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)
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
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
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
Preliminaries
 The system consists of a decision rule (an algorithm) A




and n agents (bidders).
Each bidder submits a bid (signal) bi  T
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
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
Pi :   T n  , 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.
Utilities

ti  Ti is the type of agent i
 vi :   T   is the valuation function of i.
 ui ( , ti , pi )  vi ( , ti )  pi is the utility of agent i of the
outcome ω and a payment pi, given that his type is ti
vi
v
'

 i
ti is the partial derivative of a valuation
function by the agent’s type.
Truthfulness
 For truthful mechanisms we will talk about payment functions of
the form Pi :   T n 1  , 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 s, t  T
vi (t , t )  Pi (t , bi )  vi (s , t )  Pi (s , bi )
Rochet condition
 Given an agent i and having all other bids b-i held fixed,
let G(i, bi )  (V , E ) be a weighted directed graph such
that V  T , E  T  T, and the weight of every edge is
w(s, t )  vi (wt , t )  vi ( ws , t ).
s
t
• An allocation algorithm admits a truthful payment
 i, bi G(i,b-i ) has no finite negative cycles.
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

p(t , bi )  inf  w(ti , ti 1 ) | k  0, t1 ,..., tk 1  T , tk 1  t 
 i 0

Weak monotonicity condition
(2-cycle inequality)
 Does the graph contain negative cycle of length 2 ?
 Formally, G(i, bi )  (V , E ) does not have negative 2-
cycles if and only if for every two types s, t  T
vi (t , t )  vi (s , t )  vi (t , s)  vi (s , s)
• This is of course a necessary, but not sufficient
condition
Single Parameter
 Definition: An agent i is a single parameter agent with respect to Ω if
there exists an interval Si   and a bijective transformation ri : T  Si
such that for any    , the function vˆi ( , si ) is continuous and
differentiable almost everywhere in si, where vˆi ( , si )  vi ( , r 1 ( si ))
 The purpose of ri() is to obtain unique representation for the same type
space
 We will ignore the ri() for simplicity, and assume vi  vˆi
 Definition: A mechanism (algorithm) is a mechanism (algorithm) for
single parameter agents if all agents are single parameter.
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:
1. For every two types s, t  T such that s  t,u  s it holds that
v'i(ωs ,u)  v'i(ωt ,u), except for a set of measure zero.
2. For every two types s, t  T such that s  t,u  t it holds that,
v'i(ωs ,u)  v'i(ωt ,u), except for a set of measure zero.
v(ωs,u)
v(ωt,u)
s
u1
t
T
u2
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 0  i  k t  ti
 Let s and u be the neighbors of t in the cycle
t
s
 Of course t ≤ u, t ≤ s
u
Proof (3)
 The length of the path from s to u through t is:
w( s, t )  w(t , u )  vi (t , t )  vi (s , t )  vi (u , u )  vi (t , u )
 vi (s , u )  vi (s , t )  vi (t , u )  vi (t , t )  vi (s , u )  vi (u , u )
u
u
t
t
  v'i (s , x)dx   v'i (t , x)dx vi (s , u )  vi (u , u )
u
  (v'i (s , x) v'i (t , x)) dx  w( s, u )  w( s, u )
t
• The last integral is non-negative because t ≤ u
and v'i (s , x)  v'i ( t , x) for all x ≥ t, due to the first
HMD condition
Proof (4)
 Hence a shorter negative cycle can be
constructed with a shortcut from s to u.
t
s
u
 By induction, a negative 2-cycle exists in the
graph
s
 Assume that s < u.
t
End of proof
 We infer from HMD, that:
w( s, u )  w(u , s )
 vi (u , u )  vi ( s , u )  vi ( s , s )  vi (u , s )
u
u
s
s
  v'i (u , x)dx   v'i ( s , x)dx
u
  (v'i (u , x) v'i ( s , x)) dx  0
s
• 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:
t
t
 v' ( , x)dx   v' ( , x)dx
s
i
s
s
i
t
vi (t , s)  vi (s , s)  vi (t , t )  vi (s , t )
 And we got violation of weak monotonicity. □
Theorem - Suitable Payment
 A suitable payment scheme for agent i in a single
parameter rationalizable decision rule A:Tn→Ω that is
HMD is
t
Pi (t , bi )  c(bi )  vi (t , t )   v'i ( x , x)dx
t0
where b-i is held fixed, t0 is an arbitrary type and c is an
arbitrary function of b-i.
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.
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.
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 (iI pi )c j where Ij is
the set of jobs assigned to processor j.
 The goal is to minimize the longest completion
time
j
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
{( iI pi )c j }
completion time, i.e. to minimize max
j
 The bidders can report incorrect values for
lowering their costs.
j
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
□
vj
s
t
cj
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 ti   

 The result is a service time i    {}
 The infinity result means that the bidder is never
served
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 bi   for
every bi, or it has a time ci, such that if
bi < ci then bi   and if bi > ci, then bi  ci .
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, vi ( I , ti )  min{ ti , jI pij }
 For simplicity assume that max j { pij }  ti   j pij
 The allocation algorithm does not have to allocate all
the items
 The objective function is total valuation of all agents
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
ω
ti
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)
ωb0 b0
x
Proof (end)
 Now if ωb0 ≥ b0:
v’i(ωb0, x) = 1 for x ≤ ωb0
 To fulfill the first HMD
b0 ωb0
condition, for each b1 > b0,
ωb1 should be at least ωb0
 This is achieved due to the piecewise
monotonicity □

x