Game-theoretic aspects of link placement in the WorldWide Web - - PowerPoint PPT Presentation

Related Work Description of the Problem Our Work

Game-theoretic aspects of link placement in the WorldWide Web

George Kouroupas

kouroupa@aueb.gr

Katia Papakonstantinopoulou

katia@di.uoa.gr

AUEB Theory Tea

November 5, 2010

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Motivation

Web models (important applications, eg. web search, advertising) The role of modern search engines (web structure, scaling) Game-theoretic aspect (users’ behavior: competition for reputation, traffic) PageRank

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Our work

A game-theoretic web model that captures the selfish nature

• f web page authors.

Joint work with M. Sideri, E. Koutsoupias, V. Markakis and

• C. Papadimitriou.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Our work

A game-theoretic web model that captures the selfish nature

• f web page authors.

Joint work with M. Sideri, E. Koutsoupias, V. Markakis and

• C. Papadimitriou.

Goals:

Explain the evolution of today’s web using graph theory and game theory, and taking into account the modern incentives such as PageRank and advertising, and Study whether it is possible to give the authors incentives to build a web of high total welfare.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Outline

1 Related Work 2 Description of the Problem 3 Our Work

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work (summarized)

About PageRank: The effect of new links on Google PageRank [1] Based on graph theory and game theory: Network Formation and the Structure of the Commercial World Wide Web [2] Network Reputation Games [3] (and also: On the alpha-Sensitivity of Nash Equilibria in PageRank-Based Network Reputation Games [4]) Game-Theoretic Aspects of Designing Hyperlink Structures [5]

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

PageRank (definition)

The PageRank of page i is: PR(i) = 1 − d N + d

• j∈Bi

PR(j) L(j) where N: total number of pages, Bi: set of backlinks of i, L(j): number of outgoing links of page j, d: damping factor.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

PageRank - Web graph

Figure: Example of PageRank computation.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

PageRank as an Incentive on the WWW

Figure: The relation between link structure and PageRank.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: The effect of new links on PageRank

• K. Avrachenkov and N. Litvak study to what extent a page can

control its PageRank. Main Results: Simple conditions that show whether new links bring benefits to a web page and its neighbors, in terms of PageRank. There exists an optimal linking strategy. A web page benefits from links inside its web community. Irrelevant links penalize the web pages and their web communities.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

The web graph

The link structure forms communities and raises reputation issues. Thus we can see the web as the outcome (equilibrium) of a game among the web authors. Which game is it?

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Formation & Structure of the Commercial WorldWide Web

• Z. Katona and M. Sarvary study a game in which heterogeneous

web pages establish reference and/or purchase advertising links. Main results:

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Formation & Structure of the Commercial WorldWide Web

• Z. Katona and M. Sarvary study a game in which heterogeneous

web pages establish reference and/or purchase advertising links. Main results: Their equilibrium structure (indegree/outdegree distributions) matches the empirical structure of the www very well.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Formation & Structure of the Commercial WorldWide Web

• Z. Katona and M. Sarvary study a game in which heterogeneous

web pages establish reference and/or purchase advertising links. Main results: Their equilibrium structure (indegree/outdegree distributions) matches the empirical structure of the www very well. Higher content sites:

establish more reference links purchase more advertising links (⇒ strong positive correlation between a site’s content and the number of its inlinks) sell less advertising links, and at higher prices.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Formation & Structure of the Commercial WorldWide Web

• Z. Katona and M. Sarvary study a game in which heterogeneous

web pages establish reference and/or purchase advertising links. Main results: Their equilibrium structure (indegree/outdegree distributions) matches the empirical structure of the www very well. Higher content sites:

establish more reference links purchase more advertising links (⇒ strong positive correlation between a site’s content and the number of its inlinks) sell less advertising links, and at higher prices.

High content sites earn revenue from the sales of content, while low content ones from the sales of traffic.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Network Reputation Games

• J. Hopcroft and D. Sheldon study a game in which authors may

place reference links but have no control over their inlinks. They consider PageRank and Hitting time. Main results: Considering hitting time, any directed graph is a Nash eq.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Network Reputation Games

• J. Hopcroft and D. Sheldon study a game in which authors may

place reference links but have no control over their inlinks. They consider PageRank and Hitting time. Main results: Considering hitting time, any directed graph is a Nash eq. Considering PageRank,

unless the links are delicately balanced, some page can increase its PageRank by replacing all its links with a single one to a carefully chosen page, every equilibrium has a core in which all edges are bidirectional, in a restricted setting, equilibria are characterized by simple properties (mainly edgewise walk-regularity), they demonstrate surprising algebraic properties of equilibria, relating eigenvalues and their multiplicities to graph structure.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Game-Theoretic Design of Hyperlink Structures

• N. Immorlica, K. Jain and M. Mahdian study the design of

hyperlink structure among the pages of a web site, in order to maximize the revenue generated from the traffic on the web site. Main results:

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Game-Theoretic Design of Hyperlink Structures

• N. Immorlica, K. Jain and M. Mahdian study the design of

hyperlink structure among the pages of a web site, in order to maximize the revenue generated from the traffic on the web site. Main results: Problem equivalent to Markov Decision Processes

• ptimal link structure: described by a linear program and exist

polynomial-time algorithms for finding it.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Game-Theoretic Design of Hyperlink Structures

• N. Immorlica, K. Jain and M. Mahdian study the design of

hyperlink structure among the pages of a web site, in order to maximize the revenue generated from the traffic on the web site. Main results: Problem equivalent to Markov Decision Processes

• ptimal link structure: described by a linear program and exist

polynomial-time algorithms for finding it. Using the framework of cooperative game theory, they study whether it is possible to give the authors and coalitions incentive to cooperate and build the optimal hyperlink design.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Related work: Game-Theoretic Design of Hyperlink Structures

• N. Immorlica, K. Jain and M. Mahdian study the design of

hyperlink structure among the pages of a web site, in order to maximize the revenue generated from the traffic on the web site. Main results: Problem equivalent to Markov Decision Processes

• ptimal link structure: described by a linear program and exist

polynomial-time algorithms for finding it. Using the framework of cooperative game theory, they study whether it is possible to give the authors and coalitions incentive to cooperate and build the optimal hyperlink design.

In the TU setting, the core of the game is non-empty. In the NTU setting, if mixed strategies are allowed, the core of the game is non-empty but can be highly inefficient.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Problem Description

Consider the set of web page authors. Each one of them has a web page and wants to get the maximum possible payoff from it. Web page intrinsic content link structure, considering 2 types of links: reference and advertising (paid), eg. Google, Adwords Payoff of author i Quality and PageRank of i’s page Total cost of links sold by i − Total cost of links bought by i

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Problem Description

Consider the set of web page authors. Each one of them has a web page and wants to get the maximum possible payoff from it. Web page intrinsic content link structure, considering 2 types of links: reference and advertising (paid), eg. Google, Adwords Payoff of author i Quality and PageRank of i’s page Total cost of links sold by i − Total cost of links bought by i Problem author i has to solve Choose the set of reference and/or advertising links that maximize the payoff from the site.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the WWW as a graph

Representation We represent the web as a directed graph: web pages: nodes of the graph

intrinsic quality of pages: node weights (Input), ni

hyperlinks: directed edges of the graph

value added through outgoing links: edge weights (Input), dij prices of links: edge weights as well. Fixed (Input) or not, pij

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game

Game Definition (strategic, finite)

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game

Game Definition (strategic, finite) Set of players: the set of nodes of the graph

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game

Game Definition (strategic, finite) Set of players: the set of nodes of the graph Available strategies of each player: all possible sets of ingoing links she can purchase from other nodes, and outgoing links she can establish to other nodes

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game

Game Definition (strategic, finite) Set of players: the set of nodes of the graph Available strategies of each player: all possible sets of ingoing links she can purchase from other nodes, and outgoing links she can establish to other nodes Payoff of each player: the value of her page (its quality combined with PageRank), minus the cost paid for the ingoing links, plus the cost earned from selling outgoing links.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game

Game Definition (strategic, finite) Set of players: the set of nodes of the graph Available strategies of each player: all possible sets of ingoing links she can purchase from other nodes, and outgoing links she can establish to other nodes Payoff of each player: the value of her page (its quality combined with PageRank), minus the cost paid for the ingoing links, plus the cost earned from selling outgoing links. Total welfare The total payoff (the sum of the payoffs of all players).

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game - Notation

Quality of page i Qi = ni +

• j

dijxij

• j

xij ≤ c for constant c

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game - Notation

Quality of page i Qi = ni +

• j

dijxij

• j

xij ≤ c for constant c Payoff of player i Pi = Qi ∗PageRanki +cost of links i sold−cost of links i bought

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Modeling the creation of WWW as a game - Notation

Quality of page i Qi = ni +

• j

dijxij

• j

xij ≤ c for constant c Payoff of player i Pi = Qi ∗PageRanki +cost of links i sold−cost of links i bought Total Welfare W =

• i

Pi

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

The link cost

The costs of links in our game may be: fixed equal, or different (coming up from the game)

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Interesting questions

What is the optimum graph of this game (the graph that maximizes the total welfare)?

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Interesting questions

What is the optimum graph of this game (the graph that maximizes the total welfare)? What are the Nash equilibria of this game? Which are their structural properties? Given a graph, (how) can we say whether it is a Nash equilibrium? Is the optimum graph a Nash equilibrium? What is the price of anarchy of the game? PoA = total welfare of worst Nash equilibrium

total welfare of optimum setting

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Interesting questions

What is the optimum graph of this game (the graph that maximizes the total welfare)? What are the Nash equilibria of this game? Which are their structural properties? Given a graph, (how) can we say whether it is a Nash equilibrium? Is the optimum graph a Nash equilibrium? What is the price of anarchy of the game? PoA = total welfare of worst Nash equilibrium

total welfare of optimum setting

For which prices of the links the game’s outcome is the

• ptimum graph (the optimum graph is achieved as a Nash

equilibrium)?

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Simplifying Assumptions

No link costs Indegree instead of PageRank Yi =

• k

xki

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

The optimum web graph

Optimization Problem (under the simplifying assumptions) Given ni and dij ∀i, j, let xij = 1 if there is a link from page i to j

• therwise

Maximize W =

i Pi = i QiYi = i(ni + j dijxij) k xki

subject to

j xij ≤ c

∀i, j ∈ 1, . . . , n

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

The optimum web graph - Observations

Some first observations (under the simplifying assumptions): If ni = 0 ∀i, finding the optimum graph is trivial: every graph with

j xij ≤ c ∀i, j is optimal (same total indegree).

For general ni, the optimum graph is the one in which:

high quality pages have the larger indegrees low quality pages have the smaller indegrees

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

The optimum web graph - NP-Hardness

We do not know how hard this problem is. Considering a relaxation of the problem with xi,j ∈ [0, 1] ∀i, j, it is formulated as a quadratic program

it is non convex ⇒ we can not use SDP to solve it. it is a special case of quadratic programming ⇒ NP-hardness is not inferred trivially.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Nash equilibrium web graph

Definition: Nash equilibrium graph A web graph is at Nash equilibrium iff no author has an incentive to change her set of in- and out-going links unilaterally. Interesting problems: Given the ni for every page i, and dij, pij for every possible link (i, j) of the web, find some/all possible Nash equilibria. Given a web graph (ni, dij, pij), is it at Nash equilibrium? What are the structural properties of equilibrium graphs (eg. in-/out-degree distributions)?

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Nash equilibrium web graph - Observations

Some first observations (under the simplifying assumptions): At Nash equilibrium, every player i puts all weight xij(c) to the link(s) of maximum dij.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Nash equilibrium web graph - Observations

Some first observations (under the simplifying assumptions): At Nash equilibrium, every player i puts all weight xij(c) to the link(s) of maximum dij. If all dijs are equal, the nis do not affect the Nash equilibria (they are the same, but some of them are not optimal). If all dijs are all equal, the optimum graph is a Nash equilibrium.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Nash equilibrium web graph - Observations

Some first observations (under the simplifying assumptions): At Nash equilibrium, every player i puts all weight xij(c) to the link(s) of maximum dij. If all dijs are equal, the nis do not affect the Nash equilibria (they are the same, but some of them are not optimal). If all dijs are all equal, the optimum graph is a Nash equilibrium. If ni = 0 ∀i, all Nash equilibria are optimal. For general ni, the price of anarchy may be high.

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Pricing the links

Fixed prices: What are the possible equilibria of the game?

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Pricing the links

Fixed prices: What are the possible equilibria of the game? Non-fixed prices (equal or not): Find the prices that lead to some desirable Nash equilibrium. Mechanism Design Problem Assign a price pij to each link (i, j), such that the resulting Nash equilibrium has maximum (high) total welfare. If we can not implement the optimum graph using prices, how well can we approximate it?

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Work in progress

Find the optimum graph

Exact or approximate solution

Find the Nash equilibrium graph

Is the optimum graph a Nash equilibrium?

Assume that we price the links. Find the prices that implement the optimum graph as a Nash equilibrium. Complexity issues of the above computations Anything better than Indegree? (Hitting time?) Prices as part of the players’ strategy?

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

References (1/2)

Konstantin Avrachenkov and Nelly Litvak. The effect of new links on Google PageRank. In Stoch. Models, 2006.

Back to slide 5

Zsolt Katona and Miklos Sarvary. Network Formation and the Structure of the Commercial World Wide Web. Marketing Science, 2008.

Back to slide 5

John Hopcroft and Daniel Sheldon. Network Reputation Games. Technical Report, 2008.

Back to slide 5 George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

References (2/2)

Wei Chen, Shang-Hua Teng, Yajun Wang and Yuan Zhou. On the alpha-Sensitivity of Nash Equilibria in PageRank-Based Network Reputation Games. In Proceedings of the Third International Workshop in Frontiers in Algorithmics (FAW), 2009.

Back to slide 5

Nicole Immorlica, Kamal Jain and Mohammad Mahdian. Game-Theoretic Aspects of Designing Hyperlink Structures. In Proceedings of the Second International Workshop in Internet and Network Economics (WINE), 2006.

Back to slide 5 George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW

Related Work Description of the Problem Our Work

Thank you!

George Kouroupas, Katia Papakonstantinopoulou Game-theoretic aspects of link placement in the WWW