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

• 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!

v

S t

u

l(x)=x

l(x)=x

l(x)=1

l(x)=1

l(x)=0

S t

u

v

0.5

0.5

1

1

S t

u

v

2

2

0

0

2

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:

vSourceTarget

u

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.

Percentage of removed nodes

Graph

diameter

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.

Path length

Clustering

coefficient

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.

tsoptimal

Everyone’s greed causes the total cost of transmission to be much larger than the optimum.

(Triangle inequality)

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.

v

S t

ul(x

)=x &

tax of 1/2

l(x)=x

&

tax of 1/2

l(x)=1

l(x)=1

l(x)=0

Latency of 3/2 + an extra cost of ½ in taxes.

S t

u

v

0.5+

0.5

0.5+

0.5

1

1

Applying Mechanism Design

• An even better taxation system could avoid costing the players too much in taxes

v

S t

u

l(x)=x

l(x)=x

l(x)=1

l(x)=1

l(x)=0

tax 1/2

Latency of 3/2 + no extra cost in taxes.

S t

u

v

0.5

0.5

1

1

“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:

• 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:

Opt

Nash

j

Gii jidsc ),(

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?– 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,El-Azouzi, 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