Mechanism design with correlated distributions Michael Albert and Vincent Conitzer malbert@cs.duke.edu and conitzer@cs.duke.edu

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.

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!

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)

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)

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

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

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 = (12,000-10000)*.2 + (2200-200)*.8 = 2000

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

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 + (2200-2200)*.8 = 0

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 + (2200-2200)*.8 = 0 Expected Revenue = 4160

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!

Cremer-McLean Condition  

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?

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

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.

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

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