Mechanism Design

1
Mechanism Design

• Traditional Algorithmic Setting
• Mechanism Design Setting

2
Shortest Path Problem

• Traditional Formulation
• Input
• Directed graph G (V,A)
• Two special nodes s,t
• Weight on each arc w(u,v)
• Output
• Shortest path p from s to t
• Solvable in polynomial time

3
Selfish Agent View

• Suppose each edge is a link in the internet and
  is controlled by separate entities
• Further suppose the weights are the costs for the
  edge to transmit a message from s to t
• What incentive does an agent have to transmit the
  message?
• Well pay them to send a message

4
Payments

• How much should we pay an edge that is not used?
• How much should we pay an edge that is used?
• What if the edge lies about its cost to transmit
  the message across its link?

1
10
3
8.9
s
t
10
1
5
Mechanism Design Formulation

• Traditional Formulation
• Input
• Directed graph G (V,A)
• Two special nodes s,t
• Weight on each arc w(e)
• Output
• Shortest path p from s to t
• Solvable in polynomial time

• New Formulation
• Input
• Graph G (V,A)
• Assumption Biconnected
• Special nodes s,t
• Arc agents with true costs w(e)
• Gather phase
• Agents report costs w(e)
• Mechanism
• Chooses s-t path
• Computes payments to each arc agent
• Algorithms for path computation and payments are
  known to agents

6
Mechanisms

• New Formulation
• Input
• Graph G (V,A)
• Assumption Biconnected
• Special nodes s,t
• Arc agents with true costs w(e)
• Gather phase
• Agents report costs w(e)
• Mechanism
• Chooses s-t path
• Computes payments to each arc agent
• Algorithms for path computation and payments are
  known to agents

• Path choice mechanism
• Find shortest path given reported costs using
  traditional s-t shortest path algorithm
• Payment mechanism
• For each edge e, compute the following
• 0 if arc is not used
• dGw(e) cost of shortest path without using
  arc e
• dGw(e) 0 cost of shortest path with arc e
  assuming arc e costs 0
• p(e) dGw(e) - dGw(e) 0
• Any incentive for e to lie?

7
Mechanism Design

• Input
• Agents with selfish interests and private values
• Other characteristics which are known
• Gather phase
• Agents report type to mechanism
• Output phase
• Provide solution to problem
• Provide payments to (or collect payments from)
  agents

8
Notation

• ti is true type of agent i (private value)
• ai is type reported by agent i to mechanism
• a is the vector of all agent types reported
• a-i is the vector of types reported minus ai
• o(a) outcome of mechanism
• vi(o,ti) is value of outcome o to agent i
• pi(a) payment to agent i
• u(i) pi(a) vi(o,ti) is utility of total
  outcome to i

9
Mechanism Properties

• Dominant strategy
• If for each agent i, there exists a reported
  value ai such that for all possible reported
  values of other agents a-i, u(i) is maximized
• We assume that agents are rational and thus
  employ dominant strategies
• Truthful (strategyproof) mechanism
• For each agent, reporting true private value is a
  dominant strategy
• Strongly truthful mechanism
• For each agent, reporting true private value is
  the only dominant strategy

10
Mechanism Goals

• Rationality assumption
• We assume that agents are rational and thus
  employ dominant strategies
• Truthful mechanism
• We can assume that the agents reveal true values
• Output optimization
• Create a solution, assuming truthful values,
  where value is optimal or approximately optimal
• Price optimization
• Minimize amount paid to agents or maximize amount
  collected from agents
• Output and price computations
• Should be done in polynomial time

11
Vickrey-Clarke-Groves (VCG) Mechanisms

• Objective function
• Summation of all agents valuation functions
• S vi(o,ti)
• Creates optimal output assuming truthful values
• Payment calculation
• pi(a) Sj?ivj(o,tj) hi(a-i) where hi() is an
  arbitrary function of a-i
• Key point pi(a) is not dependent on ai

12
Notes to add

• Monotone fcts
• Price of Anarchy
• nash equilibrium
• Price of Stability
• Drawbacks of VCG
• The VCG framework is a general method for
  creating truthful mechanisms.  We address the
  following four drawbacks  (1) VCG selects the
  outcome that maximizes the total social welfare,
  whereas often the decision-maker wants to
  maximize some other function. (2) In the case
  where the decision-maker is purchasing something,
  VCG must sometimes pay an unacceptably high
  premium to induce truthtelling.  (3) Sometimes
  the decision-maker would like to use the VCG
  mechanism, but cannot because computing it is
  NP-hard. (4) VCG resists manipulation by single
  agents, but, in general, multiple agents could
  collude to cheat the mechanism. Quote from Aaron
  Archer