Truthful and near-optimal mechanism design via linear programming Chaitanya Swamy Caltech and U. Waterloo Joint work with Ron Lavi Caltech

Mechanism Design Environment with “selfish” players who hold the inputs “selfish”  players only care about maximizing their utility will “lie” if that increases their utility  Cannot assume algorithm knows the true inputs. Goal: want that algorithms “work” with respect to the true input Mechanism = algorithm + prices to affect players’ utilities and motivate them to reveal the true inputs

Combinatorial Auctions (CAs) m indivisible non-identical items for sale n players that compete for subsets of these items Player i has a valuation v i (S) for each subset S of items –v i (  ) = 0 –v i (.) is non-decreasing: v i (S) ≤ v i (T) for S  T items { 1,…, m} indivisible item players

Combinatorial Auctions (CAs) m indivisible non-identical items for sale n players that compete for subsets of these items Player i has a valuation v i (S) for each subset S of items –v i (  ) = 0 –v i (.) is non-decreasing: v i (S) ≤ v i (T) for S  T Goal: Compute an allocation (S 1,…, S n ) of items to players, where S i  S j = , that maximizes social welfare: ∑ i v i (S i ). Called Social-Welfare Maximization (SWM) problem items { 1,…, m} indivisible item players

SWM for CAs: Main Issues Complexity issues –Communication: v i (.)s have exponential length. Solving SWM exactly may need exponential communication, polynomial communication forces approximation –Computation: SWM problem is NP-hard even when each v i (.) has small length, e.g., v i (S) = v i for S  T i for some set T i, 0 o/w.

SWM for CAs: Main Issues Complexity issues –Communication: Exact solution  exponential communication –Computation: SWM problem is NP-hard even with “short” v i (.). Strategic issues Players are strategic: may report valuation b i (.) different from true input v i (.) Approach problem via Mechanism Design. Mechanism M = (f, {p i } i= 1,…,n ) specifies allocation, f: (v 1,…, v n )  (S 1,…, S n ) that maximizes ∑ i v i (S i ). p i specifies price charged to player i, p i : (v 1,…, v n )  

SWM for CAs: Main Issues Complexity issues –Communication: Exact solution  exponential communication –Computation: SWM problem is NP-hard even with “short” v i (.). Strategic issues Mechanism M = (f, {p i } i= 1,…,n ) Want: compute f(.) on true input v = (v 1,…, v n ); v i not publicly known Mechanism gets “bids” b=(b 1,…, b n ) and computes Allocation: f(b) = (S 1,…, S n ) Price: p i (b) to charge to each player i Player i’s goal: maximize its utility(v i ; (b i, b -i )) = v i (S i ) – p i (b i, b -i ) Truthful mechanism: for any i, for any true input v i, i maximizes its utility by saying v i regardless of others’ bids b -i : utility(v i ; (v i, b -i )) ≥ utility(v i ; (b i, b -i )) for every b i

Yes. VCG (Vickrey-Clarke-Groves) prices ensure truthfulness. Let f(b) = (S 1,…, S n ) (allocation maximizing ∑ i b i (S i )) set p i (b) = – ∑ j≠ i b j (S j ) + h i (b -i ) VCG mechanisms Basic question: Given SWM function f; Are there prices {p i } such that (f, {p i }) is a truthful mechanism? That is, are there prices that truthfully implement f? Ignore this term

Yes. VCG (Vickrey-Clarke-Groves) prices ensure truthfulness. Let f(b) = (S 1,…, S n ) (allocation maximizing ∑ i b i (S i )) set p i (b) = – ∑ j≠ i b j (S j ) Proof of truthfulness: fix i, true valuation v i, others’ bids b -i. Let f(v i, b -i ) = (T 1,…, T n )  utility(v i ; (v i, b -i )) = v i (T i ) + ∑ j≠ i b j (T j ) = max {v i (U i ) + ∑ j≠ i b j (U j ) : allocations (U 1,…, U n )}  utility(v i ; (b i, b -i )) is maximized at b i = v i. VCG mechanisms Basic question: Given SWM function f; Are there prices {p i } such that (f, {p i }) is a truthful mechanism? That is, are there prices that truthfully implement f?

VCG (contd.) VCG mechanisms/payments: classic positive result for truthful implementation. ensure truthfulness in a more general setting. any function f: V 1 x … x V n  A with f(b) = argmax a  A ∑ i b i (a) A = set of outcomes, for CAs A = {allocations (S 1,…, S n ): S i  S j =  } V i = {valid valuations v i : A  }, for CAs V i = {v i (.): v i depends only on S i, v i (  ) = 0, v i is monotone} VCG prices are p i (b) = – ∑ j≠ i b j (a) + h i (b -i ), where a = f(b) Only general tool for truthful mechanism design. (SWM function)

Complexity vs. Strategic Issues VCG prices truthful mechanism Strategic players exact algorithm for SWM, f(.) VCG requires computing f(v), i.e., solving SWM problem: this is NP-hard! Want computationally efficient, truthful mechanisms –VCG mechanism is not (computationally) efficient

Complexity vs. Strategic Issues approx. algorithm for SWM, f(.) VCG prices truthful mechanism Strategic players ? VCG requires computing f(v), i.e., solving SWM problem: this is NP-hard! Want computationally efficient, truthful mechanisms –VCG mechanism is not (computationally) efficient Hope: can use approximation algorithm for f(.) to get truthful, approximation mechanism Cannot just “plug in” approx. algorithm and use VCG prices –VCG won’t work with any sub-optimal “reasonable” algorithm [NR00]

Complexity vs. Strategic Issues VCG requires computing f(v), i.e., solving SWM problem: this is NP-hard! Want computationally efficient, truthful mechanisms –VCG mechanism is not (computationally) efficient Hope: can use approximation algorithm for f(.) to get truthful, approximation mechanism Cannot just “plug in” approx. algorithm and use VCG prices –VCG won’t work with any sub-optimal “reasonable” algorithm [NR00] More bad news: for many approx. algorithms/techniques, there are no prices that will ensure truthfulness. approx. algorithm for SWM, f(.) truthful mechanism Strategic players ? other price scheme

What is known for CAs? Multi-unit CA (MUCA) General B, arbitrary v i (.): nothing is known ( v i (  ) =0, v i (.) monotone “multi-parameter” domain ) m items, n players B copies of each item, player wants at most 1 copy (CA: B= 1 )

What is known for CAs? Multi-unit CA (MUCA) General B, arbitrary v i (.):nothing is known Special cases –B ≥ 3: Bartal-Gonen-Nisan give O(B. m ( 1 /(B-2)) )-approx. mechanism m items, n players B copies of each item, player wants at most 1 copy (CA: B= 1 ) Bounds in non-strategic setting (no truthfulness requirement) O(m ( 1 /(B+ 1 )) )-approx. ratio, cannot do better unless P = NP B= 1 (CA): O(m 1 /2 )-approx.B=Ω(log m): ( 1 +  )-approx. Large B: big gap between bounds with/without truthfulness.

What is known for CAs? Multi-unit CA (MUCA) General B, arbitrary v i (.):nothing is known Special cases –B ≥ 3: Bartal-Gonen-Nisan give O(B. m ( 1 /(B-2)) )-approx. mechanism –Single-minded players ( i wants T i : v i (S) = v i for S  T i, and 0 o/w ) : B = 1 – Lehman-O’Callaghan-Shoham give O(m 1 /2 )-approx. B = Ω(log m) – Archer-Papadimitriou-Talwar-Tardos give ( 1 +  )-approx. mechanism for “known” single-minded case B ≥ 2 – Briest-Krysta-Vocking give O(m 1 /B )-approx. mechanism –Subadditive valuations: Dobzinski-Nisan-Schapira give O(m 1 /2 )- approx. mechanism for CA (B= 1 ) m items, n players B copies of each item, player wants at most 1 copy (CA: B= 1 ) Results obtained by cleverly tailoring approx. algorithm or technique to ensure that prices yielding truthfulness exist.

Our Results Give the first truthful, approximation mechanisms for many multi-parameter problems –O(m 1 /2 )-approx. for CA –O(m 1 /(B+ 1 ) )-approx. for MUCA for all B  ( 1 +  )-approx. when B = Ω(log m) –2-approx. for multi-parameter knapsack auctions For CA, MUCA, ratios match approximation bounds when truthfulness is not required Exponential valuations: get iterative mechanisms; same approximation ratios, strategic concept is truthfulness as ex-post Nash equilibirium. for “short” general valuations

Our Results (contd.) General technique to “export” approx. algorithm for packing SWM problem into a randomized, truthful-in- expectation, approximation mechanism c-approx. algorithm i) LP-based, ii) proves an integrality gap of c c-approximation, truthful-in-expectation mechanism VCG + … + randomization –simple, quite general: get (versatility of VCG as MD tool) + (tractability of approx. algorithms) –no need to “tailor” the algorithm: can decouple algorithmic and strategic parts of mechanism design

Truthfulness in expectation Deterministic mechanism M = (f, {p i }), f: (v 1,…,v n )  outcome a  A, p i : (v 1,…v n )  Truthfulness: for any i, any true v i, any bids b -i of others, utility(v i ; (b i, b -i )) = v i (f(b i, b -i )) – p i (b i, b -i ) is maximized at b i = v i. Randomized mechanism M = (f, {p i }) outcome f(v), price p i (v) are random variables Truthfulness-in-expectation [Archer-Tardos’0 1 ]: for any i, any true v i, any b -i, E[utility(v i, (b i, b -i ))] is maximized at b i = v i.

An Integer Program for CA v i (.): valuation of player i,v i (S) : i’s value for set S x i,S : indicates if i gets set S Maximize ∑ i,S v i (S)x i,S subject to ∑ S x i,S ≤ 1 for each player i ∑ i ∑ S:j  S x i,S ≤ 1 for each item j x i,S  {0, 1 }for each i, S.

A Linear Program for CA v i (.): valuation of player i,v i (S) : i’s value for set S x i,S : indicates if i gets set S Maximize ∑ i,S v i (S)x i,S (CA-P) subject to ∑ S x i,S ≤ 1 for each player i ∑ i ∑ S:j  S x i,S ≤ 1 for each item j x i,S  {0, 1 }for each i, S. x i,S ≥ 0for each i, S. “Short” valuations  (CA-P) is a poly-size linear program  (CA-P) can be solved in polynomial time.

Construction Overview Let P =feasible region of (CA-P) {y k } k  I =all integer points in P, i.e., y k  valid allocation (CA-P) is an SWM problem: outcome set A = points in P, player i’s value for x  P is v i (x) = ∑ S v i (S)x i,S  v i. x ( v i = {v i (S)} S ) (CA-P) can be efficiently solved  can use VCG 1 ) Use VCG to get a truthful, fractional mechanism M F = (g, {q i }) g(v 1,…, v n )= x * (v) = optimal solution to (CA-P) for (v 1,…, v n ) {q i (v)} = VCG prices.

Construction Overview Given: LP-based c-approximation algorithm A : maps any v=(v 1,…, v n ) to integer soln. y k s.t. ∑ i v i. y k ≥ ( ∑ i v i. x * (v) )/ c. Decomposition Lemma [based on Carr-Vempala02]: Given any x  P, can use A to compute in poly-time { k } k  I values s.t. a) ∑ k  I k y k = x/c,b) k ≥ 0, ∑ k  I k = 1 Exploits packing property: y k integer solution  y k with some components set to 0 is also an integer solution P  v,  y k s.t. ∑ i v i. y k ≥ ∑ i ( v i. x * (v) ) /c

Construction Overview Given: LP-based c-approximation algorithm A : maps any v=(v 1,…, v n ) to integer soln. y k s.t. ∑ i v i. y k ≥ ( ∑ i v i. x * (v) )/ c. Decomposition Lemma [based on Carr-Vempala02]: Given any x  P, can use A to compute in poly-time { k } k  I values s.t. a) ∑ k  I k y k = x/c,b) k ≥ 0, ∑ k  I k = 1 2) Using lemma express x * (v)/c = ∑ k  I k y k. { k } values give a distribution over integer solutions. Gives randomized mechanism M R = (h, {r i }) where h(v) = y k with probability k (random) price r i (v) is set so that E[r i (v)] = q i (v)/c.

Construction Overview VCG mechanism M F = (g, {q i });g(v) = x * (v), {q i (v)} = VCG prices Randomized mechanism M R = (h, {r i }), h(v) = y k with probability k where x * (v)/c = ∑ k  I k y k (random) price r i (v) is set so that E[r i (v)] = q i (v)/c. M R is truthful in expectation: for every true value v i, bids (b i, b -i ), expected value of i = E[v i. h(b i, b -i )] = v i. g(b i, b -i ) /c expected price of i = E[r i (b i, b -i )] = q i (b i, b -i ) /c M F truthful  ( v i. g(b i, b -i ) – q i (b i, b -i ) ) is maximized at b i = v i  E[v i. h(b i, b -i ) – r i (b i, b -i )] is maximized at b i = v i Also shows M R is a c-approximation mechanism (in expectation).

Getting a convex decomposition Given: x  P, LP-based c-approx. algorithm A Want k values s.t.∑ k  I k y k = x/c ∑ k  I k = 1  k ≥ 0 Linear inequalities in k variables – turn into a convenient LP

Getting a convex decomposition Min. ∑ k  I k (P) s.t. ∑ k  I k y k = x/c ∑ k  I k ≥ 1  k ≥ 0 for every k  I (P) is LP with exponential variables, but polynomial constraints. Max. w. x/c + z(D) s.t. w. y k + z ≤ 1 for every k  I z ≥ 0, w is free Dual (D) – polynomial variables, exponential number of constraints. Have to show the following: (P) is feasible optimal value of (P) is 1 (P) can be solved efficiently will show that optimal value of (D) is 1 will show that (D) can be solved using the ellipsoid method –Algorithm A can be used to obtain a separation oracle Running ellipsoid gives an equivalent, compact dual program (D') with only polynomial constraints. It’s dual is a program of the form (P) but with polynomial k variables – solve to get desired k values.

Getting a convex decomposition Min. ∑ k  I k (P) s.t. ∑ k  I k y k = x/c ∑ k  I k ≥ 1  k ≥ 0 for every k  I (P) is LP with exponential variables, but polynomial constraints. Max. w. x/c + z(D) s.t. w. y k + z ≤ 1 for every k  I z ≥ 0, w is free Dual (D) – polynomial variables, exponential number of constraints. To show: optimal value of (D) is 1 w = 0, z = 1 is a feasible solution  OPT (D) ≥ 1 Suppose there is a feasible solution (w,z) with w. x/c+z > 1 Interpret w = (w i,S ) as a valuation vector   integer soln. y k such that w. y k ≥ w. x/c > 1 –z contradiction Also yields oracle for use in ellipsoid method: w. x/c+z > 1 : use algorithm A to find above y k – get violated inequality w. x/c+z ≤ 1 : use w. x/c+z ≥ 1 hyperplane to cut ellipsoid (obj. f’n. cut)

Getting a convex decomposition Min. ∑ k  I k (P) s.t. ∑ k  I k y k = x/c ∑ k  I k ≥ 1  k ≥ 0 for every k  I (P) is LP with exponential variables, but polynomial constraints. Max. w. x/c + z(D) s.t. w. y k + z ≤ 1 for every k  I z ≥ 0, w is free Dual (D) – polynomial variables, exponential number of constraints. Also yields oracle for use in ellipsoid method: w. x/c+z > 1 : use algorithm A to find above y k – get violated inequality w. x/c+z ≤ 1 : use w. x/c+z ≥ 1 hyperplane to cut ellipsoid (obj. f’n. cut) One last hurdle: w = (w i,S ) may not be a valid valuation vector  cannot “run” A on w directly Easy fix: simply “monotonize” w; set w + (i,S) = max(0, max T  S w i,S ) and run A on w +

To summarize, Move from integer problem to fractional domain –Use VCG to get fractional truthful mechanism Use decomposition lemma to express scaled-down optimal solution as convex combination of integer solutions Interpret this as a randomized mechanism –Obtain a truthful-in-expectation, approx. mechanism General way to exploit VCG for packing SWM problems using approximation algorithms.

Open Questions Packing SWM problems: how to exploit structure of valuation class, e.g., submodular valuations? Non-SWM social-choice f’ns f: V 1 x … x V n  A: ways to “export” approximation algorithms to truthful mechanisms? General techniques to get deterministic truthful mechanisms for multi-parameter problems? Or truthfulness with high probability?

Thank You.