Network Games
Aviv Zohar
Critical MAS -March 2005
Introduction and Motivations
• Understanding and Optimizing the internet (and other networks)
– A fertile field for application of game theory and agents.
• The Internet is one big mess.
– Routing is mostly deterministic, and is heavily based on long detailed
agreements that are negotiated between service providers.
– Many of the links in the internet are unknown. (Some are kept secret on
purpose
– TCP congestion control – not in equilibrium.
– P2P no consideration of distances in the underlying network.
• Providing some quality of service assurances
• Cost effectiveness of the infrastructure.
• The network is very large and complex.
– Too many routers for people to configure manually very often. Especially
when implementing complex policies
– Very dynamic environment.
– The solution: Agents
Application Areas
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Flows
Broadcast and Multicast
Content Distribution
P2P
Ad-Hoc Networks
Network resilience to attack/failure
Routing
Quality of service
Transportation Networks
• Studied as far back as the 1950’s
• Wardrop Equilibrium –The travel time for all
used paths is the same, and is better than the
time for any unused path.
– Very similar to the Nash equilibrium.
• In a model with an infinite number of travelers,
the equilibrium exists and is unique.
– When the number of travelers is finite, things are not
that great…
The Braes Paradox
• When selfish routing of packets is applied,
The equilibrium in the game may not be
efficient.
• Adding an edge to the network may make
things even worse!
u
u
S
S
l(x)=
0
t
v
u
t
2
v
S
t
v
Selfish Routing
• Greedy selfish optimization is just like greedy
optimization in other fields. It brings us to local
maxima only.
• Bounds for the Braes paradox have also been found
(local maxima are close to global maximum).
For example, [Roughgarden,Trados] show a bound of
4/3 in latency in case of linear latency functions on
edges, and for the general case show that the overhead
from selfish routing is at most the cost optimally
routing twice the traffic.
More Selfish routing
• Another problem in selfish routing arises in
cases where only minimal effort is made by the
routers. For example:
u
Target
Source
v
v is eager to get rid of the packets and routes
them through u instead of doing the work and
sending them to the target directly.
Formation of “Good” Networks
• The ideal networks we imagine are:
–
–
–
–
–
Fault and attack tolerant
With a small diameter
Have balanced load
Many routes to every target
No bottle-necks
• Some of life’s ironies:
– Random networks have with high probability all of these
properties. (ie MOST of the possible graphs are “good”)
– Expanders: recognizing them is CONP-complete. Not many
explicit constructions are known.
– networks we see in the real world are not usually random.
Scale Free Networks
• Natural formation models – Connect with higher
probability to the more connected. (The rich get richer)
• Found everywhere.
• Simpler to attack.
Graph
diameter
Percentage of removed nodes
From: “Error and Attack Tolerance of Complex Networks” R.Albert, H. Jeong, A.L.Barabasi
Small world Networks
• It is enough to have a very small percentage of the
links wired globally to achieve many properties.
Clustering
coefficient
Path length
From: “Collective Dynamics of ‘small world’ networks” D.J. Watts & S.H. Strogatz
Networks in metric spaces
• It seems that there is a connection between many good
properties of networks and how easy they are to embed
into metric spaces.
• Random graphs need a large amount of distortion.
• This also relates to a lot of hardness results for
networks.
• For example: sparsest cut problem (Last week’s theory
seminar) is conjectured to be hard to approximate. A lot
of the approximation algorithms in the field use metric
embeddings of graphs.
Networks in metric spaces
• The internet (currently) inhabits only the 2D surface of
our planet.
• Cost of building a link is affected by geography.
• Transmission time between nodes depends on the length
of the link (Limited by speed of light) and the number
of hops (Delay in router queues)
• Thus, when building links we take distance into
account. This is certainly not random behavior.
• In order to build a more robust network , we’ll need to
make some direct links between far places. (e.g.
connect Europe to the west cost of the US directly)
• Does a single node have enough incentive to do this
alone?
Networks and Metric Spaces
from http://mappa.mundi.net/maps/maps_001
Networks and Metric Spaces
from http://www.nd.edu/~networks/gallery.htm University of notre dame physics dept
Wireless routing
• Transmitting to larger distances consumes more
power.
• Selfish nodes will want to conserve battery
power.
Everyone’s greed causes the total
cost of transmission to be much
larger than the optimum.
(Triangle inequality)
s
optimal
t
Applying Mechanism Design
• We can try and change the rules in order to make
the greedy choices of players match the optimal
actions we want them to make. For example:
taxes.
u
u
S
l(x)=
0
t
v
S
t
v
Latency of 3/2 + an extra cost of ½ in
taxes.
Applying Mechanism Design
• An even better taxation system could avoid
costing the players too much in taxes
u
u
S
l(x)=0
tax 1/2
v
t
S
t
v
Latency of 3/2 + no extra cost in taxes.
“How much can taxes help selfish routing?” R. Cole, Y. Dodis, T. Roughgarden
Designing networks is hard.
[Roughgarden et. Al]
• Like most good things in life, approximating an
optimal taxation scheme is Hard (unless P=NP)
– there is no approximation possible with a
better factor than O(n) in the general case.
• Finding the best combination of edges to remove
from the graph in order to get an optimal average
latency is also hard to approximate in the general
case – an n/2 approximation is all that can be
done. This approximation is achieved by simply
removing no edges (doing nothing).
A network Creation game
[Fabrikant, Papadimitriou, Luthra, Maneva, Shenker]
• The price of anarchy [Koutsoupias, Papadimitriou] is defined as
the ratio between the worst case Nash equilibrium and
the optimal social solution: Nash
Opt
• A simple model for formation of networks:
– n players which correspond to vertices in a graph
– Each player can decide to add edges to the graph.
– The cost of function for each player is:
ci    si   d G (i, j )
j
Just a
constant
Number of
edges built by i
Distances of i from
the rest of the graph
A network Creation game
• For α<1 The complete graph is optimal and is the only
equilibrium.
• For 2>α>1 The complete graph is still optimal but the
worst Nash equilibrium is the star. This give a cost of
anarchy of under 4/3.
• For α>2 The star is the social optimum, and is also in
equilibrium but it may not be the worst one.
– The price of anarchy in this case is greater than 3-ε for any ε.
• Conjecture: for α which is large enough, all graphs in
equilibrium are trees.
A network Creation game
• This model is clearly very simple.
• The trees and complete graphs seem to be the
only reasonable results.
• The model can be extended in many ways:
– Congestion as another costly demand
– Non uniform costs for edges
– Interaction between both vertices that participate in a
link to share it’s cost or even veto it’s existence.
–…
• Will it still be tractable?
Thoughts and Questions
• Lies in networks?
– What about white lies?
• Why use Nash-eq. as the solution concept?
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–
–
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Defection of an AS from equilibrium.
Are mixed strategies any good in network formation?
Local Pareto optimality?
Agreement by both ends of an edge.
• Is a repetitive game more realistic?
• Adding a temporal aspect to the game.
• What about partial information? We can’t see the entire
Internet.
References
• “A survey on networking games in telecommunications” Altman, Boulogne,ElAzouzi, Jimenez & Winter. Computer and Operations Research.
• “How bad is selfish routing?” Roughgarden & Trados. JACM 2002.
• “Designing networks for selfish users is hard” Roughgarden. FOCS’01.
• “On a network creation game” Fabrikant, Papadimitriou, Luthra, Maneva &
Shenker. PODC’02
• “Error and Attack Tolerance of Complex Networks” Albert, Jeong & Barabasi.
Nature 406, 378 – 382, 2000.
• “Collective Dynamics of ‘small world’ networks” Watts & Strogatz. Nature
393, 440 – 442. 1998.
• “How much can taxes help selfish routing?” Cole, Dodis & Roughgarden.
ACM EC’03