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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)

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

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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

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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

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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 Testing Truthfulness of Decision Rule (2)

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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 Halfway Monotone Derivative (HMD) condition

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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 ω b i = A(b i, b -i ) denotes the outcome when i bids b i Applicable whenever it is clear that A and b -i are fixed

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Definitions A decision rule is a function A:T n →Ω that given a vector b of n bids returns an outcome A payment scheme P is a set of payment functions, where P i 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.

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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 p i, given that his type is t i is the partial derivative of a valuation function by the agent’s type.

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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

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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 st An allocation algorithm admits a truthful payment has no finite negative cycles.

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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 t 0. The payment from agent i to the mechanism if it bids t is:

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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

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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 s i, where The purpose of r i () is to obtain unique representation for the same type space We will ignore the r i () for simplicity, and assume Definition: A mechanism (algorithm) is a mechanism (algorithm) for single parameter agents if all agents are single parameter.

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Halfway Monotone Derivative (HMD) Definition: A valuation function v i satisfies HMD condition with respect to a given decision rule, if for every fixed bid vector b -i, one of the following holds: stu1u1 u2u2 T v(ω t,u) v(ω s,u)

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

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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 t 0, t 1,…,t k, t k+1 =t 0 We show first that there is a negative 2-cycle and then infer that the condition is violated

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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 t su

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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

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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. st t su

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

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

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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 v’ i (ω t, x) for some x. It follows that for every s ≤ x ≤ t, v’ i (ω s, x) > v’ i (ω t, x)

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

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Theorem - Suitable Payment A suitable payment scheme for agent i in a single parameter rationalizable decision rule A:T n →Ω that is HMD is where b -i is held fixed, t 0 is an arbitrary type and c is an arbitrary function of b -i.

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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

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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 t i – 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 t i for losing, the value is 0.

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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. Single Commodity Auctions (2)

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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 t i, and v’ i = 1 When losing, the value is 0, and v’ i = 0 For any type t i, 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 c i, for which i loses if t i c i □

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Processor Scheduling n jobs, m processors c 1,…,c m – processors’ costs per unit p 1,…,p n – jobs’ processing requirements Running the i-th job on the j-th machine requires p i *c j time. The cost for processor j is where I j is the set of jobs assigned to processor j. The goal is to minimize the longest completion time

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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

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Mechanism Design The processors’ costs c j 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.

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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.

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Application of HMD Theorem: A scheduling algorithm is monotone iff it is HMD. Proof: v j = -c j W j, where W j is the total weight of the jobs assigned to j-th processor. v’ j = -W j HMD requires that –W j would increase if reported cost increases, which is equivalent to monotonicity condition □ cjcj vjvj v j (ω t,c j ) v j (ω s,c j ) st

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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

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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 b i, or it has a time c i, such that if b i c i, then.

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Limit (Budget) Constraints n items, m bidders p ij – the valuation of i-th bidder for the j-th item t i – 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

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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

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Strategic Limits (Budgets) Assume that all the p ij (valuations) are known The budgets are privately known to the agents

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Piecewise Monotonicity Definition: An allocation scheme for auctions with limit constraints is piecewise monotone if for every agent i and every limit t 0 such that v i (ω t 0, t 0 ) = t 0, it holds that for every t 1 > t 0, ω t 1 ≥ ω t 0.

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Rationalizability Theorem: Any piecewise monotone allocation rule is rationalizable. Proof: Denote by ω the total value of items assigned to i-th agent For ω fixed: If t i < ω: v i (ω, t i ) = t i, v’ i = 1 If t i ≥ ω: v i (ω, t i ) = ω, v’ i = 0 titi ω v i (ω, t i )

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Proof (cont.) We prove that piecewise monotonicity leads to first HMD condition. We need that for any b 0 < b 1, v’ i (ω b 0, x) ≤ v’ i (ω b 1, x) for every b 0 ≤ x First assume that ω b 0 ≤ b 0. For each x > b 0, v’ i (ω b 0, x) = 0 and so no constraints are induced for v’ i (ω b 1, x) x ωb0ωb0 v i (ω b 0, x) b0b0

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Proof (end) Now if ω b 0 ≥ b 0 : v’ i (ω b 0, x) = 1 for x ≤ ω b 0 To fulfill the first HMD condition, for each b 1 > b 0, ω b 1 should be at least ω b 0 This is achieved due to the piecewise monotonicity □ x ωb0ωb0 v i (ω b 0, x) b0b0

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