# On Optimal Multi-dimensional Mechanism Design Constantinos Daskalakis EECS, MIT Joint work with Yang Cai, Matt Weinberg.

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On Optimal Multi-dimensional Mechanism Design Constantinos Daskalakis EECS, MIT costis@mit.edu Joint work with Yang Cai, Matt Weinberg

Multi-Item Multi-Bidder Auction … 1 j n … 1 i m … …

Revenue Maximization? Additional Constraints: Demands, Budgets if the v ij ’s known exactly, can compute optimal allocation and extract full surplus. can go around computational hardness with randomization. … 1 j n … 1 i m … …

Multi-Item Multi-Bidder Auction if the v ij ’s are unknown, then all bets are off... Revenue Maximization? Additional Constraints: Demands, Budgets can go around computational hardness with randomization Natural approach: online optimization [wait till 10.30] … 1 j n … 1 i m … …

Multi-Item Multi-Bidder Auction Revenue Maximization? Additional Constraints: Demands, Budgets can go around computational hardness with randomization Suppose Bayesian information is known for bidders’ values Optimal Mechanism Design … 1 j n … 1 i m … …

Single-Parameter Optimal Mechanisms ► [Myerson ’81]: If setting is single-parameter and bidders have independent values, there exist closed-form revenue-optimal mechanisms.  single-parameter: each bidder has a fixed value for winning an item, no matter what item she gets; that value is drawn from known distribution;  in our running example: if bidder wins painting, her value for the painting is uniform in [\$10,\$100], no matter what painting she gets…

Multiple Copies of Same Item … 1 j n … 1 i m … … i.e. one (unknown) parameter determines each bidder’s values

Single-Parameter Optimal Mechanisms ► ► [Myerson ’81]: If setting is single-parameter and bidders have independent values, there are closed-form revenue-optimal mechanisms.   single-parameter: each bidder has a fixed value for winning an item, no matter what item she gets; that value is drawn from known distn’   In our running example: if bidder wins painting, her value for the painting is uniform in [\$10,\$100], no matter what painting she gets   closed-form: revenue optimization reduces to VCG ► ► Closed-form ≠ Computationally Efficient

Multi-dimensional Optimal MD ► Central Open Problem: Are there closed-form, efficient revenue-optimal mechanisms, when the bidders are multi- dimensional, i.e. have different values for different allocations?  large body of work in Economics for restricted settings; see, e.g., survey paper by [Vincent-Manelli ’07]  Constant-factor approximations are known: ► multi-item auctions, and certain matroidal settings; poly-time for regular distn’ [Chawla, Hartline, Malec, Sivan ’10] ► multi-item auctions + budget constraints [Bhattacharya, Goel, Gollapudi, Munagala ’10]

Multi-dimensional Optimal MD ► [D-Weinberg ’11]: Closed-form, efficiently computable, nearly optimal revenue mechanisms when the number of items or the number of bidders is a constant.  In the first case, we allow the values of each bidder for the items to be arbitrarily correlated, but assume that bidders are i.i.d.  In the second case, we require each bidder to have i.i.d. values for the items, but allow different bidders to have different distributions.

Multi-dimensional Optimal MD Arbitrary joint distribution 1 j n 1 2 3 … … (i.i.d) possibly different per bidder 1 2 3 … 1 i m … constant #itemsconstant #bidders

Multi-dimensional Optimal MD ► [D-Weinberg ’11]: Closed-form, efficiently computable, nearly optimal revenue mechanisms when the number of items or the number of bidders is a constant.  In the first case, we allow the values of each bidder for the items to be arbitrarily correlated, but assume bidders are i.i.d.  In the second case, we require each bidder to have i.i.d. values for all items, but allow different bidders to have different distributions. NOTES: a.Can do arbitrary budget, demand constraints. Also explicitly price bundles of items. b.Solution concept: Bayesian Incentive Compatibility (BIC), or ε-Incentive Compatibility (IC). c.Nearly optimal: OPT- ε, for any desired ε >0, when support of distributions is normalized to [0,1]; or (1- ε)OPT if distn’s MHR.

a glimpse of the techniques

Optimal MD it’s just an LP… Let be the joint distribution of all bidders’ values for the items (supported on a subset of ): specifies all v ij ’s. Suppose is discrete (deal with continuous distributions later). For every point in the support of, mechanism needs to determine a (randomized) allocation of items to bidders and the charged prices. giant Write down an LP that looks for such (allocation distribution, price list) pair for every point in, and enforces incentive compatibility, individual rationality, budget, demand constraints, etc. LP lives in dimensions both are exponential [Birkhoff–von Neumann theorem]: Sufficient to compute the marginals of the allocation distribution for each valuation vector. Improved LP lives in dimensions

Ingredient 1: The Role of Symmetries Theorem: Let be the distribution of bidders’ values for the items (supported on a subset of ). Let also be a set of permutations such that: Then there exists an optimal randomized mechanism M, respecting all the symmetries in. i.e. c.f. Nash’s symmetry theorem: “In every game there exists a Nash equilibrium that simultaneously satisfies all symmetries satisfied by the game.” NOTES: a.Above symmetry does not hold for deterministic mechanisms. b.Certifies existence of a mechanism of small description complexity, if setting is sufficiently symmetric. However, not clear how to modify LP to locate the succinct mechanism…

Ingredient 2: Monotonicity Theorem: If mechanism M is truthful and item-symmetric, then it satisfies a natural monotonicity property. Theorem: Monotonicity + Symmetry => succinct LP, if enough symmetry.

Ingredient 3: Continuous to Discrete Continuous distributions -> discretize to get finite support Previous Approach results in ε-BIC Use mechanism computed by LP as a back-end, and design a VCG front-end whereby continuous bidders “buy” discretized representatives. Transfer approximation from truthfulness to revenue. (cf recent surrogate-replica construction of [Hartline-Kleinberg- Malekian ’11])

Ingredient 4: Extreme Value Theorems So far, OPT – ε, when distributions normalized to [0,1]. From additive ε-approx. to multiplicative (1-ε)-approx? Theorem [Cai-D ’11]: Let X 1,…,X n be independent (but not necessarily identically distributed) MHR random variables. Pr[max X i ≥ β] = Ω(1) contribution to E[max X i ] from values here is ≤ ε β Then there exists anchoring point β such that: X1X1 X2X2 X3X3 XnXn Corollary: (1-ε)OPT is extracted from values in (ε β, 1/ε log1/ε β).

Beyond symmetries?

Single Unit-Demand Bidder … If the v i ’s are i.i.d., already know how to find optimal mechanism.

Multidimensional Pricing … If the v i ’s are independent (but not necessarily i.i.d.), can we at least find optimal prices? [CHK’07]

Optimal Multi-dimensional Pricing ► [Cai-D ’11]: Nearly-optimal, efficient pricing algorithms for a single unit-demand bidder whose values are independent (but not necessarily identically distributed). NOTES: a.Nearly optimal: OPT- ε, for any desired ε >0, when support of distributions is normalized in [0,1]; or (1- ε)OPT if distn’s are MHR or regular. b.Efficient = PTAS, and quasi-PTAS for regular.

Techniques ► Symmetry lemma does not apply to prices. ► Not enough symmetry, to find optimal randomized mechanism with previous approach. ► Our approach:  Search space: price vectors.  Instead focus attention on revenue distributions induced by price vectors.  Identify appropriate distance measure in this space, so that closeness reflects closeness in revenue.   implicit polynomial-size cover of this space.

Structural Results (e.g. MHR) A Constant Number of Prices Suffices For all, distinct prices suffice to get revenue, where is an increasing function that does not depend on F i or n. Theorem[Cai-D ’10] A Single Price Suffices for i.i.d. Theorem[Cai-D ’10] For all, if, then a single price suffices to get revenue, where is an increasing function that does not depend on F i or n.

Summary 1 2 3 … 1 i m … Arbitrary joint distribution constant #items 1 j n 1 2 3 … … (i.i.d) possibly different per bidder constant #bidders 1 j n 1 … … single unit-demand bidder, pricing

Open Problems Complexity of the exact problem. Conjecture: #P-hard to find optimal mechanism even for single bidder with independent values for the items. NOTE: If values are correlated, it is already known that the problem is highly-inapproximatble. [Briest ’08] Beyond item/bidder symmetric settings Combine LP approach with probabilistic covering theorems.

Multidimensional Pricing Our approach: … … C Revenue Distribution

Multidimensional Pricing Smoothness properties that would be useful: - what happens to the objective if I replace with where ? - what happens to the objective if I restrict the prices to the support of the value distributions? A: may be catastrophic

Multidimensional Pricing Our approach: … … C Revenue Distribution

Geometric Approach distance function:  implicit polynomial-size - cover of this space “Implicit”: it is output by an algorithm, given {F i } i TV

Geometric Approach distance function:

Multidimensional Pricing ► [Chawla,Hartline,Kleinberg ’07]: poly-time constant factor approxi-mations for regular distributions; even the i.i.d. case is not easier. ► [Cai-D ’10]: PTAS, for independent MHR distributions, or when support is balanced …

Multi-dimensional Optimal MD ► [Cai-D ’10]: Closed-form, efficiently computable, (1-ε)- optimal revenue mechanism for the SINGLE bidder case.

Symmetries in Nash’s paper Symmetric Games:Suppose each player p has - the same strategy set: S = {1,…, s} - the same payoff function: u = u (σ ; n 1, n 2,…,n s ) number of the other players choosing each strategy in S strategy of p E.g. : - 1000 drivers from Pasadena to Westwood Nash ’50: Always exists an equilibrium in which every player uses the same randomization. - prisoner’s dilemma Description size: s n s-1 (instead of: n s n ) Does this make computation easier?

Symmetrization R, C R T, C T C, R x y x y xy Symmetric Equilibrium Equilibrium 0, 0 Any Equilibrium Equilibrium In fact […] [Gale-Kuhn- Tucker 1950]

R, C R T,C T C, R x y x y xy Symmetric Equilibrium Equilibrium 0,0 Any Equilibrium Equilibrium In fact […] Hence, PPAD to solve symmetric 2-player games Open: - Reduction from 3-player games to symmetric 3-player games - Complexity of symmetric 3- player games Symmetrization

Multi-player symmetric games If n is large, s is small, a symmetric equilibrium x = (x 1, x 2, …, x s ) can be found as follows: - guess the support of x, 2 s possibilities - write down a set of polynomial equations an inequalities corresponding to the equilibrium conditions, for the guessed support - polynomial equations and inequalities of degree n in s variables can be solved approximately in time n s log(1/ε) (recall description complexity is s n s-1 )

Multidimensional Pricing … revenue? If v i ’s unknown can we set prices to optimize revenue ? (if v i ’s known trivial to optimize)

Multidimensional Pricing … expected revenue: If v i ’s unknown can we set prices to optimize revenue ?

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