CSCI 3210: Computational Game Theory Inefficiency of Equilibria - PDF document

12/5/19 CSCI 3210: Computational Game Theory Inefficiency of Equilibria & Routing Games Ref: Ch 17, 18 [AGT] Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan 1 Split or steal game u NE outcome vs. socially best/optimal outcome Lucy Payoff Split Steal matrix $0+fr., Tony $33K, $33K Split $66K $66K, $0, $0 Steal $0+fr. 2 1

12/5/19 Prisoner's "dilemma" game u Again: NE outcome vs socially optimal outcome Suspect 2 Payoff Not Confess Confess matrix Not 1, 1 10, 0 Suspect 1 Confess 0, 10 5, 5 Confess Costs (negative of payoffs) 3 Measuring the inefficiency of NE u What is the objective function to compare different outcomes? u Utilitarian u Egalitarian u How to deal with multiplicity of NE? u Inefficiency of which NE? u Price of anarchy vs. price of stability 4 2

12/5/19 Price of Anarchy (PoA) u PoA = Worst objective function value among all NE Objective function value of optimal outcome 5 Price of Stability (PoS) u PoS = Best objective function value among all NE Objective function value of optimal outcome 6 3

12/5/19 Example u Calculate PoA and PoS Suspect 2 Payoff Not Confess Confess matrix Not 1, 1 10, 0 Suspect 1 Confess 0, 10 5, 5 Confess Costs (negative of payoffs) 7 Example u Calculate PoA and PoS Column player Payoff L R matrix 21, -1 10, 0 U Row player 100, 10 7, 8 D Costs (negative of payoffs) 8 4

12/5/19 PoA vs. PoS u Consider costs u PoA and PoS will be >= 1 u PoA = PoS when all NE have the same cost (e.g., unique NE) u In general, PoA >= PoS 9 PoA vs. PoS u PoA: worst case guarantee in a system of independent agents u PoS: measures benefit of a protocol or proposed outcome 10 5

12/5/19 Pigou’s Example PoA and PoS 11 Routing Games 13 6

12/5/19 Model: nonatomic selfish routing u Multicommodity flow network u Directed network with multiple (source, sink) pairs u Each (source, sink) pair is called a commodity u r i amount of traffic for each commodity i u Each edge e has a delay or cost function c e u Every car going through an edge gets same delay u Cost of a path = sum of edge costs u Note: cost doesn't depend on identity of players u Congestion games 14 Equilibrium flow u Let f be a feasible flow (combining all commodities) u f is equilibrium flow if u All detours have higher (or equal) delay equilibrium flow 15 7

12/5/19 More complex graphs s 1 t 1 s 2 t 2 u Cost of the whole flow (all red and green) u Total delay on each edge 16 Surprise! u Price of anarchy for any nonatomic routing game with linear costs <= 4/3 19 8

12/5/19 Example: nonlinear Pigou u Consider large number p; 1 unit of traffic u Equilibrium cost = ? Optimal cost = ? PoA = ? equilibrium flow PoA à + ∞ 20 Example: Braess' paradox u 1 unit of traffic equilibrium flow and cost? PoA = ? 21 9

12/5/19 Braess' paradox u New super highway between v and w PoA = 4/3 22 10