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Efficient Contention Resolution Protocols for Selfish Agents

Efficient Contention Resolution Protocols for Selfish Agents. Amos Fiat, Joint work with Yishay Mansour and Uri Nadav Tel-Aviv University, Israel. Workshop on Algorithmic Game Theory, University of Warwick, UK. Deadlines:. “Alright people, listen up. The harder you push,

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Efficient Contention Resolution Protocols for Selfish Agents

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  1. Efficient Contention Resolution Protocols for Selfish Agents Amos Fiat, Joint work with Yishay Mansour and Uri Nadav Tel-Aviv University, Israel Workshop on Algorithmic Game Theory, University of Warwick, UK

  2. Deadlines: “Alright people, listen up. The harder you push, the faster we will all get out of here.” Tax deadline

  3. Deadline Analysis: 2 Symmetric Agents / 2 Time slots / Service takes 1 time slot Both agents are aggressive with prob. q, and polite with prob. 1-q   Deadline Slot #16 Slot #17 Bart is polite: With probabilityqLisa will get service and depart Bart is aggressive: With probability1-qLisa will be polite and Bart will be successful

  4. 2 agents 1 Slot before deadline And Samson said, "Let me die with the Philistines!" Judges 16:30 Let Lisa be polite with prob. q If Bart is: • polite - cost is 1 • aggressive - expected cost is q Aggression is dominant strategy Deadline Slot #17

  5. Solving with MATHEMATICA q20(t): Prob. of aggression when 20 agents are pending as a function of the time t , in equilibrium “Aggression” Probability 19 Blocking no one gets served 0.05 Time deadline

  6. Solving with MATHEMATICA qk(4k): “Aggression” prob. when k agents are pending before deadline in 4k time slots (Deadline: when lunch trays are removed at U. Warwick, CS department) # agents

  7. Deadline Cost – Few slots Theorem: In a symmetric equilibrium, whenever there are more agents than time slotsuntil deadline,agents transmit (transmission probability 1)

  8. Efficiency of a linear deadline Theorem: There exists a symmetric equilibrium for D-deadline cost function such that: if the deadline D > 20n then, the probability that not all agents succeed prior to the deadline is negligible (e-cD) If there is enough time for everyone, a “nice” equilibrium

  9. Switch Subject: Broadcast Channel / Latency • n agents (with a packet each) at time 0 • No arrivals • Known number of agents time Slot #1 Slot #5 Slot #2 Slot #6 Slot #3 Slot #4

  10. Broadcast Channel time Slot #1 Slot #5 Slot #2 Slot #6 Slot #3 Slot #4 Transmission probability 1/n is not in equilibrium • Symmetric solution: every agent transmits with probability 1/n, the expected waiting time is O(n) slots. (Social optimum) • If all others transmit with probability 1/n, agent is better off transmitting all the time and has constant latency

  11. Related Work: Strategic MAC (Multiple Access Channel) • [Altman et al 04] • Incomplete information: number of agents • Stochastic arrival flow to each source • Restricted to a single retransmission probability • Shows the existence of an equilibrium • Numerical results • [MacKenzie & Wicker 03] • Multi-packet reception • Transmission cost [due to power loss] • Characterize the equilibrium and its stability • Also [Gang, Marbach & Yuen]

  12. Protocol in Equilibrium Agent utility: Minimize latency Agent strategy: Transmission probability is a function of the number of pending agents k and current waiting time t Protocol in equilibrium: No incentive not to follow protocol Symmetry: All agents are symmetric

  13. Summary of (Latency) Results • All protocols where transmission probabilities do not depend on the time have exponential latency • We give a “time-dependent” protocol where all agents are successful in linear time

  14. Time-Independent Equilibrium Theorem: There is a unique time-independent, symmetric, non-blocking protocol in equilibrium for latency cost with transmission probabilities: Very high “Price of Anarchy” • Expected Delay of the first transmitted packet: • Probability even one agent successful within polynomial time bound is negligible • Compare to social optimum: • All agents successful in linear time bound, with high probability

  15. T Deadline Translate Latency Minimization to Deadline Effectively, no message gets through here Cost Time • Fight for every slot • Cooperation is more important when trying to avoid a large payment (deadline) • How can one create a sudden jump in cost? • Using external payments • Agents go “crazy”: everyone continuously transmits • Time dependence • Analyze step cost function (Deadline)

  16. Deadline Cost Function Cost D (Deadline) Time Deadline utility (scaled): • Success before deadline – cost 0 • Success after deadline – cost 1

  17. Equilibrium Equations (Deadline, Latency, etc.) Probability one of the other k-1 agents leaves = Quiescence Transmit (t+1) +(1- ) Ck,t+1  Ck-1,t+1+ (1 - ) Ck,t+1 = Probability the other k-1 agents are silent * Ck,t = expected cost of k agents at time t (t)= cost of leaving at timet

  18. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  19. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1= k,t Ck-1,t+1 + (1- k,t )Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  20. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  21. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  22. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  23. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  24. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1 –Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  25. Equilibrium Equations k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1) k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1 (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1) (1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1) (1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1) (1-qk,t)Fk,t+1 = (k-1)qk,t(Fk,t+1-Fk-1,t+1)

  26. Transmission probability when k players at time t Transmission Probability in Equilibrium Lemma (Manipulating equilibrium equations): Benefit from losing one agent 2/k > <1/2 1/k < > 1/2 >0 Observation: • Either transmission probability in [1/k,2/k] • Or, limited benefit from loosing one agent *Fk,t = Ck,t - (t) ; expected future cost Ck,t = expected cost of k agents at time t

  27. Analysis of Deadline utility We seek an upper bound forCn,0 = Fn,0 Fk,t = Fk-1,t+1 + (1- ) Fk,t+1 Recall: Observation: • Either transmission probability in [1/k,2/k] • Or, limited benefit from getting rid of one agent Consider a tree of recursive computation for Fn,0

  28. Fn,t Fn,t+1 Fn,t Fn-1,t+1 Fn-1,t+1 Fn,t+1 Fn,t / Fn-1,t+1 Upper Bound on Cost One descendant Two descendants Fn,t+1 < 2 Fn-1,t+1 Transmission probability (Fn,t+1 > 2 Fn-1,t+1 ) < 2 1- < 0.8  < 0.3 Fn,t =  Fn-1,t+1 + (1-) Fn,t+1 Fn,t < Fn,t+1 < 2 Fn-1,t+1 Good edges Doubling edges

  29. Fn,0 Fn,1 Fn-1,1 Fn-4,4 Fn-2,2 Fn-3,3 Fn-3,4 L1 cost=1 F1,D-9 = 0 cost=0 Upper Bound on Cost #Agents F17,D = 1 Deadline Time

  30. 1 cost=0 Upper Bound on Cost • The weight of such a path: • At least D-n good edges • Weight at most (1-β)D-n2n • Number of paths at most: Set D > 20n to get an upper bound of e-c n on cost

  31. Protocol Design: from Deadline to Latency Embed artificial deadline into “deadline” protocol Deadline Protocol: • Before time 20n transmission probability as in equilibrium • If not transmitted until 20n: • Set transmission probability = 1 (blocking) • For exponential number of time slots • Equilibrium • Sub-game perfect equilibrium • Social optimum achieved with high probability

  32. Summary • Unique non-blocking equilibrium for Aloha like Protocols • Exponential latency • Deadlines: • If enough (linear) time, equilibrium is “efficient” • Protocol Design: • Make “ill behaved” latency cost act more “polite” • Using virtual deadlines • No monetary “bribes” or penalties

  33. Future Research • General cost functions • Does the time-independent equilibrium induces an optimal expected latency? • Protocol in equilibrium for an arrival process • Arrival times / duration in general congestion games: • Atomic traffic flow: don’t leave home until 9:00 AM and get to work earlier

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