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Efficient Contention Resolution Protocols for Selfish Agents Amos Fiat, Joint work with Yishay Mansour 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, the faster we will all Deadlines: “Alright people, listen up. The harder you push, the faster we will all get out of here. ” Tax deadline

Deadline Analysis: 2 Symmetric Agents / 2 Time slots / Service takes 1 time 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 Slot #16 Slot #17 Deadline Bart is polite: With probability q Lisa will get service and depart Bart is aggressive: With probability 1 -q Lisa will be polite and Bart will be successful

2 agents 1 Slot before deadline And Samson said, 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 Slot #17 Deadline

Solving with MATHEMATICA q 20(t): Prob. of aggression when 20 agents are pending as Solving with MATHEMATICA q 20(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

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

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

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

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

Broadcast Channel Slot #1 Slot #2 Slot #3 Slot #4 Slot #5 Slot #6 Broadcast Channel Slot #1 Slot #2 Slot #3 Slot #4 Slot #5 Slot #6 time 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

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

Protocol in Equilibrium Agent utility: Minimize latency Agent strategy: Transmission probability is a function 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

Summary of (Latency) Results 1. All protocols where transmission probabilities do not depend on Summary of (Latency) Results 1. All protocols where transmission probabilities do not depend on the time have exponential latency 2. We give a “time-dependent” protocol where all agents are successful in linear time

Time-Independent Equilibrium Theorem: There is a unique time-independent, symmetric, nonblocking protocol in equilibrium for Time-Independent Equilibrium Theorem: There is a unique time-independent, symmetric, nonblocking 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

Translate Latency Minimization to Deadline Cost Effectively, no message gets through here Deadline T Translate Latency Minimization to Deadline Cost Effectively, no message gets through here Deadline T 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? – Agents go “crazy”: everyone continuously transmits – • Using external payments Time dependence Analyze step cost function (Deadline)

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

Equilibrium Equations (Deadline, Latency, etc. ) = Transmit Quiescence Probability one of the other Equilibrium Equations (Deadline, Latency, etc. ) = Transmit Quiescence Probability one of the other k-1 agents leaves (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 time t

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, Equilibrium Equations k, t( (t+1))+(1 - k, t )Ck, t+1 = k, t Ck-1, t+1 + (1 - k, t ) Ck, t+1 k, t( (t+1)-Ck, t+1) = k, t(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)

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

Analysis of Deadline utility We seek an upper bound for Recall: Cn, 0 = Analysis of Deadline utility We seek an upper bound for Recall: Cn, 0 = Fn, 0 Fk, t = Fk-1, t+1 + (1 - ) Fk, t+1 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

Upper Bound on Cost Two descendants One descendant Transmission probability Fn, t+1 < 2 Upper Bound on Cost Two descendants One descendant Transmission probability Fn, t+1 < 2 Fn-1, t+1 (Fn, t+1 > 2 Fn-1, t+1 ) Fn, t 1 - Fn, t+1 < 0. 8 Fn, t <2 Fn, / t Fn- Fn, t+1 1, t+1 <0 . 3 Fn-1, t+1 Fn, t = Fn-1, t+1 + (1 - ) Fn, t+1 Good edges Fn-1, t+1 Fn, t < Fn, t+1 < 2 Fn-1, t+1 Doubling edges

Upper Bound on Cost Fn, 0 Fn, 1 # Agents Fn-1, 1 Fn-2, 2 Upper Bound on Cost Fn, 0 Fn, 1 # Agents Fn-1, 1 Fn-2, 2 Fn-3, 3 F n-3, 4 L 1 cost=1 Fn-4, 4 F 17, D = 1 F 1, D-9 = 0 Deadline cost=0 Time

Upper Bound on Cost • The weight of such a path: – – • Upper Bound on Cost • The weight of such a path: – – • At least D-n good edges Weight at most (1 -β)D-n 2 n Number of paths at most: Set D > 20 n to get an upper bound of e-c n on cost 1 cost=0

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

Summary • Unique non-blocking equilibrium for Aloha like Protocols – • Deadlines: – • Summary • Unique non-blocking equilibrium for Aloha like Protocols – • Deadlines: – • Exponential latency 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 –

Future Research • General cost functions • Does the time-independent equilibrium induces an optimal 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