A Sufficient Condition for Truthfulness with Single Parameter Agents

1
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)

2
Agenda

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

3
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

4
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

5
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

6
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

7
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
• ?bi A(bi, b-i) denotes the outcome when i bids
  bi
• Applicable whenever it is clear that A and b-i
  are fixed

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Definitions

• A decision rule is a function ATn?O 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 ? and the bid vector b.
• A mechanism M (A,P) is a combination of a
  decision rule A and a payment scheme P.

9
Utilities

• is the type of agent i
• is the valuation function of
  i.
• is the
  utility of agent i of the outcome ? and a payment
  pi, given that his type is ti
• is the partial derivative of a
  valuation function by the agents type.

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Truthfulness

• For truthful mechanisms we will talk about
  payment functions of the form
  , which dont 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

11
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

• 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 t0.
• The payment from agent i to the mechanism if it
  bids t is

13
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

14
Single Parameter

• Definition An agent i is a single parameter
  agent with respect to O 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.

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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

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Main Theorem

• Theorem A single parameter decision rule
  A(b)Tn?O is rationalizable when all valuation
  functions are HMD.

17
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, tk1t0
• We show first that there is a negative 2-cycle
  and then infer that the condition is violated

18
Proof (2)

• If k 1 then negative 2-cycle exists
• If k gt 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

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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

20
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 lt u.

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End of proof

• We infer from HMD, that

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

22
Necessity for Special Case

• Theorem If for every i, fixed vector b-i, and
  bid bi, vi(?bi,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 lt
  t, s.t. vi(?s, x) gt vi(?t, x) for some x.
• It follows that for every s x t, vi(?s, x) gt
  vi(?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 ATn?O
  that is HMD is
• where b-i is held fixed, t0 is an arbitrary
  type and c is an arbitrary function of b-i.

26
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 agents 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.

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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.

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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 ti,
  and
• vi 1
• When losing, the value is 0, and vi 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 lt ci, and wins
  if ti gt ci ?

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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
  picj 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

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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 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.

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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
• vj -cjWj, where Wj is the total weight of the
  jobs assigned to j-th processor.
• vj -Wj
• HMD requires that Wj would increase if reported
  cost increases, which is equivalent to
  monotonicity condition
  ?

vj
s
t
cj
vj(?t,cj)
vj(?s,cj)
35
Scheduling with Timing Constraints (STC)

• n agents apply to get a service from central
  mechanism
• An agents 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

36
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 lt ci then and if bi gt ci,
  then .

37
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

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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

39
Strategic Limits (Budgets)

• Assume that all the pij (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 t0 such that
  vi(?t0, t0) t0, it holds that for every t1 gt
  t0, ?t1 ?t0.

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Rationalizability

• Theorem Any piecewise monotone allocation rule
  is rationalizable.
• Proof
• Denote by ? the total value of items assigned to
  i-th agent
• For ? fixed
• If ti lt ? vi(?, ti) ti, vi 1
• If ti ? vi(?, ti) ?, vi 0

42
Proof (cont.)

• We prove that piecewise monotonicity leads to
  first HMD condition.
• We need that for any b0 lt b1,
• vi(?b0, x) vi(?b1, x) for every b0 x
• First assume that ?b0 b0.
• For each x gt b0, vi(?b0, x) 0
• and so no constraints are
• induced for vi(?b1, x)

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Proof (end)

• Now if ?b0 b0
• vi(?b0, x) 1 for x ?b0
• To fulfill the first HMD
• condition, for each b1 gt b0,
• ?b1 should be at least ?b0
• This is achieved due to the piecewise
  monotonicity ?