CSCI 3210: Computational Game Theory Inefficiency of Equilibria - - PDF document
5/4/17 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 Split or steal game u NE outcome vs. socially
u NE outcome vs. socially best/optimal outcome Payoff matrix
u Again: NE outcome vs socially optimal outcome Payoff matrix
u What is the objective function to compare
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
u PoA =
u PoS =
u Calculate PoA and PoS Payoff matrix
u Calculate PoA and PoS Payoff matrix
u Consider costs u PoA and PoS will be >= 1 u PoA = PoS when all NE have the same cost
u In general, PoA >= PoS u PoA: worst case guarantee in a system of
u PoS: measures benefit of a protocol or
u Assume one unit of traffic from s to t u Objective function: average travel time u NE: all traffic in lower edge. Avg cost = 1. u Optimal: ½ and ½ . Avg cost = ½ * 1 + ½ * ½. u PoA, PoS = ?
x = amount of traffic on edge
u Multicommodity flow network
u Directed network with multiple (source, sink) pairs u Each (source, sink) pair is called a commodity u ri amount of traffic for each commodity i
u Each edge e has a delay or cost function ce
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
u Let f be a feasible flow (combining all
u f is equilibrium flow if
u A commodity uses a path P in its flow within f
=> All detours have higher (or equal) delay
equilibrium flow
u What is the traffic flow on an (s,t) path?
u Amount of traffic using that path fully (not
partially) to go from s à t u What's the cost (or delay) of a path?
u Total delay on that path– traffic using part of that
path also contribute to delay
s1 s2 t1 t2
u Sum of edge costs (or delays) on that path
flow (all the flows, not just on P) cost function
flow on edge e path
u Cost (or delay) of a flow
flow cost function
flow on edge e
u Consider large number p; 1 unit of traffic u Equilibrium: All down. cost = 1. u Optimal: ε up, (1-ε) down
u Cost = ε * 1 + (1-ε) * (1-ε)p à 0
equilibrium flow PoA à +∞
u 1 unit of traffic
equilibrium flow ½ ½
Equilibrium cost = ½ * ( ½ + 1) + ½ * (1 + ½) =3/2
PoA = 1
u New super highway between v and w
PoA = 4/3
u Restricted version of multicommodity flow
u ri amount of traffic for each commodity i u Each s à t flow must go through a single path
(difference with nonatomic selfish routing) u Each player/commodity chooses the
u Equilibrium flow: No better path for any
u There may not exist any equilibrium flow u 2 players: both s à t u r1 = 1, r2 = 2 u Proof: If player 2 goes s à t, then player 1's
u Prove that equilibrium flow exists
u Consider linear cost functions only u Pigou's example gives a lower bound of 4/3 u NE: all traffic in lower edge. Avg cost = 1. u Optimal: ½ and ½ .
u PoA = 1 / (3/4) = 4/3
x = amount of traffic on edge
u Price of anarchy for any nonatomic routing