SECOND PART:

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• SECOND PART
• Algorithmic Mechanism Design

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Suggested readings

• Algorithmic Game Theory, Edited by Noam Nisan,
  Tim Roughgarden, Eva Tardos, and Vijay V.
  Vazirani, Cambridge University Press.
• Algorithmic Mechanism Design for Network
  Optimization Problems, Luciano Gualà, PhD Thesis,
  Università degli Studi dellAquila, 2007.
• Web pages by Éva Tardos, Christos Papadimitriou,
  Tim Roughgarden, and then follow the links
  therein

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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.

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The implementation problem

• Given
• An economic system comprising of self-interested,
  rational agents, which hold some secret
  information
• A set of system-wide goals
• Question
• Does there exist a mechanism that can implement
  the goals, by enforcing (through suitable
  economic incentives) the agents to cooperate with
  the system by revealing their private data?

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Mechanism Design

• Informally, designing a mechanism means to define
  a game in which a desired outcome must be reached
• However, games induced by mechanisms are
  different from games in standard form
• Players hold independent private values
• The payoff matrix is a function of these types
• ? each player doesnt really know about the other
  players payoffs, but only about its one!
• ? Games with incomplete information
• ? Dominant Strategy Equilibrium is used

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Mechanism Design Problem ingredients

• N agents each agent has some private information
  ti?Ti (actually, the only private info) called
  type
• A set of feasible outcomes F
• For each vector of types t(t1, t2, , tN), a
  social-choice function f(t)?F specifies an output
  that should be implemented (the problem is that
  types are unknown)
• 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

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Example the Vickrey Auction

• Assume that the system-wide goal is to allocate a
  job by a sealed-bid auction. The set of feasible
  outcomes is given by all the bidders. The
  social-choice function is to allocate to the
  bidder with lowest true cost
• f(t)arg mini (t1, t2, , tN)
• Each agent knows its cost for doing the job
  (type), but not the others one
• Ti 0, ? The agents cost may be any positive
  amount of money
• ti 80 Minimum amount of money the agent i is
  willing to be paid
• ri 85 Exact amount of money the agent i bids to
  the system for doing the job (not known to other
  agents)

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Mechanism Design Problem ingredients (2)

• For each feasible outcome x?F, each agent makes a
  valuation vi(ti,x) (in terms of some common
  currency), expressing its preference about that
  output
• Vickrey Auction If agent i wins the auction then
  its valuation is equal to its actual costti for
  doing the job, otherwise it is 0
• For each feasible outcome x?F, each agent
  receives a payment pi(x) in terms of the common
  currency payments are used by the system to
  incentive agents to be collaborative. Then, the
  utility of outcome x will be
• ui(ti,x) pi(x) - vi(ti,x)
• Vickrey Auction If agents cost for the job is
  80, and it gets the contract for 100 (i.e., it is
  paid 100), then its utility is 20

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Mechanism Design Problem the goal

• Implement (according to a given equilibrium
  concept) the social-choice function, i.e.,
  provide a mechanism Mltg(r), p(x)gt, where
• g(r) is an algorithm which computes an outcome
  xx(r) as a function of the reported types r
• p(x) is a payment scheme specifying a payment
  w.r.t. an output x
• such that xf(t) is provided in equilibrium
  w.r.t. to the utilities of the agents.

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Mechanism Design a picture

Private types
Reported types
Output which should implement the social choice
function
Mechanism
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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Game induced by a MD problem

• This is a game in which
• The N agents are the players
• The payoff matrix is given (in implicit form) by
  the utility functions

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Implementation with dominant strategies

• Def. A mechanism is an implementation with
  dominant strategies if there exists a reported
  type vector r(r1, r2, , rN) such that
  f(t)x(r) in dominant strategy equilibrium,
  i.e., 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).

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Mechanism Design Economics Issues

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

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Strategy-Proof Mechanisms

• If truth telling is the dominant strategy in a
  mechanism then it is called Strategy-Proof
• Agents report their true types instead of
  strategically manipulating it
• Utilitarian Problems A problem is utilitarian if
  its objective function is such that f(t) ?i
  vi(ti,x)
• The Vickrey Auction is utilitarian

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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 minx?F ?i vi(ri,x)
• Payment function
• pi (x) hi(r-i) - ?j?i vj(rj,x)
• where hi(r-i) is an arbitrary function of the
  types of other players
• What about non-utilitarian problems? We will see

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VCG-Mechanisms are Strategy-Proof

• Proof (Intuitive sketch)
• Payment given to agent i
• pi (x) hi(r-i)-?j?i vj(rj,x)
• and both the terms above are independent of the
  type, strategy and valuation of agent i
• So it is best for agent i to report its true
  value. Strategic behavior does not lead to a
  beneficial outcome.

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Clarke Mechanisms

• This is a special VCG-mechanism (known as Clarke
  mechanism) in which
• hi(r-i)?j?i vj(rj,x(r-i))
• pi ?j?i vj(rj,x(r-i)) -?j?i vj(rj,x)
• In Clarke mechanisms, agents utility are always
  non-negative

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Clarke mechanism for the Vickrey auction

• The VCG-mechanism is
• xarg minx?F ?i vi(ri,x) ? allocate to the bidder
  with lowest reported cost
• pi ?j?i vj(rj,x(r-i)) -?j?i vj(rj,x) ? pi
  ?j?i vj(rj,x(r-i)) ? pay the winner the second
  lowest offer, and pay 0 the losers
• Let us convince ourself it is strategy-proof by
  case analysis. For a player i, let Tminj?i rj
• tiltT then, if riltti, he still wins, but he keeps
  on to be paid T, while if rigtti, he may still win
  (again being paid T), but he may also lose (if
  rigtT), by getting a null utility
• tigtT then, if rigtti, he keeps on not to win,
  while if riltti, he may win, but he will be paid
  Tltti, by getting a negative utility.
• Remark the difference between the second lowest
  offer and the lowest 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
• System has to calculate N1 functions
• Once with all agents (for g(r)) 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 compute
  the underlying social-choice function?

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Algorithmic mechanism design for graph problems

• Following 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
• ? The agents type is the true weight of the edge
• Classic optimization problems on G become
  mechanism design optimization problems!
• Many basic network design problems have been
  faced shortest path (SP), single-source shortest
  paths tree (SPT), minimum spanning tree (MST),
  minimum Steiner tree, and many others

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Summary of forthcoming 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 problems, the time complexity of
the mechanism equals that of the canonical
centralized algorithm!