Mechanism Design: Online Auction or Packet Scheduling

1. 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

2. Reminder of previous results • Upper bound of 2 • Lower bound of φ = (√5 + 1)/2 ≈ 1.618

3. (3,2) (3,3) (2,3) (4,4) Time: 1, packet (4,4) sent 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

4. Lower Bound: φ = (√5 + 1)/2 Figures from “Online Scheduling with Partial Job Values: Does Timesharing or Randomization Help?” by Chin and Fung, Algorithmica, 37, 149-164, 2003.

5. 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

6. 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) < pi(b-i). Can agent i ever get item i? • Suppose v(i) > pi(b-i) but doesn’t win item 1 • Now have M agents all arrive at time 1 with deadlines M and values in the range of (1, 1+ε). • 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

7. Restricted lower bound of 2 • Hajiaghayi, Kleinberg, Mahdian, Parkes (EC 2005) and no 2-ε mechanism • Describe what happens in the following scenarios • (1,1,infinity) and (1,2,1) (ar, dep, value). • (1,1,1+δ) and (1,2,1) (what about price?) • (1,2,1+δ) and (1,2,1) and (2,2,infinity) • (1,2,1+δ) and (1,1,1) and (2,2,infinity) • (1,2,1+δ) and (1,1,1)

8. 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-ε), (2,2,1)

9. 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 agent’s value by extra 2δ where δ 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

11. Application • Japanese auction: tradition incremental auction (i.e. bids raise by ε 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