Mechanism Design and Truthful Algorithms

Transcription

1 Mechanism Design and Truthful Algorithms Ocan Sankur 13/06/2013 Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

2 Mechanism Design Mechanism design is about designing games in which some desired objective is achieved when all players play selfishly. Reverse game theory: Rather than analyzing a given game, design one that fits your needs. In this talk, we will see several mechanisms where the best strategy is to tell the truth. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

3 Motivation Public Project Problem A public good is to be constructed, with cost c. There are n agents who will enjoy this good, each with appreciation v i 0. If the project is accepted, everyone pays c/n; the utility of agent i is v i c/n. If the project is rejected, no cost, and all utilies are 0. Decision rule: Accept iff n i=1 v i c. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

4 Motivation Public Project Problem A public good is to be constructed, with cost c. There are n agents who will enjoy this good, each with appreciation v i 0. If the project is accepted, everyone pays c/n; the utility of agent i is v i c/n. If the project is rejected, no cost, and all utilies are 0. Decision rule: Accept iff n i=1 v i c. Possible manipulation: If v i > c/n, then Agent i should declare c and guarantee that the project is accepted! Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

5 Motivation Public Project Problem A public good is to be constructed, with cost c. There are n agents who will enjoy this good, each with appreciation v i 0. If the project is accepted, everyone pays c/n; the utility of agent i is v i c/n. If the project is rejected, no cost, and all utilies are 0. Decision rule: Accept iff n i=1 v i c. Possible manipulation: If v i > c/n, then Agent i should declare c and guarantee that the project is accepted! If v i < c/n, and if everyone else is truthful, then Agent i could try declaring 0. Mechanism design: design games that selfish players cannot manipulate. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

6 Direct Revelation Mechanisms An outcome is to be chosen by a central authority from a set A. There are n agents with different preferences. 0. Each agent i has a true valuation v i : A R, 1. Each agent i declares a valuation v i : A R, 2. A social choice function f : V 1... V n A is applied on (v i ) i. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

7 Direct Revelation Mechanisms An outcome is to be chosen by a central authority from a set A. There are n agents with different preferences. 0. Each agent i has a true valuation v i : A R, 1. Each agent i declares a valuation v i : A R, 2. A social choice function f : V 1... V n A is applied on (v i ) i. Currency To prevent manipulations, we need a common currency in which agents can pay taxes or be paid. Given outcome a, if Player i pays some quantity m of money, then his utility is v i (a) m. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

8 Direct Revelation Mechanisms An outcome is to be chosen by a central authority from a set A. There are n agents with different preferences. 0. Each agent i has a true valuation v i : A R, 1. Each agent i declares a valuation v i : A R, 2. A social choice function f : V 1... V n A is applied on (v i ) i. 3. Each agent i pays an amount determined by a payment function p i : V 1... V n R. Currency To prevent manipulations, we need a common currency in which agents can pay taxes or be paid. Given outcome a, if Player i pays some quantity m of money, then his utility is v i (a) m. Mechanism: (f, p 1,..., p n ). Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

9 Example Sell an item to the highest bidder. Objective: No one should increase his utility by lying. Regular Auction An item is to be sold to the highest bidder. A = set of bidders. Each player has a private value v i, and declares a value v i. Payment functions: The highest bidder i 0 pays v i 0, others pay 0. Utilities: For the highest bidder i 0 the utility is v i0 v i 0, for others it is 0. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

10 Example Sell an item to the highest bidder. Objective: No one should increase his utility by lying. Regular Auction An item is to be sold to the highest bidder. A = set of bidders. Each player has a private value v i, and declares a value v i. Payment functions: The highest bidder i 0 pays v i 0, others pay 0. Utilities: For the highest bidder i 0 the utility is v i0 v i 0, for others it is 0. Assume every one is telling the truth, and v 1 < v 2 <... < v n 1 < v n. Outcome: n gets the item and pays v n. Utility: 0. But he would rather lie and declare v n 1 + ɛ, and have utility v n v n 1 ɛ. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

11 Example Sell an item to the highest bidder. Objective: No one should increase his utility by lying. Regular Auction An item is to be sold to the highest bidder. A = set of bidders. Each player has a private value v i, and declares a value v i. Payment functions: The highest bidder i 0 pays v i 0, others pay 0. Utilities: For the highest bidder i 0 the utility is v i0 v i 0, for others it is 0. Assume every one is telling the truth, and v 1 < v 2 <... < v n 1 < v n. Outcome: n gets the item and pays v n. Utility: 0. But he would rather lie and declare v n 1 + ɛ, and have utility v n v n 1 ɛ. Open to manipulation! Players will try to learn others valuations. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

12 Objective & Constraints Social Welfare We focus on social choice functions that maximize social welfare, i.e. f ( v) argmax a A n v i (a). i=1 Truthfulness Given f, we want to find p 1,..., p n such that v i (f ( v)) p i ( v) v i (f ( v i, v i )) p i ( v i, v i ). Telling the truth should be a dominant strategy (may not be unique). Such a function is implementable: agents do not have incentive to cheat. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

13 Without payments Without payments, what social choice functions are truthful? Gibbard-Satterthwaite Let f be a truthful social choice function onto A, with A 3, then f is a dictatorship. (the choice is dictated by one of the players). Money and payments, or other assumptions are necessary to truthfully implement non-trivial functions. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

14 Vickrey-Clarke-Groves (VCG) Mechanisms VCG payments Consider a social choice function f that maximizes social welfare. Define payments by p i ( v) = h i ( v i ) j i v j(f ( v)), where h i are arbitrary functions (not depending on v i ) Intuition: - Each player is paid the social welfare of the other players. has incentive to increase others welfare. - A player s own declaration does not affect h i. The payment is reduced only by increasing others welfare. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

15 Vickrey-Clarke-Groves (VCG) Mechanisms - 2 Theorem Consider a mechanism (f, p 1,..., p n ) which maximizes social welfare, and payments are given by VCG (for any choice of h i ). The mechanism is truthful. Proof For any player i with valuation v i, and declarations v i of others, prove that declaring v i is as good as declaring any other v i. Denote a = f (v i, v i ) and a = f (v i, v i). Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

16 Vickrey-Clarke-Groves (VCG) Mechanisms - 2 Theorem Consider a mechanism (f, p 1,..., p n ) which maximizes social welfare, and payments are given by VCG (for any choice of h i ). The mechanism is truthful. Proof For any player i with valuation v i, and declarations v i of others, prove that declaring v i is as good as declaring any other v i. Denote a = f (v i, v i ) and a = f (v i, v i). The utility of declaring v i is v i (a) + j i v j(a) h i ( v i ). Declaring v i yields v i (a ) + j i v j(a ) h i ( v i ). Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

17 Vickrey-Clarke-Groves (VCG) Mechanisms - 2 Theorem Consider a mechanism (f, p 1,..., p n ) which maximizes social welfare, and payments are given by VCG (for any choice of h i ). The mechanism is truthful. Proof For any player i with valuation v i, and declarations v i of others, prove that declaring v i is as good as declaring any other v i. Denote a = f (v i, v i ) and a = f (v i, v i). The utility of declaring v i is v i (a) + j i v j(a) h i ( v i ). Declaring v i yields v i (a ) + j i v j(a ) h i ( v i ). But a = f (v i, v i ) maximizes social welfare, so v i (a) + j i v j (a) h i ( v i ) v i (a ) + j i v j (a ) h i ( v i ). Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

18 Example: Auction - 2 Vickrey Auction An item is to be sold to the highest bidder. Each player has a private value v i, and declares a value w i. Payment functions: The highest bidder i 0 pays max i i0 w i, others pay 0. Utilities: For the highest bidder i 0 the utility is v i0 w i0, for others it is 0. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

19 Example: Auction - 2 Vickrey Auction An item is to be sold to the highest bidder. Each player has a private value v i, and declares a value w i. Payment functions: The highest bidder i 0 pays max i i0 w i, others pay 0. Utilities: For the highest bidder i 0 the utility is v i0 w i0, for others it is 0. Assume every one is telling the truth, and v 1 < v 2 <... < v n 1 < v n. Outcome: n gets the item and pays v n 1. Utility: v n v n 1. Would lying help? If n declares v n > v n, he would still get the item and pay v n 1. same utility. If n declares v n (v n 1, v n ), same utility. If n declares v n v n 1, he won t get the item. utility 0. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

20 Example: Auction - 2 Vickrey Auction An item is to be sold to the highest bidder. Each player has a private value v i, and declares a value w i. Payment functions: The highest bidder i 0 pays max i i0 w i, others pay 0. Utilities: For the highest bidder i 0 the utility is v i0 w i0, for others it is 0. In fact, payments in Vickrey s auction are given by a VCG mechanism. Giving the item to the highest bidder maximizes social welfare i.e. i v i(a). Highest bidder n pays Others pay 0. p n ( v) = h i ( v i ) j i v j(f ( v)) = v n 1 +0 Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

21 Rationality & Feasability The choice of h i is rather arbitrary. A mechanism is individually rational if always v i (f ( v)) p i ( v) 0. A mechanism is feasible if always i p i( v) 0 (mechanism does not need external financing). Clarke s Pivot Rule: Choose p i ( v) = max b v j (b) j i j i v j (a), where a = f ( v). Each player i pays the damage they cause to others. Clarke s Pivot Rule all VCG mechanisms are ind. rational and feasible. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

22 Computer Science Example: Routing Goal: Allocate the shortest (cheapest) path from s to t. - Edges are owned by different selfish agents. - Each edge e costs c e to the owner if it is taken, 0 otherwise. Maximizing social welfare: finding the shortest path ( e p c e). VCG: To each e 0 p, pay e p c e e p\{e 0 } c e, where p is the shortest path, and p the shortest path not using e 0. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

23 Extensions of VCG Affine maximizer A social choice function f is an affine maximizer if for some A A, and weights w 1,..., w n > 0, and c a R for a A, we have f (v 1,..., v n ) arg max a A (c a + i w i v i (a)), Payment functions can be adapted naturally with the weights. Theorem [Roberts 1979] If A 3 and f is onto, V i = R A, and (f, p 1,..., p n ) is truthful, then f is an affine maximizer. Not true for A 2. Open question: Relaxing the hypothesis V i = R A. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

24 Why Payments? Reminder: Gibbard-Satterthwaite Let f : V 1... V n A be a truthful social choice function onto A, with A 3, and V i = R A, then f is a dictatorship. For A = 2? Affine maximizers are not the only truthful functions if V i R A. In the rest: restrictions of V i. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

25 Direct Characterization of Truthfulness (Relevant when A 2 or V i R A ). A mechanism (f, p 1,..., p n ) is truthful iff the following holds for each i and v, 1 The payment p i only depends on f ( v) and v i (and not on v i in any other way). If we fix v i, the payment p a only depends on the outcome a. 2 The mechanism optimizes for each player: Proof. where a ranges over f (, v i ). f (v i, v i ) arg max a (v i(a) p a ) Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

26 One-dimensional Valuations With Payments In computer science, objectives different than maximizing social welfare arise, e.g. truthful scheduling. Several tasks need to be scheduled on different machines. - Each machine declares its available processing time, - and is a selfish agent, trying to avoid work. We want to schedule with minimum makespan. If A is the set of schedulings for machines M: min max load(i). a A i M Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

27 A Scheduling on Related Machines There n jobs to be assigned to m machines. Job j consumes p j time units, and machine i has speed c i. (So machine i requires p j c i time to complete job j). The load of machine i, is l i = p j. j j assigned toi Under payments P i, the utility of machine i is Machines are agents; they declare c i. l i c i P i. Objective: Design a truthful mechanism selecting a scheduling with makespan min a max i li ac i Valuations are one-dimensional (c i determines the valuation). Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

28 Characterization of Truthfulness: Weak Monotonicity Weak monotonicity A social choice function satisfies weak monotonicity (WMON) iff for all i and v i, f (v i, v i ) = a b = f (v i, v i) implies v i (a) v i (b) v i (a) v i (b). Recall that mechanism (f, p 1,..., p n ) is truthful iff v i (f ( v)) p i ( v) v i (f ( v i, v i )) p i ( v i, v i ). Theorem - If (f, p 1,..., p n ) is truthful, then f satisfies WMON. - If f satisfies WMON, there exist p 1,..., p n such that the mechanism is truthful. (Holds whenever valuation sets are convex sets). Proof. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

29 Application of WMON Fix c i, costs declared by all players but i. Then, l i (c i ) is a function of c i. Consider c i < c i. WMON means l i (c i )c i + l i (c i )c i l i (c i )c i + l i (c i )c i l i (c i ) l i(c i ). which means that the work load should be nonincreasing! Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

30 WMON implies implementability (Proof by figure) Assume an algorithm with nonincreasing load curves is given. Consider payments p i (c) = c 0 l i(x)dx c l i (c). Utility (with declared and true cost c): c l i (c) p i (c) = c 0 l i(x)dx Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

31 WMON implies implementability (Proof by figure) Assume an algorithm with nonincreasing load curves is given. Consider payments p i (c) = c 0 l i(x)dx c l i (c). Utility (with declared and true cost c): c l i (c) p i (c) = c 0 l i(x)dx Utility (with declared cost c ): c l i (c ) (p i (c) + A) Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

32 Overall Theorem To prevent utilities to be negative, consider. Theorem [Archer, Tardos FOCS 01] A scheduling algorithm is truthfully implementable iff its load functions are nonincreasing. In this case, the following payments functions yield individually rational dominant strategy implementations: p i (c) = c 0 [l i (x) l i (c)]dx + c l i (x)dx. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

33 Overall Theorem To prevent utilities to be negative, consider. Theorem [Archer, Tardos FOCS 01] A scheduling algorithm is truthfully implementable iff its load functions are nonincreasing. In this case, the following payments functions yield individually rational dominant strategy implementations: p i (c) = c 0 [l i (x) l i (c)]dx + c l i (x)dx. How to design an efficient algorithm with nonincreasing load functions? (Efficient computation of a scheduling and payments) Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

34 Existing Algorithms and The New One The problem is NP-complete [Hochbaum & Shmoys 88]. We want everything to be computable in PTIME. A PTAS is known (for all ɛ > 0, an ɛ-approximation algorithm) but is not nonincreasing! Truthful Algorithm (sketch): Fix a time bound T Create a bin of size T /c i for for machine i Greedily solve the bin packing problem: assign longest job to fastest machine upto time T, and cut fractionally Randomly round fractional jobs For an efficiently computable T, this gives a factor-2 approximation alg. that is truthful in expectation. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

35 Single-Peaked Preferences Without Payments Let us fix A = [0, 1]. A relation on A is single-peaked if there exists p A such that x A \ {p}, x λx + (1 λ)p. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

36 Single-Peaked Preferences Without Payments Let us fix A = [0, 1]. A relation on A is single-peaked if there exists p A such that Examples 1 Average of peaks: x A \ {p}, x λx + (1 λ)p. f ( 1,..., n ) = 1 p i. n i Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

37 Single-Peaked Preferences Without Payments Let us fix A = [0, 1]. A relation on A is single-peaked if there exists p A such that Examples 1 Average of peaks: x A \ {p}, x λx + (1 λ)p. f ( 1,..., n ) = 1 p i. n i 2 Dictatorship: f ( 1,..., n ) = p i0. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

38 Single-Peaked Preferences Without Payments Let us fix A = [0, 1]. A relation on A is single-peaked if there exists p A such that Examples 1 Average of peaks: x A \ {p}, x λx + (1 λ)p. f ( 1,..., n ) = 1 p i. n i 2 Dictatorship: f ( 1,..., n ) = p i0. 3 Median of peaks: f ( 1,..., n ) = median{p i } i. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

39 Single-Peaked Preferences: Characterization Theorem [Moulon 1980, Ching 1997] Under single-peaked preferences, a social choice function f is onto, anonymous, and truthful if and only if y 1,..., y n 1 [0, 1] such that f ( 1,..., n ) = median(p 1,..., p n, y 1,..., y n 1 ). anonymous: invariant under permutation of indices. A characterization is available as generalized median voter schemes for onto and truthful functions. Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25

40 Conclusion & Further Reading Truthfulness is not the only solution concept. One can design games with e.g. Nash equilibria. And apply it on richer game structures Combinatorial auctions! (e.g. Google Ads) Nisan, Ronen. Algorithmic mechanism design. STOC 99. Archer, Tardos. Truthful Mechanisms for one-parameter Agents. FOCS 01 Roberts. Characterization of implementable choice rules Nisan et al. Algorithmic Mechanism Design. (LSV library). Ocan Sankur (ULB) Mechanism Design and Truthful Algorithms June 13, / 25