A Sufficient Condition for Truthfulness with Single Parameter Agents
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Transcript of 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