On Optimal Multi-dimensional Mechanism Design

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 1 … … j i … n … m

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

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

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

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 1 … … j i … n … 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 aremulti-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 constant #items constant #bidders 1 1 1 1 … … 2 j i 2 … … 3 3 m n (i.i.d) Arbitrary joint distribution possibly different per bidder

Multi-dimensional Optimal MD NOTES: Can do arbitrary budget, demand constraints. Also explicitly price bundles of items. Solution concept: Bayesian Incentive Compatibility (BIC), or ε-Incentive Compatibility (IC). 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. • [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.

a glimpse of the techniques

giant Optimal MD it’s just an LP… Suppose is discrete (deal with continuous distributions later). Let be the joint distribution of all bidders’ values for the items (supported on a subset of ): specifies all vij’s. For every point in the support of , mechanism needs to determine a (randomized) allocation of items to bidders and the charged prices. both are exponential 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 [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 . c.f. Nash’s symmetry theorem: “In every game there exists a Nash equilibrium that simultaneously satisfies all symmetries satisfied by the game.” i.e. NOTES: Above symmetry does not hold for deterministic mechanisms. 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 X1,…,Xn be independent (but not necessarily identically distributed) MHR random variables. Then there exists anchoring pointβ such that: X2 Corollary: (1-ε)OPT is extracted from values in (εβ, 1/εlog1/ε β). Xn X3 X1 • contribution to E[max Xi] from values here is ≤ ε β • Pr[max Xi ≥ β] = Ω(1)

Beyond symmetries?

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

Multidimensional Pricing • If the vi’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, efficientpricing algorithms for a single unit-demand bidder whose values are independent (but not necessarily identically distributed). NOTES: Nearly optimal: OPT- ε, for any desired ε>0, when support of distributions is normalized in [0,1]; or (1- ε)OPT if distn’sare MHR or regular. 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 Fior n . For all , if , then a single price suffices to get revenue, where is an increasing function that does not depend on Fior n . Theorem[Cai-D ’10] • A Single Price Suffices for i.i.d. Theorem[Cai-D ’10]

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

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.

Thank you for your attention

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 replace with where ? A: may be catastrophic • what happens to the objective if I restrict the prices to the support of the value distributions?

Multidimensional Pricing Our approach: … C Revenue Distribution …

Geometric Approach “Implicit”: it is output by an algorithm, given {Fi}i implicit polynomial-size - cover of this space distance function: 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 phas - the same strategy set: S = {1,…, s} - the same payoff function: u = u (σ ; n1, n2,…,ns) Description size: s ns-1 strategy of p (instead of: nsn) number of the other players choosing each strategy in S E.g. : - prisoner’s dilemma - 1000 drivers from Pasadena to Westwood Nash ’50: Always exists an equilibrium in which every player uses the same randomization. Does this make computation easier?

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

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

Multi-player symmetric games If n is large, s is small, a symmetric equilibriumx = (x1, x2, …, xs) can be found as follows: - guess the support of x, 2spossibilities can be solved approximately in time ns log(1/ε) - 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 (recall description complexity is s ns-1)

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

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

On Optimal Multi-dimensional Mechanism Design