Mechanism Design: Online Auction or Packet Scheduling

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Mechanism Design Online Auction or Packet
Scheduling

• Online auction of a reusable good (packet slots)
• Agents types (arrival, departure, value)
• Agents can lie about value
• Agents can lie about arrival departure
• Restrict to later arrival, earlier departure
• Goals
• Maximize value of agents who receive good
• Maximize revenue generated by auctioneer

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Reminder of previous results

• Upper bound of 2
• Lower bound of f (v5 1)/2 1.618

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Upper bound of 2

• Greedy
• Always send feasible packet with maximum value
• Greedy is 2-competitive
• Come up with a 2 packet instance which gives
  lower bound of 2

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Lower Bound f (v5 1)/2
Figures from Online Scheduling with Partial Job
Values Does Timesharing or Randomization Help?
by Chin and Fung, Algorithmica, 37, 149-164, 2003.
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Mechanism Design Bounds

• Agents can lie about values, arrival time, and
  departure time
• Unbounded
• Can create 3-competitive mechanism using Set Nash
  concept
• Agents can lie about values, arrival time, and
  early departure time
• How could we enforce such a mechanism?
• Bound of 2 exactly

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General Lower Bound

• Lavi and Nisan (SODA 2005)
• Must have restriction on deadline or else cannot
  guarantee bounded competitive ratio
• Key observation
• Consider price pi(b-i) faced by agent i at time 1
• Suppose v(i) lt pi(b-i). Can agent i ever get
  item i?
• Suppose v(i) gt pi(b-i) but doesnt win item 1
• Now have M agents all arrive at time 1 with
  deadlines M and values in the range of (1, 1e).
• Only one agent wins item in first time slot
• Optimal allocation is all agents win an item in
  some slot
• M can be arbitarily large so no bound on
  competitive ratio

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Restricted lower bound of 2

• Hajiaghayi, Kleinberg, Mahdian, Parkes (EC 2005)
  and no 2-e mechanism
• Describe what happens in the following scenarios
• (1,1,infinity) and (1,2,1) (ar, dep, value).
• (1,1,1d) and (1,2,1) (what about price?)
• (1,2,1d) and (1,2,1) and (2,2,infinity)
• (1,2,1d) and (1,1,1) and (2,2,infinity)
• (1,2,1d) and (1,1,1)

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Restricted upper bound of 2

• Based on greedy 2-competitive algorithm
• Allocation
• In each time slot, give item to highest bidder
• Price computation
• Second price auction
• Price can drop in later rounds if it could have
  gotten the item cheaper in a later round
• Example
• (1,2,2), (1,1, 2-e), (2,2,1)

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Variations

• k copies of each item available in each time slot
• Basically the same except the k top bidders in
  each time slot get the item
• Asynchronous time slots
• Item is needed for 1 unit of time but not all
  arrivals/deadlines are at integer time points
• 5 competitive mechanism
• Weight currently running agents value by extra
  2d where d is how long it has run for

10
Set Nash Idea

• Identify a set of recommended strategies for
  all players
• Set-Nash Equilibria A best response to all other
  agents playing a recommended strategy is to
  employ some recommended strategy
• Truthful mechanism set of strategies is 1,
  truthfulness
• Not as powerful as truthful strategy is best
  response to ANY combination of strategies from
  other agents
• Any game set could be all strategies and then
  this is trivially true

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Application

• Japanese auction tradition incremental auction
  (i.e. bids raise by e until there is a winner)
• Sequential Japanese auction use Japanese auction
  at each time t
• Players observe dropouts
• Myopic strategy
• Drop out when price reaches v(i) or when there
  are exactly d(i)-t players that did not drop yet
• Semi-myopic strategy
• Drop no later than when price reaches v(i) and,
  satisfying first condition, no earlier than when
  only d(i)-t players did not drop yet
• If all players employ a semi-myopic strategy,
  3-approximation
• Set-Nash Equilibria The set of semi-myopic
  strategies forms a set Nash equilibrium