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Revenue Optimization in Multi- Dimensional Settings Constantinos Daskalakis EECS, MIT Reference:

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Revenue-optimization, so far ► ► Jason’s talk: Bayesian mechanism design: Auction design in presence of stochastic information about the bidders. Analysis tools: Bayes-Nash equilibrium. Single-dimensional setting: 1 type of item, many bidders, constraints on who can be given an item simultaneously. [Myerson’81]: Revenue-optimal auction in all single-dimensional settings. ► ► In fact, a reduction from auction- design to algorithm- design: Revenue optimization reduces to (virtual) welfare optimization. ► ► Tim’s talk: Single-dimensional settings are not only optimally solvable, but also amenable to simple approximations. “Exist constant-factor approximations which use little information about the bidders.”

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online marketplaces sponsored search online inventory allocation spectrum auctions Beyond Single-Dimensional Settings - heterogeneous items - complicated allocation constraints

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-Bidders have values on items and bundles of items. -Bidder’s valuation (or type) encodes that information. Multi-dimensional Auction Setting … 1 j n … 1 i m … … revenue/social welfare/other objective

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-Example 1: Additive bidder -described by vector of values (one value per item) -their valuation is: -Example 2: Single-minded combinatorial bidder -described by the subset of items they desire S, and their value v -their valuation is: For simplicity, throughout talk assume bidders are additive. Example Valuations

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-Bidders have values on items and bundles of items. -Bidder’s valuation (or type) encodes that information. -additive bidder: -Bidders’ types (t 1,…,t m ) come from known product distribution. Multi-dimensional Auction Setting … 1 j n … 1 i m … … revenue/social welfare/other objective universe of possible valuations for bidder i

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-Bidders have values on items and bundles of items. -Bidder’s valuation (or type) encodes that information. -additive bidder: -Bidders’ types (t 1,…,t m ) come from known product distribution. -Auctioneer will decide some allocation A [m] x [n], and charge prices. -There are (possibly combinatorial) constraints on what allocations are allowed. Multi-dimensional Auction Setting … 1 j n … 1 i m … … revenue/social welfare/other objective universe of possible valuations for bidder i

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-Items are paintings. -No painting should be given to more than one bidder -so: … 1 j n … 1 i m … … Example 1: selling paintings

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-Items are houses. -No house should be given to more than one bidder. -No bidder should receive more than one house. -so: Example 2: selling houses … 1 j n … 1 i m … …

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-Items are slots with doctors in a hospital. -No slot should be given to more than one bidder. -No bidder should get more than one slot with same doctor, or overlapping slots with different doctors. Example 3: selling doctor appointments … 1 j n … 1 i m … …

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-Items are possible locations for building a bridge L = {l 1, l 2, …,l n }. -If a location is given to one bidder, it is given to all bidders (as every bidder will use a bridge if it is built). -so: Example 4: building bridges … 1 i m …

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-Bidders have values on items and bundles of items. -Bidder’s valuation (or type) encodes that information. -Bidders’ types (t 1,…,t m ) come from known product distribution. -Auctioneer will decide some allocation A [m] x [n], and charge prices. -There are (possibly combinatorial) constraints on what allocations are allowed. -INPUT: m, n, T 1,…,T m,, and some access to. Multi-dimensional Auction Setting … 1 j n … 1 i m … … revenue/social welfare/other objective universe of possible valuations for bidder i

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Auction in Action Auctioneer: Each Bidder: -Uses as input: the auction specification, her own type, and her beliefs about the types of the other bidders; -Chooses how to play; -Goal: optimize her own utility (= value for allocated bundle – price charged). -Commits to an auction design, specifying (i) allowed bidder actions, (ii)the allocation and (iii)the price rule; -Asks bidders to play the auction; -Implements the allocation and price rule specified by the auction; -Goal: Optimize revenue/welfare. … 1 j n … 1 i m … … expected welfare: over bidders’ types t 1, …, t m, the randomness in the auction, and the bidders’ strategies in Bayes Nash equilibrium outcome in chosen by auction expected revenue: over bidders’ types t 1, …, t m, the randomness in the auction, and the bidders’ strategies in Bayes Nash equilibrium payment made by bidder i to the auctioneer

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background welfare vs revenue optimization in multi- dimensional settings

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Welfare-Optimization ► ► [Vickrey-Clarke-Groves]: Mechanism design for welfare-optimization is no harder than algorithm design for welfare-optimization. ► ► the VCG auction (as a reduction): bidders are asked to report their types: t 1, t 2,…, t m ; the mechanism chooses the allocation ; ► ► obtained via a call to a welfare optimization algorithm bidders are charged so that: reported types = true types. ► ► truthfulness-inducing payments can be computed via calls to a welfare optimization algorithm (e.g. Clarke pivot payments) ► ► Corollary: the only bottleneck to tractable welfare-optimizing mechanisms is whether there is a computationally efficient algorithm for the underlying welfare optimization problem. ► ► Approximation preserving mechanism-to-algorithm reduction ? yes! [Hartline-Lucier’10, Hartline-Kleinberg-Malekian’11, Bei-Huang’11] Spoiler Alert: Nicole will present these results.

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Revenue-Optimization ► ► [ Myerson ’81 ] : In all single-dimensional (i.e. single-item / single-item multi-unit) settings, mechanism design for revenue optimization reduces to algorithm design for welfare optimization. ► ► Myerson’s auction (as a reduction): bidders are asked to report their types ; reported types are transformed to virtual-types ; the virtual-welfare maximizing allocation is chosen; ► ► this is a call to a welfare optimization algorithm and prices are charged to make sure bidders report truthfully. ► ► truthfulness-inducing payments can be computed via calls to a welfare optimization algorithm ► ► Corollary: If the underlying welfare-maximization problem is tractable, then so is the revenue-optimal auction. ► ► Unanswered: Beyond single-item settings? Robustness to approximation; ► ► Myerson’s proof is “mystical;” result comes mysteriously out of algebra…

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Multi-Dimensional Auction Phenomena optimal mechanism: sell item for 1/2 expected revenue: 1/4 … … additive optimal mechanism ? idea 1: run n different auctions as above expected revenue: n/4 idea 2: offer the grand bundle of all the items at price expected revenue: moral of the story: bundling helps

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Multi-Dimensional Auction Phenomena (2) - optimal deterministic mechanism: post price of $5.097 on each item expected revenue: $ a better randomized mechanism additionally sell for $5.057 the lottery (1/2,1/2) moral of the story: randomization helps unit demand expected revenue: $5.06 [Thanassoulis’04]: [Briest-Chawla-Kleinberg-Weinberg’10]: gap may be arbitrarily high in general [Chawla-Malec-Sivan’10]: for independent values, unit-demand bidders, gap is at most a factor of 34.

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Multi-dimensional Mechanisms (known results prior to FOCS’12) ► Large body of work in Economics: [McAfee-McMillan’88], [Wilson’93], [Armstrong’96], [Rochet-Chone’98], [Armstrong’99],[Zheng’00], [Basov’01], [Kazumori’01], [Thanassoulis’04],[Vincent-Manelli’06], [Vincent-Manelli’07], … Progress sporadic. ► Recently (2007-today), algorithmic tools enabled progress. constant-factor approximations (limited settings) ► [Chawla-Hartline-Kleinberg ’07], [Chawla et al’10], [Bhattacharya et al’10], [Alaei’11], [Hart-Nisan ’12], [Kleinberg-Weinberg ’12], [Alaei et al’12] exact solutions (limited settings) ► [Cai-D ’11], [D-Weinberg ’12], [Cai-D-Weinberg ’12a], [Alaei et al ’12], [Cai-Huang’13] “limited settings”: ► additive bidders and no allocation constraints ► unit-demand bidders and matroid constraints on who can be served multi- to single- dimensional reduction multi- to single- bidder reductions

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Setting Previous Results OPT: [Cai-D-Weinberg STOC’12] APPX: [A,BGGM,CHMS,KW] OPT: OPEN APPX: OPEN OPT: OPEN APPX: OPEN OPT: OPEN paintings houses doctor appointments bridges (matroid) (matching) (downwards-closed)

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Challenge ► Revenue optimization in general multi-item settings. ideally: (i) unified solution for all settings (ii) Robustness of solutions to approximation, complexity.

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A General Solution ► A generic reduction: Mech. for Revenue Optimization Alg. for Welfare Optimization ► [Cai-D-Weinberg FOCS’12]: Suppose that: bidder types are independent; [the number of bidders m, items n, and the set-system of feasible allocations are unrestricted.] ► then the revenue-optimal auction can be computed with queries to a welfare-optimization algorithm A for. ► The optimal auction has the following form: bidders are asked to report their types; reported types are transformed into virtual types via bidder-specific functions; the virtual-welfare optimizing allocation in is chosen with a call to A; prices are charged to enforce truth-telling in Nash equilibrium. in Myerson’s theorem: virtual function = deterministic, closed-form here, randomized, computed during execution of Ellipsoid.

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A General Solution ► A generic reduction: Mech. for Revenue Optimization Alg. for Welfare Optimization ► [Cai-D-Weinberg FOCS’12]: Suppose that: bidder types are independent; [the number of bidders m, items n, and the set-system of feasible allocations are unrestricted.] ► then the revenue-optimal auction can be computed with queries to a welfare-optimization algorithm A for. ► The optimal auction is a virtual welfare maximizer. ► Corollary: If the underlying welfare-maximization problem for is tractable (ignoring incentives), then so is the revenue-optimal auction.

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Today’s menu General Auction Setting Challenge, General Result Background on Optimal MD Some Technical Ideas Combinatorial Optimization Viewpoint

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Specifying an auction (explicit but expensive) ► ► via the ex-post allocation & price rule: ► ► description size: ► ► feasibility trivial to check (in fact hardwired into A) ► ► Folklore: can write LP on A and P to find a revenue-optimal auction ► ► why? expected revenue is linear in P, truthfulness constraints linear in A, P ► ► trouble: exponentially many variables solving LP takes exponential time solution is bad: “laundry-list” auction : price that bidder i pays under type profile : probability distribution over feasible sets from which allocation is chosen when bidders’ types are

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► ► via the ex-post allocation probabilities: ► ► description size still prohibitive: ► ► and feasibility is hard to check now efficiently: 1. 1.is there a distribution over feasible allocations with these marginals, in view of the possibly combinatorial constraints on feasible allocations given by ? 2. 2.bright exception: for simple allocation constraints, e.g. matching constraints, checking feasibility is easy using the Birkhoff-von Neumann theorem. ► ► can’t write LP anymore and, even if I could, its solution would be useless. : marginal probability that item j is allocated to bidder i when bidders’ types are : price that bidder i pays when bidders’ types are Specifying an auction (lossy and still expensive)

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the reduced form of an auction ► ► a.k.a. the interim allocation & price rule : ► ► description size: ; ► ► c.f. description complexity of ex-post allocation rule ; ► ► feasibility hard to check: 1. 1.Can the per-bidder marginal probabilities be reconciled? 2. 2.…in a way that also respects the allocation constraints given by ? i.e. when can interim probabilities be converted to a feasible mechanism? ► ► wishful thinking: what if we could check feasibility efficiently? : marginal allocation probability of item j to bidder i when his type is t i (over the randomness in the other bidders’ types, and the randomness in mechanism) : expected price paid by bidder i when his type is t i (over the other bidders’ types, and the randomness in the mechanism)

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Variables: Constraints: Objective: - the expected revenue Mechanism Design with the Reduced Form Bayesian Nash Equilibrium: - Need: (i) Separation oracle for feasible reduced forms - (ii) Efficient map from feasible reduced form to mechanism (optimal feasible reduced form is useless in itself) the reduced form of sought auction expected value of bidder i of type for being given (additivity of bidders)

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feasibility of reduced forms

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Feasibility of Interim Rules (example) ► ► easy setting: single item, two bidders with types uniformly distributed in T 1 ={A, B} and T 2 ={C, D} respectively ► ► allocation constraints = item cannot be given to more than one bidder ► ► Question: Are the following interim allocation probabilities feasible? whenever types are A, C: A needs to get item whenever types are A, D: A needs to get item whenever types are B, C: C needs to get item whenever types are B, D: type A satiated type C satiated B needs to get item with prob. 0.4 and D needs to get item with prob. 0.8 so infeasible ! bidder 1 A B ½ ½ bidder 2 C D ½ ½

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Feasibility of Interim Rules - Single-item reduced forms: - [Border ’91, Border ’07, CKM ’11]: Necessary and sufficient conditions. - linear constraints OR separation oracle w/ runtime. - [Cai-D-Weinberg’12, Alaei et al ’12]: SO w/ runtime. - Any hope for multi-item reduced forms, w/ arbitrary allocation constraints ? - [Cai-D-Weinberg FOCS’12 ]: Given black-box access to max-welfare algorithm for can decide feasibility of reduced-forms efficiently. Moreover, given feasible reduced form can efficiently find a mechanism with this reduced form. - the combinatorial optimization perspective - geometric view:

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Claim 1: Claim 2: Given max-welfare algorithm for allocation constraints can find a separation oracle for (and vice versa). Claim 3: Every vertex of the polytope is the reduced form of a virtual welfare maximizing allocation rule. Feasibility of Interim Rules set of feasible interim rules MD version of Grötschel-Lovász-Schrijver equivalence of optimization and separation

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Characterizing the Vertices

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interpretation: virtual value derived by bidder i when given item j, if his type is A expected virtual welfare achieved by allocation rule with interim rule virtual welfare maximizing interim rule “in direction ”

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A: The VCG mechanism w/ virtual functions f 1,…, f m Characterizing the Vertices Q: OK understood what corner does, but what mechanism has this ? interpretation: virtual value derived by bidder i when given item j, if his type is A A virtual VCG allocation rule is defined by virtual functions, where, for all i. It takes as input a type-vector t 1, t 2, …, t m - transforms it into the virtual type-vector - then optimizes welfare using virtual types instead of true ones

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Characterizing the Vertices is a polytope whose corners are implementable by virtual VCG allocation rules. [CDW ’12]: The interim allocation rule of any feasible mechanism can be implemented as a distribution over virtual VCG allocation rules.

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Claim 1: Claim 2: Given max-welfare algorithm for allocation constraints can find a separation oracle for (and vice versa). Claim 3: Every vertex of the polytope is the reduced form of a virtual welfare maximizing allocation rule. Feasibility of Interim Rules set of feasible interim rules MD version of Grötschel-Lovász-Schrijver equivalence of optimization and separation We can optimize over the set of feasible reduced forms, using the ellipsoid algorithm. The optimal reduced form can be converted to a convex combination of max-welfare computations.

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► ► Connection to Grötschel-Lovász-Schrijver Wishful thinking: given algorithm for welfare optimization under allocation constraints, can get linear optimization algorithm for. Reality: Can’t do this efficiently; but can get additive-approx FPTAS. Trouble: GLS requires exact linear optimization, and its extensions multiplicative approximations. Solution: Specialized for MD. Our result provides an extension of GLS to additive approximation algorithms, exploiting the structure of. ► ► End product: FPRAS OPT-ε revenue, in time poly(1/ε). [This still happens at BNE and not ε-BNE] Reality Check

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► ► Unexpected properties of Myerson’s optimal single-item auction: the auction is deterministic (no internal coin-flips) ; while it is optimal among all auctions, it is itself a Dominant Strategy Truthful auction. ► ► Through our combinatorial optimization perspective we can easily show that for all single-dimensional settings; and for all objective functions that are linear in the interim rule (e.g. welfare, revenue, linear combinations thereof) the optimal auction is deterministic, DST, and a virtual welfare maximizer. ► ► Why? The optimal solution of the LP must be a corner. (No regularity assumption or ironing is needed for the argument.) De-mystifying Myerson

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Summary ► ► Welfare optimization is a well-understood problem: the VCG auction provides a reduction from mechanism to algorithm design ► ► The same is not true for revenue: Myerson’s auction optimizes revenue in single-dimensional settings; but multi-item settings are not well understood. ► ► I showed a general result providing the natural generalization of Myerson’s theorem to all multi-item settings. “The revenue optimal auction is a virtual-welfare maximizer; which can be computed with polynomially many queries to a welfare-maximizing algorithm.” ► ► Techniques: geometry, ellipsoid algorithm; can optimize over reduced forms using welfare algorithm as a separation oracle.

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Further Results/Open Problems ► ► Approximation Preserving reduction? Given a-approximation for welfare, can obtain a-approximation for revenue? Yes [Cai-D-Weinberg SODA’13]; Complication: Can’t get separation oracle for polytope of reduced forms from approximation algorithm for welfare. ► ► Implicit type-distributions? our running-time is polynomial in ; but suppose type-distribution is given implicitly ► ► e.g. consider additive bidder whose value for item j is uniform in {a j, b j } independently of other items Q: can running time be improved for such implicit disn’s? [D-Deckelbaum-Tzamos]: Problem is #P-hard even when there is a single additive bidder whose values are independent, rational, of support two. But FPTAS still not precluded. size of support of bidder i’s type-distribution

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Further Results/Open Problems ► ► Nature of Virtual Functions both Myerson’s auction and its generalization are virtual welfare maximizers; Myerson provided a closed form formula for each bidder’s virtual function; in the generalized auction, the virtual functions are computed by an algorithm; any structure for simple multi-item settings? ► ► Value Oracle Model? in our work valuations in support are described explicitly; what if valuations are implicit? ► ► i.e. given S, a circuit outputs value of bidder for subset S of items Problem is NP-hard in general [Dobzinski-Fu-Kleinberg’11] but what if valuations have structure? Thanks for listening

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