 2013)
In 2014, NASA awarded contracts to Boeing and Space-X worth $4.2 billion and $2.6 billion through an auction process (NASA 2014)
In 2014, $10 billion of ad revenue was generated through auctions (IAB 2015)
The FCC spectrum auction, currently in the final round, expects to allocate between $60 and $80 billion worth of broadcast spectrum
It is important that the mechanisms we use are revenue optimal!

4. Do current techniques get us “close enough”?
Standard simple mechanisms do very well with large numbers of bidders
VCG mechanism revenue with n+1 bidders ≥ optimal revenue mechanism with n bidders, for IID bidders (Bulow and Klemperer 1996)
For “thin” markets, must use knowledge of the distribution of bidders
We use the distribution to set the reserve price for a Myerson auction
Thin markets are a large concern
Sponsored search auctions with rare keywords or ad quality ratings
Of 19,688 reverse auctions by four governmental organizations in 2012, one third had only a single bidder (GOA 2013)

5. What if Types are Correlated?
This result is for all possible distributions over bidder valuations
Specifically, the impossibility of efficient allocation and revenue maximization must encompass the case where the agents types are independent.
This is unlikely to hold in many situations
Oil drilling rights
Sponsored search auctions
Anything with resale value
Anything with a common value component (like similar inputs)
Under correlation, we can break this impossibility result
Cremer and McLean (1985, 1988), Albert, Conitzer, Lopomo (2016)

6. Example: Divorce arbitration
Outcomes:
Each agent is of high type w.p. .2 and low type w.p. .8
Preferences of high type:
u(get the painting) = 11,000
u(museum) = 6,000
u(other gets the painting) = 1,000
u(burn) = 0
Preferences of low type:
u(get the painting) = 1,200
u(museum) = 1,100
Distribution under independent valuations H L H
.2*.2 = .04
.8*.2 = .16 L
.2*.8 = .16
.8*.8 = .64

7. Perfectly Correlated Distribution
high
low
high .2
low .8
Maximum Social Welfare = 12,000*.2 + 2,200*.8 = 4,160

8. Clarke (VCG) mechanism
high
low
high
Both pay 5,000
Husband pays 200
low
Wife pays 200
Both pay 100
Expected sum of divorcees’ utilities = (12, )*.2 + ( )*.8 = 2000

9. Mechanism with Perfect Correlation
high
low
high
Both pay nothing
Both pay nothing
low
Both pay nothing
Both pay nothing
Expected sum of divorcees’ utilities = (12,000)*.2 + (2200)*.8 = 4,160

10. Maximum Revenue with Perfect Correlation
high
low
high
Both pay $6000
Both pay nothing
low
Both pay nothing
Both pay $1100
Expected Revenue = 4160
Expected sum of divorcees’ utilities = (12,000 – 12,000)*.2 + ( )*.8 = 0

11. Clarke (VCG) mechanism + side payments
high
low
high
Husband pays 200
Both pay 5,000
& husband pays 1,100,
Wife pays 1,000
& both pay 1,000
low
Wife pays 200
Both pay 100
& husband pays 1,000,
Wife pays 1,100
& both pay 1,100
Expected sum of divorcees’ utilities = (12,000 – 12,000)*.2 + ( )*.8 = 0
Expected Revenue = 4160

12. How much correlation do we need to maximize revenue?
Need to look at ex-interim individually rational (IR) mechanisms:
Σθ-i π(θ-i| θi) [vi(θi, o(θ1, θ2, …, θi, …, θn)) - xi(θ1, θ2, …, θi, …, θn)] ≥ 0
For now we will use dominant strategy (ex-post) incentive compatible:
vi(θi, o(θ1, θ2, …, θi, …, θn)) - xi(θ1, θ2, …, θi, …, θn) ≥
vi(θi, o(θ1, θ2, …, θi’, …, θn)) - xi(θ1, θ2, …, θi’, …, θn)
Nearly any correlation will do! In fact, for bidders with two types each, any correlation at all will do!

13. Cremer-McLean Condition

20. Can we do better than Cremer-McLean?
The Cremer-McLean condition is sufficient, but not necessary
While the condition is generic for two (or more) bidders with the same number of types, is this always going to be the case?
What if we really have an external signal that we are using to condition payments, so that there is only one bidder?
Ad auctions with click through rates of related ads
Prices of commodities that are used as part of the production process
What if we don’t know the distribution well?
Maybe we want to “bin” the other bidders bids in order to estimate a smaller distribution
What is both necessary and sufficient?

26. Necessary and Sufficient Condition for Ex-Interim IR and Dominant Strategy IC

27. Why restrict ourselves to Dominant Strategy IC?
While dominant strategy IC is sufficient to give us a generic condition when there are sufficient bidders, we’ve already seen that is not necessarily the case.
Can we relax the necessary conditions if we consider BNE incentive compatibility?
Σθ-i π(θ-i| θi) [vi(θi, o(θ1, θ2, …, θi, …, θn)) - xi(θ1, θ2, …, θi, …, θn)] ≥
Σθ-i π(θ-i| θi) [vi(θi, o(θ1, θ2, …, θi’, …, θn)) - xi(θ1, θ2, …, θi’, …, θn)]
This gives us the ability to have multiple lotteries over the external signal.

37. Necessary and Sufficient Condition for Ex-Interim IR and BNE IC

38. Impossibility results from mechanism design with independent valuations
Myerson-Satterthwaite Impossibility Theorem [1983]:
We would like a mechanism that:
is efficient,
is budget-balanced (all the money stays in the system),
is BNE incentive compatible, and
is ex-interim individually rational
This is impossible! v(
) = x v(
) = y