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

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 p i (b -i ) faced by agent i at time 1 Suppose v(i) < p i (b -i ). Can agent i ever get item i? Suppose v(i) > p i (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.(1,1,infinity) and (1,2,1) (ar, dep, value). 2.(1,1,1+δ) and (1,2,1) (what about price?) 3.(1,2,1+δ) and (1,2,1) and (2,2,infinity) 4.(1,2,1+δ) and (1,1,1) and (2,2,infinity) 5.(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 2 δ 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