Algorithmic Mechanism Designalgo2.iti.kit.edu/.../courses/amd/mechanism_design_part2.pdf ·...

Organization Randomized Mechanisms Mechanisms with Verification Summary

Algorithmic Mechanism DesignNoan Nisan, Amir Ronen

- 2nd Part -

Seminar Talkby

Paul D. Dutting

Fakultat fur InformatikUniversitat Karlsruhe (TH)

May 3, 2007

Algorithmic Mechanism Design Fakultat fur Informatik Universitat Karlsruhe (TH)


Overview

Organization

Randomized MechanismsModel of RandomizationA Randomized Mechanism for Task Scheduling

Mechanisms with VerificationModel of VerificationThe Compensation-and-Bonus MechanismPolynomial-Time Mechanisms

Summary



Model of Randomization

Randomized Mechanisms

Definition (Randomized Mechanism)

A randomized mechanism is a probability distribution over a family{mr |r ∈ I} of mechanisms, all sharing the same sets of strategiesand possible outputs.The outcome of such a mechanism is a probability distribution overoutputs and payments; the problem specification must specify whatputput distributions are required. For the case of optimizationproblems, the objective function on such a distribution is taken tobe the expectation g(a, t) = Er∈I (g(omr (a), t)).




Dominant Strategies and Truthful Mechanisms

Definition (Universally Dominant Strategy)

A strategy ai is called universally dominant for agent i if it is adominant strategy for every mechanism in the support of therandomized mechanism. A randomized mechanism is calleduniversally truthful if truth-telling is a dominant strategy, andstrongly truthful if it is the only one.




Task Scheduling Problem Reviewed

Let’s recall the following key results:

Theorem (Min Work Mechanism)

Min Work is a strongly truthful n-approximation mechanism for thetask scheduling problem.

Theorem (Lower Bound for Deterministic Mechanisms)

There does not exist a mechanism that implements ac-approximation for the task scheduling problem for any c < 2.

⇒ for 2 agents:Min Work is a 2-approximation and no mechanism can do better



A Randomized Mechanism for Task Scheduling

A Simple Example

Example

Consider two kind of tasks j = 1, 2 and two agents i = 1, 2 suchthat t1

1 = 1, t21 = 1 + ε, t1

2 = 10 + ε, t22 = 10. If there are two tasks

of each kind, the allocations of Min Work and Opt look like this:

agent 1

agent 2 2

time20

agent 1

agent 2

time1 10 11+e




Evaluating the Simple Example

Let’s look at the makespans:

tMinWork=20tOpt=11+ε } ⇒

tMinWorktOpt

= 2011+ε ≈ 1.81 for small enough ε

Gets worse as the number of tasks or the size discrepancy grows!

QuestionIs it possible to achieve a similar behavior as Opt with arandomized mechanism? Should we allow some tasks to beallocated to the less efficient agent (as long as it is not much lessefficient)?




Biased Min Work Mechanism

The Biased Min Work Mechanism (for 2 agents)

Parameters: A real number β ≥ 1 and a bit vector s ∈ {1, 2}Input: The reported type vectors t = (t1, t2)Output: An allocation x = (x1, x2) & a payment p = (p1, p2)Mechanism: x1 ← ∅; x2 ← ∅, p1 ← 0; p2 ← 0

For each task j = 1..k do:Let i = sj and i ′ = 3− iIf t i

j ≤ β t i ′j

Then x i ← x i ∪ {j}; pi ← pi + β t i ′j

Else x i ′ ← x i ′ ∪ {j}; pi ′ ← pi ′ + β t ij




The Randomly Biased Min Work Mechanism

Definition (Randomly Biased Min Work Mechanism)

The Randomly Biased Min Work Mechanism is the distribution onbiased min work mechanisms given by β = 4/3, and a uniformdistribution of s ∈ {1, 2}k .

Let’s prove the following main result:

Theorem (Randomly Biased Min Work Mechanism)

The Randomly Biased Min Work Mechanism is a (polynomial timecomnputable) strongly truthful implementation of a7/4−approximation for task scheduling with two agents.




Proving the Theorem - Step 1

We will prove the preceding theorem in two steps.

Step 1

The Randomly Biased Min Work Mechanism is strongly truthful.

ProofSee chalkboard!




Proving the Theorem - Step 2

Step 2

The allocation obtained by the Randomly Biased Min WorkMechanism is a 7/4− approximation for the task schedulingproblem.

First, we will distillate the hardest case...




Proving the Theorem - Distillating the Hardest Case

ClaimIt is enough to consider the following case:

1. For each k-task the efficiency discrepany between the agentsis arbitrarily close to β.

2. If opt allocates an l-task j to agent i , then t3−ij /t i

j = β.

3. Under opt both agents have the same finishing time.

4. One of the agents is more efficient than the other on allk-tasks.

5. There are at most four tasks, where at most one k-task and atmost one l-task is allocated by opt to each agent.




Proving the Theorem - The Hardest Case

Due to the preceding considerations we can concentrate on thefollowing case:

t1j t2

j opt-alloc bmw-alloc

k1 a β a 1 1k2 b β b 2 1

l1 c β c 1 rndl2 β d d 2 rnd

ProofSee chalkboard!




Intermediate Summary

So far...

◦ we introduced randomized mechanisms and transferred thenotions dominant strategy and (strongly) truthful mechanismto this kind of mechanisms.

◦ we devised a randomized mechanism for the task schedulingproblem with two agents that beats the lower bound fordeterministic mechanisms.



Model of Verification

Mechanisms with Verification - The Basic Idea

IdeaUp to now the sole information the mechanisms are based uponare the agents’ declarations. In many settings though, one couldtake advantage of various extra information available at varyingpoints in time.

We will focus on:

Mechanisms with 2 Stages

Declaration Phase: agents talk decision (eg. allocation)Execution Phase: agents execute the agreed output payment




Mechanism with Verification

Definition (Mechanism with Verification 1/2)

◦ An agent’s strategy is composed of two separate parts: adeclaration d i and an execution e i .

◦ Each declaration d i is chosen by the agent, based on its typet i , in an unrestricted manner.

◦ The decision k of the mechanism must be a function of justthe declarations d1, .., dn.

◦ The agent’s execution e i may depend on t i as well as on k.The problem specification specifies, for each t i , the possiblee()’s an agent of type t i may choose.




Mechanism with Verification

Definition (Mechanism with Verification 2/2)

◦ The output of the mechanism is the result of the decision kand the agents’ executions e1(k), .., en(k). The outputfunction o(k, e) is a part of the problem specification.

◦ The output o, determines both the objective function g(o, t)and the agents’ valuations’ v i (t i , o).

◦ The payment pi that the mechanism provides depends onboth, the declarations d1, .., dn and the executionse1(k), .., en(k).




Truthful Mechanism with Verification

Definition (Truthful Mechanism with Verification)

A mechanism with verification is called truthful if

1. The agents’ declarations are simply to report their types.

2. For each agent i of type t i , there is a dominant strategy ofthe form ai = (t i , e i ()).

We say that the mechanism is strongly truthful if it is the onlydominant strategy.




Task Scheduling Reformulated

Definition (Task Scheduling with Verification)

The problem is the same as before, except that the mechanismknows the the times t = t1, .., tk the tasks were actually performedin and may base its payment p(t, t) = p1(t, t), .., pn(t, t) on thisextra information.

◦ Mechanism: (x(t), p) where x(t) is the allocation function

◦ Feasible output: (x , t) where x = x1, .., xn is the allocation

◦ Objective function: g(x , t) = maxi∑

j∈x i tj (make-span)

◦ Agent i ’s valuation: v i (x , t) = −∑

j∈x i tj

◦ Agent i ’s utility: ui (t, t) = pi (t, t) + v i (x , t)



The Compensation-and-Bonus Mechanism


The crucial aspect is to choose the payments adequately:

Definition (Compensation-and-Bonus Mechanism)

The Compensation-and-Bonus Mechanism is defined by an optimalallocation function x(t) and the payment functionspi (t, t) = c i (t, t) + bi (t, t) with

◦ Compensation: c i (t, t) =∑

j∈x i tj

◦ Corrected time vector: corr i (x , t, t) = {t lj if j∈x l and l 6=i

tj if j∈x i

◦ Bonus: bi (t, t) = −g(x(t), corr i (x , t, t))




Example 1: Truthtelling and Eager Agents

j 1 2 3 Remarks

Agent 1 t1j 10 30 45 opt.alloc.

t1j 10 30 45 act.exec.times

Agent2 t2j 100 60 100 opt.alloc.


Compensation: c i (t, t) =∑

j∈x i tj ⇒ c1(..) = 55 and c2(..) = 60

Corr. time vec.: corr i (x , t, t) = {t lj if j∈x l and l 6=i

tj if j∈x i ⇒ corr1,2(..) = t

Bonus: bi (t, t) = −g(x(t), corr i (x , t, t))⇒ b1,2(..) = −60Payment: pi (t, t) = c i (t, t) + bi (t, t)⇒ p1(..) = −5 & p2(..) = 0Utilities: ui (t, t) = pi (t, t) + v i (x , t)⇒ u1(..) = u2(..) = −60




Example 2: Lying but Eager Agents

j 1 2 3 Remarks





Compensation: c1(..) = 40 and c2(..) = 100Corr. time vec.: corr1,2(..) = t = (10, 30, 100) Bonus:b1,2(..) = −100Payment: p1(..) = −60 & p2() = 0Utilities: u1(..) = u2(..) = −100




Example 3: Truthtelling but Lazy Agents

j 1 2 3 Remarks





Compensation: c1(..) = 40 and c2(..) = 100Corr. time vec.: corr1(..) = (20, 60, 80) & corr2(..) = (10, 60, 45)Bonus: b1(..) = −100 and b2(..) = −60Payment: p1(..) = −60 and p2(..) = 0Utilities: u1(..) = −100 and u2(..) = −60




Truthfulness of the Compensation-and-Bonus Mechanism

Theorem (Compensation-and-Bonus Mechanism)

The Compensation-and-Bonus Mechanism is a strongly truthfulimplementation of the task scheduling problem.

Proof.See chalkboard!




The Generalized Compensation-and-Bonus Mechanism

The preceding findings can be generalized as follows:

Definition (Generalized Compensation-and-Bonus Mechanism)

The Generalized Compensation-and-Bonus Mechanism is definedby an optimal allocation function x(t) and the payment functionspi (t, t) = c i (t, t) + bi (t, t) with

◦ Generalizd Compensation: c i (t, t)≤∑

j∈x i tj

◦ Generalized Bonus: bi (t, t) = mi (t−i ,− g(x(t), corr i (x , t, t)))where mi (t−i ,w) is any positive real-valued function, that ismonotonically increasing in w




Further Findings

Theorem (Generalized Compensation-and-Bonus Mechanism)

The Generalized Compensation-and-Bonus Mechanism is astrongly truthful implementation of the task scheduling problem.

AnnotationThe Generalized Compensation-and-Bonus Mechanism can be usedto satisfy

◦ Participation Constraints: If an agent is truth-telling, itsutility is non-negative.

◦ Budget Limits: The total payments may not exceed a certainmaximum amount specified in advance.





So far...

◦ We introduced an extended model, formalizing situationswhere the mechanism has to make a descision based on theagents’ declarations but delays their payments until it knowstheir actual bahavior.

◦ We devised two optimal mechanisms for the adapted taskscheduling problem. The key idea was to calculate an optimalallocation of tasks based on the agents’ declarations and todetermine their payments based on the actual executiontimes. This allows cheating agents to be punished.



Polynomial-Time Mechanisms

Motivation

The rest of this talk is concerned with the following problem:

ProblemComputing an optimal allocation is known to be NP-hard.

IdeaOne could replace the optimal allocation algorithm in theCompensation-and-Bonus Mechanism with a knownpolynomial-time approximation algorithm.




Our First Idea Fails

Unfortunately, this does not work as the following theorem shows:

Theorem (Approximation Lacks Truthfulness)

Any Compensation-and-Bonus Mechanism that is based on anon-optimal allocation algorithm is not truthful.





Bounded Scheduling Problem

Let’s look at another variant of the task scheduling problem!

Definition (Bounded Scheduling Problem)

The problem is the same as in task scheduling with verification,except that the number of agents n is fixed to a constant and thereexist fixed constants a, b > 0 such that a ≤ t i

j ≤ b for all i and j .

AimDesign a (1 + ε)−approximation mechanism for this restrictedvariant of the problem.




The Rounding Algorithm of Horowitz and Sahni

IdeaUse the following approximation algorithm for the allocation oftasks (see Horowitz and Sahni [1] for details).

The Rounding Algorithm of Horowitz and Sahni

◦ Round up the entries t ij to integer multiples of δ (a parameter

chosen as a function of a and ε).

◦ Solve this rounded problem exactly in polynomial time usingdynamic programming.




The Payment Function of Nisan and Ronen

IdeaUse the following payment function.

The Payment Function of Nisan and Ronen

Let t denote the vector where all entries of t are rounded up to aninteger multiple of δ. Let g(..) be the make-span function anddenote by g(x , t) the make-span g(x , t). The payment function isgiven by pi (t, t) = c i (t, t) + bi (t, t), where

◦ c i (t, t) =∑

j∈i (t) tj is the Compensation and

◦ bi (t, t) = −g(x(t), corr i (x(t), t, t)) is the Bonus.




The Rounding Mechanism

TheoremThe Rounding Mechanism uses the Rounding Algorithm ofHorowitz and Sahni for the allocation of tasks and the PaymentFunction of Nisan and Ronen. For every fixed ε > 0 it is apolynomial time mechanism with verification that truthfullyimplements a (1 + ε)−approximation for the Bounded TaskScheduling problem.






So far...

◦ we pointed out that computing the optimal allocation of tasksis NP-hard. As a consequence the optimalCompensation-and-Bonus Mechanism is computionallyintractable. Simply replacing the optimal allocation algorithmby an approximation algorithm does not work in general.

◦ we introduced a restricted variant of the task scheduling withverification problem. For this problem we devised apolynomial time (1 + ε)−approximation mechanism.



Summary (of the 2nd Part)

◦ We designed a randomized mechanism for the task schedulingproblem that beats the lower bound for deterministicmechanisms.

◦ We proposed an extended model, mechanisms withverification, and presented two (computionally intractable)optimal mechanisms.

◦ For a restricted variant of the task scheduling problem wedevised a (1 + ε)−approximation mechanism.



That’s it. Thank you!Are there any questions?



Literature

[1] Ellis Horowitz and Sartaj Sahni. Exact and ApproximateAlgorithms for Scheduling Nonidentical Processors. In Journalof the Association for Computing Machinery, p.317-327, 1976.

[2] Noam Nisan and Amir Ronen. Algorithmic MechanismDesign. In Games and Economic Behavior 35, p.166-196, 2001.

[3] Noam Nisan and Amir Ronen. Algorithmic MechanismDesign (extended abstract). In The Thirty-First Annual ACMSymposium on Theory of Computing, 1999.


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