SECOND PART:

1

• SECOND PART
• Algorithmic Mechanism Design

2
Implementation theory

• Imagine a planner who develops criteria for
  social welfare, but cannot enforce the desirable
  allocation directly, as he lacks information
  about several parameters of the situation. A mean
  then has to be found to implement such
  criteria.
• Subfield of economic theory with an engineering
  perspective Designs economic mechanisms like
  computer scientists design algorithms, protocols,
  systems,

3
The implementation problem

• Given
• An economic system comprising of self-interested,
  rational agents, which hold some private
  preferences
• A system-wide goal (social-choice function)
• Question
• Does there exist a mechanism that can enforce
  (through suitable economic incentives) the
  selfish agents to behave in such a way that the
  desired goal is implemented?

4
An example auctions
Social-choice function the winner should be the
guy having in mind the highest value for the
painting
t110
r111
ri is the amount of money player i bids (in a
sealed envelope) for the painting
t212
r210
t37
r37

• The mechanism tells to players
• How the item will be allocated (i.e., who will be
  the winner)
• (2) The payment the winner has to return, as a
  function of the received bids

ti is the maximum amount of money player i is
willing to pay for the painting
If player i wins and has to pay p his utility is
uiti-p
5
A simple mechanism no payment
t110
r1?
?!?
t212
r2?
t37
r3?
Mechanism The highest bid wins and the price of
the item is 0
it doesnt work
6
Another simple mechanism pay your bid
Is it the right choice?
t110
The winner is player 1 and hell pay 9
r19
t212
r28
t37
r36
Mechanism The highest bid wins and the winner
will pay his bid
Player i may bid rilt ti (in this way he is
guaranteed not to incur a negative utility)
and so the winner could be the wrong one
it doesnt work
7
An elegant solution Vickreys second price
auction
I know they are not lying
t110
The winner is player 2 and hell pay 10
r110
t212
r212
t37
r37
Mechanism The highest bid wins and the winner
will pay the second highest bid
every player has convenience to declare the
truth! (we prove it in the next slide)
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Theorem
In the Vickrey auction, for every player i, riti
is a dominant strategy
proof
Fix i, ti, r-i, and look at strategies for player
i. Let R maxj?i rj
Case ti R
Declaring riti gives utility ui ti-R 0
(player wins)
declaring ri gt ti R gives the same utility
(player wins)
declaring R lt ri lt ti gives the same utility
(player wins)
declaring ri lt R ti yields ui0 (player loses)
Case ti lt R
Declaring riti gives utility ui 0 (player
loses)
declaring ri lt ti lt R gives the same utility
(player loses)
declaring ti lt ri lt R gives the same utility
(player loses)
declaring ri gt R gt ti yields ui ti-R lt 0 (player
wins)
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Mechanism Design Problem ingredients

• N agents each agent i, i1,..,N, has some
  private information ti?Ti (actually, the only
  private info) called type
• Vickreys auction the type is the painting value
  the agents have in mind, and so Ti is the set of
  positive reals
• A set of feasible outcomes X
• Vickreys auction X is the set of agents
  (bidders)

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Mechanism Design Problem ingredients (2)

• For each vector of types t(t1, t2, , tN), and
  for each feasible outcome x?X, a social-choice
  function f(t,x) measures the quality of x as a
  function of t. This is the function that the
  mechanism aims to implement (i.e., it aims to
  select an outcome x that minimizes/maximizes
  it), but the problem is that types are unknown!
• Vickreys auction f(t,x) is the type associated
  with x (namely, a bidder x), and the objective is
  to maximize f, i.e., allocate to the bidder with
  highest type
• Each agent has a strategy space Si and performs a
  strategic action we restrict ourself to direct
  revelation mechanisms, in which the action is
  reporting a value ri from the type space (with
  possibly ri ? ti), i.e., Si Ti
• Vickreys auction the action is to bid a value
  ri

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Mechanism Design Problem ingredients (3)

• For each feasible outcome x?X, each agent makes a
  valuation vi(ti,x) (in terms of some common
  currency), expressing its preference about that
  output x
• Vickreys auction if agent i wins the auction
  then its valuation is equal to its type ti,
  otherwise it is 0
• For each feasible outcome x?X, each agent
  receives/gives (this depends on the problem) a
  payment pi(x) in terms of the common currency
  payments are used by the system to incentive
  agents to be collaborative.
• Vickreys auction if agent i wins the auction
  then he returns a payment equal to -rj, where rj
  is the second highest bid, otherwise it is 0
• Then, for each feasible outcome x?X, the utility
  of agent i coming from outcome x will be
• ui(ti,x) pi(x) vi(ti,x)
• Vickreys auction if agent i wins the auction
  then its utility is equal to
• ui -rjti 0, where rj is the second highest
  bid, otherwise it is ui 000

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Mechanism Design Problem the goal

• Given all the above ingredients, design a
  mechanism Mltg(r), p(x)gt, where
• g(r) is an algorithm which computes an outcome
  xx(r)?X as a function of the reported types r
• p(x) (p1,,pN) is a payment scheme w.r.t.
  outcome x specifying a payment value for each
  player
• which implements the social-choice function f in
  equilibrium (according to a given solution
  concept, e.g., dominant strategy equilibrium,
  Nash equilibrium, etc.), w.r.t. players
  utilities.

(In other words, there exists a reported type
vector r for which the mechanism provides a
solution x(r) and a payment scheme p(x(r)) such
that players utilities ui(ti, x(r)) pi(x(r))
vi(ti, x(r)) are in equilibrium, and f(t,
x(r)) is optimal (either minimum or maximum))
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Mechanism Design a picture
Output which should implement the social choice
function in equilibrium w.r.t. agents utilities

• System

Private types
Reported types
I propose to you the following mechanism
Mltg(r), pgt
t 1
r 1
p1
Agent 1
tN
r N
pN
Agent N
Payments
Each agent reports strategically to maximize its
well-being
in response to a payment which is a function of
the output!
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Implementation with dominant strategies

• Def. A mechanism is an implementation with
  dominant strategies if the s.c.f. f is
  implemented in dominant strategy equilibrium,
  i.e., there exists a reported type vector
  r(r1, r2, , rN) such that for each agent i
  and for each reported type vector r (r1, r2, ,
  rN), it holds
• ui(ti,x(r-i,ri)) ui(ti,x(r))
• where x(r-i,ri)x(r1, , ri-1, ri, ri1,, rN),
  and f(t,x(r)) is optimized.

15
Strategy-Proof Mechanisms

• If truth telling is the dominant strategy in a
  mechanism then it is called Strategy-Proof or
  truthful ? rt.
• ? Agents report their true types instead of
  strategically manipulating it
• ? The algorithm of the mechanism runs on the
  true input

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Truthful Mechanism Design Economics Issues

• QUESTION How to design a truthful mechanism? Or,
  in other words
• How to design g(r), and
• How to define the payment scheme
• in such a way that the underlying social-choice
  function is implemented truthfully? Under which
  conditions can this be done?

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A prominent class of problems

• Utilitarian problems A problem is utilitarian if
  its social-choice function is such that f(t,x)
  ?i vi(ti,x)

notice the auction problem is utilitarian, and
so they are all problems where the s.c.f. is
separately-additive w.r.t. agents valuations, as
in many network optimization problems
Good news for utilitarian problems there is a
class of truthful mechanisms ?
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Vickrey-Clarke-Groves (VCG) Mechanisms

• A VCG-mechanism is (the only) strategy-proof
  mechanism for utilitarian problems
• Algorithm g(r) computes
• x arg maxy?X ?i vi(ri,y)
• Payment function for player i
• pi (x) hi(r-i) ?j?i vj(rj,x)
• where hi(r-i) is an arbitrary function of the
  reported types of players other than player i.
• What about non-utilitarian problems?
  Strategy-proof mechanisms are known only when the
  type is a single parameter.

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Theorem
VCG-mechanisms are truthful for utilitarian
problems
proof
Fix i, r-i, ti. Let r(r-i,ti) and consider a
strategy ri?ti
xg(r-i,ti) g(r) xg(r-i,ri)
?jvj(rj,x)
hi(r-i)
?j?ivj(rj,x) vi(ti,x)
hi(r-i)
ui(ti,x)
ui(ti,x)
?jvj(rj,x)
hi(r-i)
?j?ivj(rj,x) vi(ti,x)
hi(r-i)
but x is an optimal solution w.r.t. r (r-i,ti),
i.e., x arg maxy?X ?i vi(r,y)
ui(ti,x) ? ui(ti,x).
?jvj(rj,x)
?jvj(rj,x)
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How to define hi(r-i)?
Remark not all functions make sense. For
instance, what happens in our example of the
Vickrey auction if we set for every agent
hi(r-i)-1000 (notice this is independent of
reported value ri of agent i, and so it obeys the
definition)? Answer It happens that players
utility become negative more precisely, the
winners utility is ui(ti,x) pi(x) vi(ti,x)
hi(r-i) ?j?i vj(rj,x) vi(ti,x) -1000012
-988 while utility of losers is ui(ti,x)
pi(x) vi(ti,x) hi(r-i) ?j?i vj(rj,x)
vi(ti,x) -1000120 -988 ?This is
undesirable in reality, since with such
perspective agents would not partecipate to the
auction!
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The Clarke payments
solution maximizing the sum of valuations when i
doesnt play

• This is a special VCG-mechanism in which
• hi(r-i)-?j?i vj(rj,x(r-i))
• pi(x) -?j?i vj(rj,x(r-i)) ?j?i vj(rj,x)
• With Clarke payments, one can prove that agents
  utility are always non-negative
• ? agents are interested in playing the game

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The Vickreys auction is a VCG mechanism with
Clarke payments

• Observe that auctions are utilitarian problem.
  Then, the VCG-mechanism associated with the
  Vickreys auction is
• xarg maxy?X ?i vi(ri,y)
• this is equivalent to allocate to the bidder
  with highest reported cost (in the end, the
  highest type, since it is strategy-proof)
• pi -?j?i vj(rj,x(r-i)) ?j?i vj(rj,x)

this is equivalent to say that the winner pays
the second highest offer, and the losers pay 0,
respectively
Remark the difference between the second highest
offer and the highest offer is unbounded
(frugality issue)
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VCG-Mechanisms Advantages

• For System Designer
• The goal, i.e., the optimization of the
  social-choice function, is achieved with
  certainty
• For Agents
• Agents have truth telling as the dominant
  strategy, so they need not require any
  computational systems to deliberate about other
  agents strategies

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VCG-Mechanisms Disadvantages

• For System Designer
• The payments may be sub-optimal (frugality)
• Apparently, the system may need to run the
  mechanisms algorithm N1 times once with all
  agents (for computing the outcome x), and once
  for every agent (for the associated payment)
• ? If the problem is hard to solve then the
  computational cost may be very heavy
• For Agents
• Agents may not like to tell the truth to the
  system designer as it can be used in other ways.

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Mechanism Design Algorithmic Issues

• QUESTION What is the time complexity of the
  mechanism? Or, in other words
• What is the time complexity of g(r)?
• What is the time complexity to calculate the N
  payment functions?
• What does it happen if it is NP-hard to implement
  the underlying social-choice function?

Question What is the time complexity of the
Vickrey auction? Answer T(N), where N is the
number of players. Indeed, it suffices to check
all the offers, by keeping track of the lowest
and second lowest one.
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Algorithmic mechanism design and network
protocols

• Large networks (e.g., Internet) are built and
  controlled by diverse and competitive entities
• Entities own different components of the network
  and hold private information
• Entities are selfish and have different
  preferences
• ? MD is a useful tool to design protocols working
  in such an environment, but time complexity is an
  important issue due to the massive network size

27
Algorithmic mechanism design for network
optimization problems

• Simplifying the Internet model, we assume that
  each agent owns a single edge of a graph G(V,E),
  and establishes the cost for using it
• ? Classic optimization problems on G become
  mechanism design optimization problems in which
  the agents type is the weight of the edge!
• Many basic network design problems have been
  faced shortest path (SP), single-source shortest
  paths tree (SPT), minimum spanning tree (MST),
  and many others

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Some remarks

• In general, network optimization problems are
  minimization problems (the Vickreys auction was
  instead a maximization problem)
• However, we have
• for each x?X, the valuation function vi(ti,x)
  represents a cost incurred by player i in the
  solution x (and so it is a negative function of
  its types)
• the social-choice function f(t,x) is negative
  (since it is the sum of negative functions), and
  so its maximization (as computed by the algorithm
  of the VCG-mechanism) corresponds to a
  minimization of the sum of absolute values of the
  valuation functions
• payments are from the mechanism to agents

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Summary of main results
Centralized algorithm Selfish-edge mechanism
SP O(mn log n) O(mn log n)
SPT O(mn log n) O(mn log n)
MST O(m ?(m,n)) O(m ?(m,n))
? For all these basic problems, the time
complexity of the mechanism equals that of the
canonical centralized algorithm!
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Exercise redefine the Vickrey auction in the
minimization version
I want to allocate the job to the true cheapest
machine
The winner is machine 3 and it will receive 10
t110
r110
t212
job to be allocated to machines
r212
t37
r37
Once again, the second price auction works the
cheapest bid wins and the winner will get the
second cheapest bid
ti cost incurred by i if he does the job
if machine i is selected and receives a payment
of p its utility is p-ti