Mechanism Design

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Mechanism Design
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Overview

• Incentives in teams (T. Groves (1973))
• Algorithmic mechanism design (Nisan and Ronen
  (2000))
• - Shortest Path
• - Task Scheduling

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Framework

• Something needs to be done with the help of n
  agents
• Is there a way of inducing them to do it (might
  lack knowledge or control)
• The way if it exists is called a mechanism
• Assumption1 The agents are rational
• Assumption2 The agents are independent (no
  communication)
• The mechanism is said to be truthful if there is
  no incentive for an agent to lie
• A lie is defined as something the agent could do
  so that the goal is not achieved.

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An Organization
CEO

Sub-unit 1
Sub-unit 2
Sub-unit n
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Pay-Fire Incentive
-Optimally performing employees are rewarded -Pay
is independent of how other employees
perform -Assumes that the CEO has complete
information
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An Organization
CEO

Sub-unit 1
Sub-unit 2
Sub-unit n
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Own Profit Incentive
-- Payment to player i is independent of the
decisions of the others --But it is dependent on
the messages --Why is there no advantage in lying
?
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Profit Sharing
--It is hard to remove message dependence without
losing truthfulness --Truthful mechanism Nobody
has incentive to lie --Strongly truthful
mechanism Truth telling is the only dominant
strategy --Dominant strategy No unilateral
incentive to deviate
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Direct Revelation Mechanisms

• The message strategy space and state space (t)
  are the same
• m(x(t),p(t))
• x(t) is a set of feasible outputs given t
• p(t) is a vector of payments to the agents
• g(t,x(t)) is the function to optimize
• m(x(t),p(t)) is a c-approximation for
  m(x(t),p(t)) if g(t,x(t))lt c . g(t,x(t))

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VGC mechanisms

• VGC (Vickrey-Groves-Clarke)
• VGC mechanisms are truthful
• x(t) is feasible iff it maximizes g (so that we
  concern ourselves with providing the correct
  incentive structure.)

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Shortest Path

• Each edge is an agent
• People want to send messages to other people
• People are at vertices
• Goal is to minimize cost
• Each edge has a cost
• Payment to each edge

Complexity is O(m n log(m))
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Task Scheduling

• k tasks
• n processors
• State of agent i
• Goal is to minimize the completion time of the
  set of tasks (make-span)
• A task need not go to the agent that does it the
  fastest.

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Min-Work Mechanism
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Min-Work (contd.)

• Min-Work is truthful
• Nisan and Ronen show it is strongly truthful
• Min-Work is an n-approximation for make-span

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Bounds on approximations
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Proof Sketch
T1 T2
1 1
1 1
1 1
1 1
T1 T2
e 1
e 1
1e 1
1e 1
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Randomized Mechanisms

• A probability distribution over a family of
  mechanisms that share the same set of strategies
  and outputs
• Optimize the GE(g)
• Payments etc. are defined as expectations over
  payments

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Randomly-Biased Min-Work
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We will first show that the mechanism is truthful.
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Weighted VGC Mechanisms
The mechanism is truthful.
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Proof Sketch
T1 T2 Opt Rbmw
1 (be) 1 1
1 (be) 2 1
1 b 1 rnd
b 1 2 rnd
g(t,opt(t))1 b e 1 4/3 7/3 , 7/4
g(t,opt(t))49/12 g(t,rbmw(t))1/4((1 1 1
b) (1 1 b) (1 1 1) (b1))
1/4(93b) 13/4
3.25 lt 49/12
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Mechanisms with Verification

• Assumption Agents actions can be verified
• Routing, Task scheduling etc.
• Check the effect of such a simplifying assumption
  both on mechanism design and computation

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Make-span with Verification
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Generalized Compensation and Bonus Mechanisms
---Participation and Bonus Constraints
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Computational Problems

• Exponential-time allocation algorithm
• Approximations tend to violate truthfulness (will
  discuss a theorem from Nisan and Ronen)
• If the no. of agents are fixed, and declarations
  are bounded a truthful polynomial time
  approximation mechanism exists. (Computing the
  exact solution is NP-hard)

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Bounded Scheduling Problems
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Rounding Mechanism

• Compensation using actual times
• Bonus using rounded times.
• All revelations that are rounded up to the same
  value as the true revelations are dominant
  strategies.

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Extensions

• Repeated games
• e-dominant strategies
• Partial verification