A Sufficient Condition for Truthfulness with Single Parameter Agents

1. A Sufficient Condition for Truthfulness with Single Parameter Agents Michael Zuckerman, Hebrew University 2006 Based on paper by Nir Andelman and Yishay Mansour (Tel Aviv University)

2. Agenda • Introduction to Truthful Mechanisms • Definitions and preliminaries • The HMD condition for truthfulness • The Suitable Payment Function • The HMD Applications

3. What is Mechanism Design • Selfish agents interact with centralized decision maker • Each agent • has his own private type • submits a bid, which signals his type • Aims to optimize his own utility • The mechanism aims to • Optimize the total result, e.g.: • Maximize the social welfare (the sum of utilities) • Maximize the maximal utility • Maximize the minimal utility • Give an incentive to the agents to signal their true type • Achieved by assigning payments to or from the mechanism

4. Testing Truthfulness of Decision Rule • How can we know whether a decision rule can be melded into truthful mechanism by adding a proper payment scheme ? • VCG mechanism is always truthful • Works only for certain optimization functions (like maximizing social welfare) • Is practical only when the optimum can be calculated

5. Testing Truthfulness of Decision Rule (2) • A criteria given by Rochet • Sufficient and necessary condition • Does not provide computationally convenient method for testing truthfulness • 2-cycle inequality = weak monotonicity • Necessary but not sufficient • Easy to work with • Mirrlees-Spence condition • Sufficient and necessary • Simple • Works only when the output of the mechanism is continuous

6. Halfway Monotone Derivative (HMD) condition • Generalization of Mirrlees-Spence condition • Does not make assumptions on algorithm output space • A sufficient condition for algorithm truthfulness • For some valuation functions is also a necessary condition • Easy to work with • Characterizes also the structure of the payment function

7. Preliminaries • The system consists of a decision rule (an algorithm) A and n agents (bidders). • Each bidder submits a bid (signal) • The outcome is calculated by an algorithm A(b), where b is the bid vector • The bid vector without the i-th bid is denoted by b-i • ωbi = A(bi, b-i) denotes the outcome when i bids bi • Applicable whenever it is clear that A and b-i are fixed

8. Definitions • A decision rule is a function A:Tn→Ω that given a vector b of n bids returns an outcome • A payment schemeP is a set of payment functions , where Pi determines the payment of agent i to the mechanism, given the output ω and the bid vector b. • A mechanism M = (A,P) is a combination of a decision rule A and a payment scheme P.

9. Utilities • is the type of agent i • is the valuation function of i. • is the utility of agent i of the outcome ω and a payment pi, given that his type is ti • is the partial derivative of a valuation function by the agent’s type.

10. Truthfulness • For truthful mechanisms we will talk about payment functions of the form , which don’t depend on the i-th bid • Definition: Algorithm A admits a truthful payment if there exists a payment scheme P such that for any set of fixed bids b-i, and for any two types

11. s t Rochet condition • Given an agent i and having all other bids b-i held fixed, let be a weighted directed graph such that , and the weight of every edge is • An allocation algorithm admits a truthful payment • has no finite negative cycles.

12. Suitable Payment Function • If the decision rule is rationalizable, then the payment function for the i-th agent is: • For every vector of fixed bids b-i choose an arbitrary type t0. • The payment from agent i to the mechanism if it bids t is:

13. Weak monotonicity condition(2-cycle inequality) • Does the graph contain negative cycle of length 2 ? • Formally, does not have negative 2-cycles if and only if for every two types • This is of course a necessary, but not sufficient • condition

14. Single Parameter • Definition: An agent i is a single parameter agent with respect to Ω if there exists an interval and a bijective transformation such that for any , the function is continuous and differentiable almost everywhere in si, where • The purpose of ri() is to obtain unique representation for the same type space • We will ignore the ri() for simplicity, and assume • Definition: A mechanism (algorithm) is a mechanism (algorithm) for single parameter agents if all agents are single parameter.

15. v(ωs,u) v(ωt,u) s u1 t u2 T Halfway Monotone Derivative (HMD) • Definition: A valuation function vi satisfies HMD condition with respect to a given decision rule, if for every fixed bid vector b-i, one of the following holds:

16. Main Theorem • Theorem: A single parameter decision rule A(b):Tn→Ω is rationalizable when all valuation functions are HMD.

17. Proof • We shall prove for the first HMD condition (the second condition is similar). • Assume by contradiction that A is not rationalizable • There is some graph G(i, b-i) with negative cycle t0, t1,…,tk, tk+1=t0 • We show first that there is a negative 2-cycle and then infer that the condition is violated

18. t s u Proof (2) • If k = 1 then negative 2-cycle exists • If k > 1 let t be the node such that • Let s and u be the neighbors of t in the cycle • Of course t ≤ u, t ≤ s

19. Proof (3) • The length of the path from s to u through t is: • The last integral is non-negative because t ≤ u • and for all x ≥ t, due to the first • HMD condition

20. t s u s t Proof (4) • Hence a shorter negative cycle can be constructed with a shortcut from s to u. • By induction, a negative 2-cycle exists in the graph • Assume that s < u.

21. End of proof • We infer from HMD, that: • And this is a contradiction to the cycle being negative. □

22. Necessity for Special Case • Theorem: If for every i, fixed vector b-i, and bid bi, v’i(ωbi,x) does not depend on x, then HMD is a necessary and sufficient condition for truthfulness.

23. Proof • This is enough to prove the necessity • Assume by contradiction, that HMD does not hold • There is an agent i, bid vector b-i and types s < t, s.t. v’i(ωs, x) > v’i(ωt, x) for some x. • It follows that for every s ≤ x ≤ t,v’i(ωs, x) > v’i(ωt, x)

24. Proof (end) • Integrate both sides of the inequality: • And we got violation of weak monotonicity. □

25. Theorem - Suitable Payment • A suitable payment scheme for agent i in a single parameter rationalizable decision rule A:Tn→Ω that is HMD is where b-i is held fixed, t0 is an arbitrary type and c is an arbitrary function of b-i.

26. HMD applications • We will talk about well known results, and see that they can be achieved by HMD condition • Single Commodity Auctions • Processor Scheduling • Then we will present new single parameter mechanisms, and apply HMD for them • Scheduling with Timing Constraints • Auctions with Limit Constraints

27. Single Commodity Auctions • We will talk about auctions, where each bidder has a unit demand • The results hold also for known single minded bidders • The agent’s private value is ti – the value of the product for the agent • For each specific bidder there are two possible outcomes: winning and losing • for winning, the value is ti • for losing, the value is 0.

28. Single Commodity Auctions (2) • Theorem: A deterministic auction is rationalizable iff for each bidder there is a critical value (determined by the other bids), s.t. the bidder wins if it bids above it, and loses otherwise (unless it has no winning bid) • Example: the second price auction.

29. Application of HMD in Single Commodity Auctions • Corollary: In deterministic auctions the critical value is equivalent to HMD. • Proof: • When winning, the value of the i-th agent is ti, and v’i = 1 • When losing, the value is 0, and v’i = 0 • For any type ti, the derivative of winning outcome is higher than the losing outcome • For b-i fixed, all deterministic HMD mechanisms must either decide that i never wins, or have a value ci, for which i loses if ti < ci, and wins if ti > ci□

30. Processor Scheduling • n jobs, m processors • c1,…,cm – processors’ costs per unit • p1,…,pn – jobs’ processing requirements • Running the i-th job on the j-th machine requires pi*cj time. • The cost for processor j is where Ij is the set of jobs assigned to processor j. • The goal is to minimize the longest completion time

31. Complexity • If all the costs and weights are known, then the it is NP-Complete • There is a PTAS to this problem • If the number of machines is constant, then there is an FPTAS to this problem

32. Mechanism Design • The processors’ costs cj are private values of their owners • The goal is to minimize the longest completion time, i.e. to minimize • The bidders can report incorrect values for lowering their costs.

33. Monotonicity • Definition: Scheduling algorithm is monotone if the amount of work it assigns to any computer does not decrease if the computer raises its speed (when the rest of the inputs remain constant). • Theorem (Archer and Tardos): Scheduling algorithm is truthful if and only if it is monotone.

34. Application of HMD • Theorem: A scheduling algorithm is monotone iff it is HMD. • Proof: • vj = -cjWj, where Wjis the total weight of the jobs assigned to j-th processor. • v’j = -Wj • HMD requires that –Wj would increase if reported cost increases, which is equivalent to monotonicity condition □ vj s t cj vj(ωt,cj) vj(ωs,cj)

35. Scheduling with Timing Constraints (STC) • n agents apply to get a service from central mechanism • An agent’s type is a timing constraint (deadline) which it must by served before, to get a positive valuation • The result is a service time • The infinity result means that the bidder is never served

36. Rationalizability for STC • Theorem: Given that a server never serves an agent after its declared deadline, then it is rationalizable iff for each agent, either for every bi, or it has a time ci, such that if bi < ci then and if bi > ci, then .

37. Limit (Budget) Constraints • n items, m bidders • pij – the valuation of i-th bidder for the j-th item • ti – the budget constraint of the i-th agent • For bundle of items I, • For simplicity assume that • The allocation algorithm does not have to allocate all the items • The objective function is total valuation of all agents

38. Some General Knowledge • This optimization problem is NP-Complete • A simple greedy algorithm gives a 2-approximation • LP-rounding gives a 1.58-approximation • There is a PTAS when the number of bidders is constant

39. Strategic Limits (Budgets) • Assume that all the pij (valuations) are known • The budgets are privately known to the agents

40. Piecewise Monotonicity • Definition: An allocation scheme for auctions with limit constraints is piecewise monotone if for every agent i and every limit t0 such that vi(ωt0, t0) = t0, it holds that for every t1 > t0, ωt1 ≥ ωt0.

41. vi(ω, ti) ti ω Rationalizability • Theorem: Any piecewise monotone allocation rule is rationalizable. • Proof: • Denote by ω the total value of items assigned to i-th agent • For ωfixed: • If ti < ω: vi(ω, ti) = ti, v’i = 1 • If ti ≥ ω: vi(ω, ti) = ω, v’i = 0

42. vi(ωb0, x) b0 x ωb0 Proof (cont.) • We prove that piecewise monotonicity leads to first HMD condition. • We need that for any b0 < b1, v’i(ωb0, x) ≤ v’i(ωb1, x) for every b0 ≤ x • First assume that ωb0 ≤ b0. • For each x > b0, v’i(ωb0, x) = 0 and so no constraints are induced for v’i(ωb1, x)

43. vi(ωb0, x) b0 ωb0 x Proof (end) • Now if ωb0 ≥ b0: • v’i(ωb0, x)= 1 for x ≤ ωb0 • To fulfill the first HMD condition, for each b1 > b0, ωb1 should be at least ωb0 • This is achieved due to the piecewise monotonicity □