Mechanisms for Matching Markets with Budgets

Transcription

1 Mechanisms for Matching Markets with Budgets Paul Dütting Stanford LSE Joint work with Monika Henzinger and Ingmar Weber Seminar on Discrete Mathematics and Game Theory London School of Economics July 3, 2014

2 Matching Markets and Strategic Behavior Many markets have matching structure abc def ghi jkl mno abc def ghi jkl mno abc def ghi jkl mno houses cars sponsored search Participants in these markets act strategically Tiny London apartment with bed next to the kitchen sink for rent at 737 a month The Telegraph, June 3, 2014 See: Edelman and Ostrovsky [2006] Paul Dütting Matching with Budgets July 3, / 24

3 Mechanisms to Prevent Strategic Behavior Mechanism is a protocol to counteract strategic behavior preferences mechanism matching, payments Mechanisms for matching markets have been analyzed thoroughly Shapley and Shubik [1972] Leonard [1983] Demange et al. [1986] Alkan [1989] Alkan [1992] Crawford and Knoer [1981] Quinzii [1984] Demange and Gale [1985] Alaei et al. [2009] Paul Dütting Matching with Budgets July 3, / 24

4 Continuous vs. Discontinuous Utilities Focus of prior work has been on continuous utilities, in practice utilities are often discontinuous utility vs. utility payment payment Our goal: Bridge gap between theory and practice by designing mechanisms for discontinuous utilities Paul Dütting Matching with Budgets July 3, / 24

5 Outline of Talk 1. Model and Definitions 2. Existence 3. Computation 4. Incentives Paul Dütting Matching with Budgets July 3, / 24

6 Input Specification I J u ij (p j ) + + Set of bidders I with I = n, set of items J with J = k For each bidder i utility for being unmatched is 0 For each bidder i utility for being matched to item j at price p j is given by utility function u i,j (p j ) (C1): Strictly monotonically decreasing in pj (C2): There exist threshold prices pi,j such that u i,j ( p i,j ) 0 (C3): Locally right-continuous, i.e., x : limɛ 0 u i,j (x + ɛ) = u i,j (x) Paul Dütting Matching with Budgets July 3, / 24

7 Mechanism (I,J,u ij (p j )) mechanism (mu,p) A mechanism is a protocol that Asks each bidder i for its utility ui,j ( ) for each item j Computes an outcome (µ, p) consisting of a matching µ and payments p = (p 1,..., p n) A mechanism is dominant strategy incentive compatible (DSIC) if Consider bidder i with utilities ui,j ( ) For any two utility matrices u and u with u i,j( ) = u i,j ( ) for i and all j and u i,j( ) = u i,j( ) for i i and all j And corresponding outcomes (µ, p ) and (µ, p ) We have u i u i Paul Dütting Matching with Budgets July 3, / 24

8 Output Specification mu p 1 p 2 Outcome (µ, p) is envy free (EF) if For all bidder-item pairs (i, j) I J : ui u i,j (p j ) Outcome (µ, p) is bidder optimal (BO) if Outcome (µ, p) is EF For all bidders i I and outcomes (µ, p ) that are EF: u i u i Note: If (µ, p) is EF and p j = 0 for all unmatched items, then (µ, p) is a Walrasian Equilibrium Paul Dütting Matching with Budgets July 3, / 24

9 Outline of Talk 1. Model and Definitions 2. Existence 3. Computation 4. Incentives Paul Dütting Matching with Budgets July 3, / 24

10 Existence Result Theorem 1. For all inputs (I, J, u i,j ( )) satisfying conditions (C1)-(C3) a bidder optimal, envy free outcome (µ, p ) exists. Additionally: If we unilaterally relax any of the three conditions, then this ceases to be true. Paul Dütting Matching with Budgets July 3, / 24

11 Proof Sketch Step 1: Existence of a EF outcome with minimal prices Lemma 2. There exist at least one EF outcome (µ, p). Lemma 3. Can combine any two EF outcomes (µ, p), (µ, p ) into EF outcome (µ, p ) with u i = max{u i, u i } and p j = min{p j, p j }. Lemma 4. If the set of EF prices {p : µ s.t. (µ, p) is EF} has a unique infimum p, then there must be a matching µ that together with prices p forms an EF outcome. Step 2: EF outcome with minimal prices is BO Lemma 5. An EF outcome (µ, p) such that p j p j for all EF outcomes (µ, p ) is BO. Paul Dütting Matching with Budgets July 3, / 24

12 A Closer Look at the Key Lemma mu mu' p 1 p' 1 p 2 + p' 2 = cannot occur p 3 p' 3 mu mu' mu'' p' 1 p 1 p 1 p 2 + p' 2 = p' 2 I + = higher utility in mu p 3 I\I + = higher utility in mu' p' 3 p' 3 Paul Dütting Matching with Budgets July 3, / 24

13 Outline of Talk 1. Model and Definitions 2. Existence 3. Computation 4. Incentives Paul Dütting Matching with Budgets July 3, / 24

14 Computational Results Theorem 6. For linear utilities with a single discontinuity a BO, EF outcome can be computed in time polynomial in the number of bidders n and the number of items k. Additional results: Polynomial-time algorithm for piece-wise linear with multiple discontinuities Approximate algorithm for general utilities via piece-wise linear approximation Direct algorithm for general utilities whose running time is polynomial in n but exponential in k Paul Dütting Matching with Budgets July 3, / 24

15 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 6 u = 11 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 Paul Dütting Matching with Budgets July 3, / 24

16 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 6 u = 11 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 Paul Dütting Matching with Budgets July 3, / 24

17 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 6 u = 11 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 Paul Dütting Matching with Budgets July 3, / 24

18 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 6 u = 11 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 Paul Dütting Matching with Budgets July 3, / 24

19 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 6 δ = 1 u = 11 δ = 4 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 Paul Dütting Matching with Budgets July 3, / 24

20 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 5 u = 10 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 1 Paul Dütting Matching with Budgets July 3, / 24

21 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 5 u = 10 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 1 Paul Dütting Matching with Budgets July 3, / 24

22 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 5 u = 10 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 1 Paul Dütting Matching with Budgets July 3, / 24

23 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 5 u = 10 u = 10 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 1 Paul Dütting Matching with Budgets July 3, / 24

24 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 5 δ = 5 u = 10 δ = 3 u = 10 δ = 3 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 1 Paul Dütting Matching with Budgets July 3, / 24

25 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 2 u = 4 u = 4 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 4 p = 3 Paul Dütting Matching with Budgets July 3, / 24

26 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; Paul Dütting Matching with Budgets July 3, / 24

27 Algorithm for Linear Utilities with Single Discontinuity Pseudocode: Initialize prices with zero and matching with empty matching; while There exists an unmatched bidder do Take unmatched bidder; Compute maximal alternating tree; while There exists no unmatched item in tree do Compute minimal δ; Raise prices of items in maximal alternating tree by δ; Augment matching along alternating path; u = 2 u = 2 u = 2 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 4 p = 3 p = 2 Paul Dütting Matching with Budgets July 3, / 24

28 Proof Sketch For termination/running time: Observation 1. Bidders can only get unmatched if a maximum price is reached. This can happen at most O(nk) times in total. Observation 2. If no maximum price is reached, then the number of items in maximal alternating tree grows by at least one. For bidder optimality: Lemma 7. An EF outcome (µ, p ) with prices p j p j must have p j p j + δ for all items j whose price gets raised by the algorithm. Lemma 8. The prices computed by the algorithm are such that p j p j for every EF outcome (µ, p ). Paul Dütting Matching with Budgets July 3, / 24

29 Outline of Talk 1. Model and Definitions 2. Existence 3. Computation 4. Incentives Paul Dütting Matching with Budgets July 3, / 24

30 Results for Incentives Proposition 1. There does not exist a DSIC mechanism that always computes a BO, EF outcome. Note: True even for linear with unit slopes and a single, publicly known discontinuity per bidder Theorem 9. For inputs in general position every mechanism that computes a BO, EF outcome for linear utilities with unit slopes and a single discontinuity is DSIC. Additional results: Same result for piece-wise linear with non-identical slopes and multiple discontinuities Also comparable result for general, discontinuous utilities Paul Dütting Matching with Budgets July 3, / 24

31 Counter Example u = 2 u = 2 u = 2 6,6 5,6 11,4 5,4 4,4 10,3 4,3 p = 4 p = 3 p = 2 u = 6 u = 4 u = 9 6,6 5,6 0,4 5,4 4,4 10,3 4,3 p = 1 all bidders report their utilities truthfully second bidder misreports his utility for first item Paul Dütting Matching with Budgets July 3, / 24

32 General Position To define general position we define the following multi graph: One node per bidder, one node per item A forward edge from bidder i to item j with weight v i,j A backward edge from bidder i to item j with weight v i,j A discontinuity edge from agent to good with weight m i,j v i,j Input is in general position if no two walks in the multi graph that meet the following requirements have the same weight: Start with the same bidder Alternate between forward and backward edges End with a distinct discontinuity edge Paul Dütting Matching with Budgets July 3, / 24

33 Counter Example (Revisited) 6,6 5,6 11,4 5,4 4,4 10,3 4,3 blue: (+4) + (-4) + (3-10) = -7 red: (11-4) = -7 Paul Dütting Matching with Budgets July 3, / 24

34 Proof Sketch Step 1: General position implies BO, EF outcome has specific structure Lemma 10. For inputs in general position outcome (µ, p) computed by our mechanism is such that (i) p j > 0 implies j is matched (ii) last item that gets matched has p j = r j Step 2: Use that BO, EF outcome has specific structure to establish DSIC Lemma 11. In no outcome (µ, p) all bidders can be strictly better off than in BO, EF outcome (µ, p ). Lemma 12. If not all but some bidders are strictly better off in EF outcome (µ, p) than in BO, EF outcome (µ, p ), then among bidders that are not strictly better off there must be one that experiences envy. Paul Dütting Matching with Budgets July 3, / 24

35 Conclusion and Future Work Conclusion: Showed the existence of a BO, EF outcome for general discontinuous utilities Gave polynomial-time mechanism for piece-wise linear utilities with multiple discontinuities Can handle more general, discontinuous utilities via piece-wise linear approximation Gave sufficient condition for implication BO, EF DSIC Future Work: Same solution concepts, but beyond unit demand Other solution concepts Paul Dütting Matching with Budgets July 3, / 24

36 References An Expressive Mechanism for Auctions on the Web Paul Dütting, Monika Henzinger, and Ingmar Weber Transactions on Economics and Computation, accepted subject to revision Bidder Optimal Assignments for General Utilities Paul Dütting, Monika Henzinger, and Ingmar Weber Theoretical Computer Science, Volume 478, Pages 2232, March 2013 Sponsored Search, Market Equilibria, and the Hungarian Method Paul Dütting, Monika Henzinger, and Ingmar Weber Information Processing Letters, Volume 113, Pages 67-73, February 2013 Paul Dütting Matching with Budgets July 3, / 24