Routing in Cost-shared Networks: Equilibria and Dynamics Debmalya - - PowerPoint PPT Presentation
Routing in Cost-shared Networks: Equilibria and Dynamics Debmalya Panigrahi (joint works with Rupert Freeman and Sam Haney; Shuchi Chawla, Seffi Naor, Mohit Singh, and Seeun Umboh) set of agents want to route traffic from their respective
Debmalya Panigrahi
(joint works with Rupert Freeman and Sam Haney; Shuchi Chawla, Seffi Naor, Mohit Singh, and Seeun Umboh)
set of agents want to route traffic from their respective source to sink vertices each edge used in routing has a fi fixed cost that is shared equally by agents using the edge minimize sum of f cost of f edges used in routing (Steiner forest)
However …
s1 s2 t2 t1 2 2 1 2 1 2 2
s1 s2 t2 t1 2 1 2 1 2 2 2
s1 s2 t2 t1 2 1 2 1 2 2 2
s1 s2 t2 t1 2 1 2 1 2 2 2
equilibrium: no agent has a less expensive routing path
equilibrium: no agent has a less expensive routing path do equilibriums alw lways exist?
equilibrium: no agent has a less expensive routing path do equilibriums alw lways exist? yes, reason coming up soon …
equilibrium: no agent has a less expensive routing path do equilibriums alw lways exist? yes, reason coming up soon …
(a (and what can th the controller do about it it?)
unfortunately, very ry suboptimal
n agents s t price of anarchy
unfortunately, very ry suboptimal what role can the controller play?
n agents s t price of anarchy
how bad is the best equilibrium? i.e., controller chooses routing paths but they need to be in in equilibrium
price of stability
how bad is the best equilibrium? i.e., controller chooses routing paths but they need to be in in equilibrium this is a potential game
(corollary: equilibrium always exists)
[Anshelevich, Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden ’04]
price of stability
edge e used by ne agents potential of edge e is φe = = ce (1 + 1/2 + 1/3 + … + 1/ne) in the example, if agent moves from 1 to 2 Δ φ = c2/(n2+1) – c1/n /n1 = dif ifference in in shared cost Initialize with optimal solution and run to equilibrium
[Anshelevich, Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden ’04]
OPEN: Can this logarithmic ratio be improved?
[Li ’09: O(log n / log log n)] [Best lower bounds are small constants]
special case: broadcast games each vertex has an agent all agents route to a common gateway destination
OPEN: Can this logarithmic ratio be improved?
[Li ’09: O(log n / log log n)] [Best lower bounds are small constants]
broadcast games
Fiat-Kaplan-Levy-Olonetsky-Shabo ’06: O(log log n) Liggett-Lee ’13: O(log log log n) Bilo-Flammini-Moscardelli ’13: O(1) v is responsible for edge ev
broadcast games
Fiat-Kaplan-Levy-Olonetsky-Shabo ’06: O(log log n) Liggett-Lee ’13: O(log log log n) Bilo-Flammini-Moscardelli ’13: O(1) v is responsible for edge ev
broadcast games what about multicast games? Main challenge Mechanism for tr transferring responsibility
v is responsible for edge ev Fiat-Kaplan-Levy-Olonetsky-Shabo ’06: O(log log n) Liggett-Lee ’13: O(log log log n) Bilo-Flammini-Moscardelli ’13: O(1) who is responsible for edge e?
recent progress multicast games on quasi-bipartite graphs pri rice of f stability is is O(1 (1)
[Freeman, Haney, P.]
agent-agent path is of length ≤ 2
exponential gap between best and worst equilibria which of these equilibria is achievable?
OPEN: Find any equilibrium in polynomial time.
changes in potential can be exponentially small
what if agents can join and leave the network?
s1 t1 1 3 1 1 1 5 3
what if agents can join and leave the network?
s1 t1 1 3 1 1 1 5 3
what if agents can join and leave the network?
s1 t1 1 3 1 1 1 5 3 s2 t2
what if agents can join and leave the network?
s1 t1 1 3 1 1 1 5 3 s2 t2
OPEN: What is the quality of the equilibrium reached if there are no departures?
if arrivals and moves are not interleaved, then O(log3 n) [Charikar, Karloff, Matheiu, Naor, Saks ’08]
theorem: if agent departures is allowed, then poly(n (n)
OPEN: What is the quality of the equilibrium reached if there are no departures?
if arrivals and moves are not interleaved, then O(log3 n) [Charikar, Karloff, Matheiu, Naor, Saks ’08]
[Chawla, Naor, P., Singh, Umboh]
theorem: if agent departures is allowed, then poly(n (n) what can th the controller do?
OPEN: What is the quality of the equilibrium reached if there are no departures?
if arrivals and moves are not interleaved, then O(log3 n) [Charikar, Karloff, Matheiu, Naor, Saks ’08]
[Chawla, Naor, P., Singh, Umboh]
if the controller suggests (improving) moves to attain equilibrium between arrival/departure phases theorem: equilibrium within lo log n of optimal
[Chawla, Naor, P., Singh, Umboh]
partition graph into subgraphs of diameter 2k, for 1 ≤ k ≤ log n (embed into a distribution of HSTs)
hope: vertices with edges of same length are well ll-separated
im improving move rem emoves an overcharge
im improving move rem emoves an overcharge but can create a dif ifferent one
im improving move rem emoves an overcharge but can create a dif ifferent one repeat
im improving move rem emoves an overcharge but can create a dif ifferent one repeat potential argument sh shows se sequence is is fi finite eventually, th there is is no overcharging
how do we extend to multiple arrivals/departures? now, overcharging on multiple subgraphs (1) overchargin ing only done by le leaves of the routing tree except possib ibly one subgraph charged by y 2 non-le leaves (2) if if there is is overchargin ing, g, then there is is an im improving move that main intains in invaria iant (1) (3) potential l decreases over tim ime (4) eventually, , there is is no overcharging
summary equilibria in network games can have linear inefficiency but the best equilibrium has lo log inefficiency
yes, for broadcast and multicast on quasi-bipartite
if agents join/leave/move arbit itrarily, inefficiency can be lin linear but controlling the moves yields lo log inefficiency