A Sufficient Condition for Truthfulness with Single Parameter Agents

A Sufficient Condition for Truthfulness with Single

Parameter Agents

Michael Zuckerman, Hebrew University 2006Based on paper by Nir Andelman and Yishay

Mansour (Tel Aviv University)

Agenda

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

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

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

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)

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

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

Tbi

Definitions

A decision rule is a function A:Tn→Ω that given a vector b of n bids returns an outcome

A payment scheme P 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.

ni TP :

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.

ii Tt Tvi :

i

ii t

vv '

iiiiii ptvptu ),(),,(

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

1: ni TP

Tts ,

),(),(),(),( isisiititi bPtvbPtv

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

),(),( EVbiG i

s t

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

TTETV ,

).,(),(),( twvtwvtsw siti

)G(i,bbi -ii ,

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:

k

ikkiii ttTttkttwbtp

01111 ,,...,,0|),(inf),(

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 ),(),(),(),( svsvtvtv sitisiti

• This is of course a necessary, but not sufficient condition

),(),( EVbiG i

Tts ,

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.

iS ii STr : ),(ˆ ii sv

ii vv ˆ

))(,(),(ˆ 1iiii srvsv

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:

zero. measure ofset afor except ,

that,holdsit such that typesevery twoFor 2.

zero. measure ofset afor except ,

thatholdsit such that typesevery twoFor 1.

,u)(ωv',u)(ωv'

tut,sTs, t

,u)(ωv',u)(ωv'

sut,sTs, t

tisi

tisi

s tu1 u2

T

v(ωt,u)

v(ωs,u)

Main Theorem

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

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

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

ittki 0

t

s u

Proof (3)

The length of the path from s to u through t is:

),(),()),('),('(

),(),(),('),('

),(),(),(),(),(),(

),(),(),(),(),(),(

uswuswdxxvxv

uvuvdxxvdxxv

uvuvtvuvtvuv

uvuvtvtvutwtsw

ti

u

t si

uisi

u

t ti

u

t si

uisititisisi

tiuisiti

• The last integral is non-negative because t ≤ u and for all x ≥ t, due to the first HMD condition

),('),(' xvxv tisi

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.

s t

t

s u

End of proof

We infer from HMD, that:

0)),('),('(

),('),('

),(),(),(),(

),(),(

dxxvxv

dxxvdxxv

svsvuvuv

suwusw

si

u

s ui

u

s si

u

s ui

uisisiui

• And this is a contradiction to the cycle being negative. □

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.

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)

Proof (end)

Integrate both sides of the inequality:

And we got violation of weak monotonicity. □

),(),(),(),(

),('),('

tvtvsvsv

dxxvdxxv

sitisiti

t

s ti

t

s si

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.

t

t xitiiii dxxvtvbcbtP0

),('),()(),(

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

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.

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)

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 □

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

jIi i cpj

)(

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

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.

}){(max jIi ij

cpj

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.

Application of HMD

Theorem: A scheduling algorithm is monotone iff it is HMD.

Proof: vj = -cjWj, where Wj is 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 □

cj

vj

vj(ωt,cj)

vj(ω

s ,cj)

s t

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

it}{

i

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 .

ib

ib ib c

i

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

Ij ijiii pttIv },min{),(

j ijiijj ptp }{max

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

Strategic Limits (Budgets)

Assume that all the pij (valuations) are known

The budgets are privately known to the agents

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.

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 ti < ω: vi(ω, ti) = ti, v’i = 1

If ti ≥ ω: vi(ω, ti) = ω, v’i = 0

tiω

v i(ω, t i)

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) xωb0

v i(ωb 0, x

)

b0

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 □

xωb0

v i(ωb 0, x

)

b0